1125899906842624 Pictures
11 New Numerical Functions in Mathematica 7
120 Rhombic Triacontahedra
1, 2, 3-Parameter Logistic Rasch and Birnbaum Models and Item Analysis
12 Rhombic Triacontahedra
12x12 Icosahedra
14-Piece Dissection of Juel's Pyramid into a Parallelepiped
2002-2008 New Zealand General Election Results under Varying Cutoff Procedures
21609 Clusters of Polyhedra
2,2 Turing Machines
26-Piece Dissection of a Cube
2-by-n Disk-Packing Paradox
2D 2C R1 Cellular Automaton on a Triangular Grid
2D and 3D Packard-Takens Autocorrelation Plots of Sinusoidal Functions
2D and 3D Views of the Weierstrass Function
2D Cellular Automata for Feature Detection
2D Cellular Automata for Texture Recognition
2D Cellular Automaton Animations
2D Cellular Automaton Growth Models
2D Cellular Automaton on a Triangulated Surface
2D Logic Game with Letters
2D Root Systems
2D Spring Pendulum
2D Totalistic Cellular Automata
2D Triangular Cellular Automata on a Distorted Grid with Holes
2D Vector Addition
2D View of a Conformal Mapping
2D Walsh Functions
2x2 Ordinal Games
30+12 Rhombic Triacontahedra
3×3 Determinants by Expansion
3×3 Determinants Using Diagonals
3-Color Left/Right Mobile Automata
3D Band Member Mix
3D Boid Model
3D Chebyshev Lissajous Figures
3D Construction Set
3D Cubic B-Spline Curves
3D Deterministic Context-Free (DOL) Systems
3D Dissection Puzzle
3D Flying Pipe-Laying Turtle
3D Graphics with 2D Overlay
3D Graph of a General Quadratic Form
3D Hermite Lissajous Figures
3D Kerr Black Hole Orbits
3D Lambdoma
3D Layers Evolution of Five-Neighbor Outer Totalistic 2D Cellular Automaton
3D Layers Evolution View of a 2D Totalistic Cellular Automaton
3D Legendre Lissajous Figures
3D Lissajous Figures
3D Multipole Shapes
3D Projectile Motion
3D Spore Creations
3D Totalistic Cellular Automata
3D Vector Fields
3D View of Binary Logical Operations
3D Views of Elementary Cellular Automata
3D Waves
3j Symbols
3n+1 Flying Saucers
3n+1 Graph
3x3 Matrix Explorer
4-Band Color Code for Resistors
5000 Demonstrations and Counting
54-Faced Space-Filling Polyhedron
5x5x5 Pentacube Puzzle
600-Cell 120-Cell Duality
600 Cell Orthogonal Scale Base Projection
60 Rhombic Dodecahedron
60 Rhombic Triacontahedra in Two Different Clusters
60 Small Rhombicosidodecahedra
80 Golden Rhombohedra
A 2-pire Map
Abacus
A Basic Property of Integrals
ABC Analysis
A Bee's Eye View of Cellular Automata
Absorption of Acetone in Air Using Water
Absorption Spectroscopy
Academic Admissions
A CA-horus Line
A'Campo's Link of a Divide
A Canadian Lottery Wheel
A Case of the Epsilon-Delta Definition of a Limit
A Catalog of Conformal Mappings
Acceleration of a Wheel
Accidentals, on Purpose
Accumulation Point
Accuracy of Series Approximations
A Cellular Automaton-Based Heat Equation
A Cellular Automaton Mapped on the Surface of a Cuboid
Acid-Base Titrations
AC Induction Motor Rotor Design
AC Induction Motor VFD Operation
A Circumcircle through the Midpoint of a Triangle's Side
A Collinearity between the Nine-Point Center, the Foot of an Altitude, and a Midpoint
A Collinearity from the Medial and Excentral Triangles
A Collinearity Involving an Excircle, the Incircle, and the Circumcircle
A Combinatorial Game
A Complex Gaussian Function
A Conceptual Model of Lapse Financed Life Insurance
A Concurrency from a Point and a Triangle's Excenters
A Concurrency from Circumcircles of Subtriangles
A Concurrency from Midpoints of Arcs of the Circumcircle
A Concurrency from Six Pedal Points
A Concurrency from the Midpoints of Line Segments through the Circumcenter
A Concurrency from the Reflection of the Incircle's Contact Points across the Incenter
A Concurrency from the Reflections of the Contact Points with the Excircles
A Concurrency Generated by Circles about a Triangle's Sides and Lines through an Internal Point
A Concurrency Generated by Lines through the Orthocenter and Circles about a Triangle's Sides
A Concurrency Generated by the Angle Bisectors
A Concurrency Involving the Midpoints of the Sides and the Altitudes
A Concurrency of Lines Joining Orthocenters
A Concurrency of Lines through Points of Tangency with Excircles
A Conditional Probability Mass Function
A Congruent Angle inside the Incircle
A Conjecture of Apoloniusz Tyszka on the Addition of Rational Numbers
A Continuous Mapping That Fixes All Boundary Points but No Interior Points
A Convergent Sequence Satisfies the Cauchy Criterion
AC Power Factor Principle
AC Rotating Magnetic Field Principle
AC Synchronous Machine Vector Diagram
AC Three-Phase Neutral Current
AC Thyristor Operation
AC Thyristor Trigger Angle versus Power Factor
Action of a Siphon
Activator-Inhibitor Cellular Automata
AC Transformers
A Cycle Index Spreadsheet
A Cyclic Quadrilateral Concurrency
Adams' Circle and the Gergonne Point
Adaptive 3D Plotting
Adaptive Contour Plotting
Adaptive Plotting
Adding Effects to Images
Adding Fractions
Adding in a Given Base
Adding Mixed Numbers
Adding Points on an Elliptic Curve
Adding Polarizations with Phase Shifts
Adding Polygonal Numbers
Adding Whole Numbers
Addition and Subtraction of Integers
Addition Bingo
Addition Formula for Sine
Addition of Fractions
Addition of N Vectors in 2D
Addition of Points on an Elliptic Curve over the Reals
Addition Slide Rule
Addition Table
Additive Cellular Automata
Additive Cipher
A Device that Illustrates Pythagoras' Theorem
Adiabatic Converter for Sulfur Trioxide Production
A Disappearing/Reappearing Coin
A Dissection of a Prolate Golden Rhombohedron
A Dissection Proof of Pythagoras's Theorem
Adjacency Matrices of Manipulable Graphs
Adjacent Flow of Two Immiscible Fluids
Adjusting RGB Color of Image
A Domino Puzzle
A Double Exponential Equation
A Double Parabola with Retroreflection Properties
Adsorption Isotherms of Ethylene in NaX (Zeolite Structure FAU)
Advanced Celestial Sphere
Adverse Selection
Aerial Photo Scale
A Face from Words
Affine Products of Locators
Affine-Scaling Interior Point Method
Affine Transform
A Fifth-Degree Polynomial and Its Derivatives
A Finite Automaton Editor
A First Morley Triangle Concurrence
A Forest Growth Curve
A Formula for Primes in Arithmetic Progressions
A Function Invariant under a Group of Transformations
A Function with a Jump Discontinuity
Age Distributions from a Leslie Model for Age-Structured Populations
A Generalization of Hansen's Theorem
A Generalization of IMO 2008 Problem 1
A Generalization of the Mean Value Theorem
A Geometrical Theorem of Leibniz
A Geometric Proof of the Square Pyramidal Number Formula
A Gothic Window
A Graph Theory Interpretation of the Sum of the First n Integers
A Grating at the End of a Tube
A Greeting from a Surface Plot
Aharonov-Bohm Effect
A Health Stories Model of Long-Term Care Insurance
A Hexagon with Opposite Sides Parallel
A Hybrid Benz-Grignani Logo
Aircraft Propeller
Aircraft Wing
Airplanes and Crosswinds
Akaike Criterion in Orbit Determination
Akrasia
Albrecht Dürer's Heptagon of 1525
Albrecht Dürer's Nonagon of 1525
Albrecht Dürer's Pentagon of 1525
A Lemma of Archimedes about a Bisected Segment
A Lever with Four Weights
Algebraic Loops (1); Properties
Algebraic Loops (2); Symmetry-Conserving Vector-Division Hoop Algebras
Algebraic Solutions to Polynomial Equations
Algebraic Values of Trigonometric Functions of Inverse Trigonometric Functions
Algebra Quiz
Algorithmic Architecture with Cellular Automata
Algorithmic Terrain with Cellular Automata
Algorithms for Egyptian Fractions
Aliasing in Time Series Analysis
A Library of Functions with Transformations
A Limit Theorem from Information Theory
A Line Associated with an Excircle
A Line Parallel to a Side of a Triangle
Allen Key
Allocation of Bandwidth by TCP
All Possible Sums and Differences of Powers
Almost-Circular Cellular Automaton
A Lockable Box
Altitudes and Incircles
A Magic Box
A Massive Bar Oscillating between Two Identical Rotating Support Rollers
Amdahl's Law
American Call and Put Option
A Minimal Circumcircle Measure of District Compactness
Ammann Chair
Ammann Tiles
A Model for Successive Radioactive Decays
A Model of Market Shares I
A Model of Market Shares II
A Monotone Sequence Bounded by e
Amortized Loan Interest and Principal
Amplitude Modulation
A Multiobjective Optimization of a Room Configuration
Anagrams
Analemmas
Analog & 24-Hour Digital Clocks
Analogous and Complementary Colors
An Altitude and Four Concyclic Points
Analysis of a Single-Span Euler-Bernoulli Beam under Different Loading Conditions
Analysis of Minimum Credit Card Payments
Analyzing the Crystallography of the SH2 Domain and Its Residue Contact Map
An Amoeba Problem
Anamorphic 2D Images That Look Three-Dimensional from a Particular Viewpoint
Anamorphic Prisms
An Analog Clock
An Angle Bisection
An Application of the Gergonne-Euler Theorem
An Application of the NestList Function
An Approximation to the n-th Prime Number
An ASCII Game of Life
An Automated Barrier
An Automatic Analog/Digital Clock
Ancient Aleppo Tiling
An Efficient Test for a Point to Be in a Convex Polygon
An Elliptic Identity
An Enneper-Weierstrass Minimal Surface
Aneroid Barometer
An Essential Singularity
A Net That Folds into an Icosahedron or into a Double Biaugmented Triangular Prism
A New Kind of Casimir Energy and Its Implications for Dark Energy
A New Kind of Sunrise
A New Kind of Yin-Yang Rotation
An Example of a Production Function
An Example of Subtraction of Negative Numbers
An Example of the Bolyai-Gerwien Theorem
An Experiment with Boyle's Law
An Expert System Specific to Improving Personal Relationships
An Exploration of the "6174 Problem" and Some Generalizations
Angle and Length Measurement of a Polygon
Angle Bisector for an Angle Subtended by a Tangent Line
Angle Bisectors in a Triangle
Angle Bisectors of Two Intersecting Lines
Angle Bisectors on the Circumcircle
Angle Bisector Theorem
Angle Measure
Angles in a Cube
Angles in a Quadrilateral Total 360°
Angles in a Triangle Total 180 Degrees
Angles Measured in Degrees and Radians
Angular Acceleration
Angular Displacement
Angular Spheroidal Functions as a Function of Spheroidicity
Anharmonic Oscillator Phase Space Trajectories 2D
Anharmonic Oscillator Spectrum via Diagonalization of Amplitudes
An Illustration of the Argument Principle
An Illustration of the Central Limit Theorem Using Chi-Square Samples
Animal Sounds
Animated Binary
Animated Butterfly
Animated Pendulum
Animated Projectile Motion
Animation of Free Kicks in Soccer
An IMO Problem Involving Concurrency
An IMO Triangle Problem
An Increasing Preference Distribution
An Infinite Knight's Tour
An Infinitely Colorable Set of 3D Regions
An Infinite Tree of Circles or Spheres
An Interval Eventually Bounding Trajectories of the Logistic Map
An Introduction to Invariant Subspaces Using a Cube
An Intuitive Proof of the Pythagorean Theorem
An Inventory Control Model
Anisotropic Elasticity
Annotated Quadratic Polynomial
Annual Sunspot Numbers
Annual Temperature Cycles
A Noncontinuous Limit of a Sequence of Continuous Functions
A Nonlinear Stage-Structured Cannibalism Model
An Ordinary Fractional Differential Equation
An Osculating Circle for Various Curvatures
Another Concurrency Generated by Circles about a Triangle's Sides and Lines through an Internal Point
Another Dissection of Two Rhombic Solids into an Icosidodecahedron and a Rhombicosidodecahedron
Another Knights and Knaves Puzzle Generator
Another Relation between the Areas of Triangles Associated with an Excircle
Another Triangle into Hexagram Dissection
Another Variant of Kosnita's Theorem
Antibody Production Fed-Batch
An Unforgettable Substitution Code
Any Quadrilateral Can Tile
Any Triangle Can Tile
A Packing of the 3 x 3 x 3 Cube
A Parabolic Mirror
A Parabolic Partial Differential Equation in Three Different Geometries
A Parallelogram Defined by the Centers of Four Circumcircles
A Parallelogram Defined by the Centers of Four Incircles
A Parallelogram Related to the Excircles
A Parallelogram with Vertices on Two Circles
A Parameterized Multistate Life Table
A Passive Cochlear Model
A Path through the Lattice Points in a Quadrant
Apéry's Rational Approximation to His Constant
Apollonian Gasket
A Polygon from Its Midpoints
A Polygon That Folds into a Regular Tetrahedron or into a Rectangular Box
Appearance of Spiral Behavior
Application of the Murphree Liquid Efficiency Method to the Study of a Binary Distillation Column
Applying a Coloring Function to an Image
Applying Graphics Filters Using Convolution
Applying the Pólya-Burnside Enumeration Theorem
Applying the Soave-Redlich-Kwong Equation of State
Approximate Bubble Temperature Distribution for Benzene, Toluene, p-Xylene Ternary Mixture
Approximating a Judge
Approximating Catalan's Constant
Approximating Pi by the Monte Carlo Method
Approximating Pi with Continued Fractions
Approximating Pi with Inscribed Non-Regular Polygons
Approximating Pi with Inscribed Polygons
Approximating the Derivative by the Symmetric Difference Quotient
Approximating the Length of Viviani's Curve
Approximating the Logarithm of Any Base with Continued Fractions
Approximating the Riemann Zeta Function with Continued Fractions
Approximating the Tangent to a Curve with Secants
Approximating Volumes by Summation
Approximation by Rationals
Approximation of Discontinuous Functions by Fourier Series
Approximations to the Distribution of Primes
A Probabilistic Forest Fire Model
A Product of Chord Lengths in a Circle
A Proof of Euler's Formula
A Proof of the Difference Identity for Cosine
A Rational Function that Iterates to the Identity
Arc Form Design Widget
Arc Form Discovery Widget
Archimedean Screw
Archimedean Solids
Archimedes' Approximation of Pi
Archimedes' Hat Box Theorem
Arc Length
Arc Length Approximation
Arc Length of Cycloid
Arc Length of the Hyperbolic Cosine
Arc Resolution
Area and Volume of n-Dimensional Spheres
Area Between Curves
Area Bisectors via Monte Carlo
Area Function for a Piecewise Curve
Area of a Circle from Segments
Area of a Disk
Area of a Hexagon Formed by the Vertices and Altitude Extensions of a Triangle
Area of an Ellipse
Area of a Normal Distribution
Area of a Parallelogram
Area of a Pentagon
Area of a Quadrilateral by Triangulation
Area of a Triangle inside the Incircle
Area of a Triangle in the Poincaré Disk
Area of Epicycloid and Hypocycloid
Area of the Arbelos
Area, Perimeter, and Diagonal of a Rectangle with Uncertain Sides
Areas in a Multiplication Table
Areas of Parallelograms
Areas of Parallelograms and Trapezoids
Areas of Triangles
Area Under a Curve
Area under a Cycloid
Area under the Exponential Curve
Area Under the Tractrix
A Recursive Sequence Convergent to e
A Relation between Altitudes of Four Triangles
A Relation between Incircles and Excircles
A Relation between the Areas of Four Triangles
A Relation between the Contact Triangles of a Triangle
A Remarkable Formula Of Ramanujan
Are the Lines Parallel?
Are These Lines Straight?
Argand Diagram
A Rhombic Dodecahedron Made of Rhombic Triacontahedra
A Ring Game
Arithmetic in Lambda Calculus
Arithmetic Operation Game
Arithmetic Operation Game 2
Arithmetic Sculpture for Addition and Multiplication
Arithmetic Sequences
Arithmetic Series
Arnold's Cat Map
A Rod Sliding in a Semicircular Well
A Rolling Reuleaux Triangle
A Rose for Valentine's Day
A Rotating Reuleaux Triangle
A Rotating Waveplate Polarization Analyzer
Arrhenius Equations for Reaction Rate and Viscosity
Art of Tangent Circles
A Sculpture Based on the Diamond Unit Lattice
A Second-Order Chemical Reaction
A Selection and Analysis Tool for Fluorescence Recovery after Photobleaching (FRAP) Region of Interest (ROI)
A Signed Rank Test of Hypotheses about a Median
Asimov's Ladder of Physical Constants
A Simple Inventory Control Model
A Simple Lever
A Simple Lofting Function
A Simple Logic Game
A Simple Metronome
A Simple Model of Magnetization
A Simple, Standard Linear Programming Scenario
A Simplified Epidemic Model
A Simplified Model for the Big Bang
A Sine/Cosine Identity
A Snowball's Rate of Change
A Solution of Euler's Type for an Exact Differential Equation
A Spatial Dynamic Jury Model
A Special Parameter Value for the Logistic Map
A Sphere Divided by Planes and a Cone
A Spider Walking on a Polygon
A Spikey Made of Tetrahedra and Octahedra
A Squaring Cellular Automaton
Assessor Model for Simulated Test Markets
Asset Allocation
A Strategy for Exploring k=2, r=2 Cellular Automata
AstroMast
AstroMesh Reflector
Astronomical Data
Astronomical Units of Length
A Study of the Dynamic Behavior of a Three-Variable Autocatalator
Asylum in the United States
Asymmetric Propeller
Asymptotic Expansions
A Tessellation from a Square Twist Fold
A Tessellation of the Sphere
A Test for the Convexity of a Quadrilateral
A Theo Jansen Walking Linkage
A Theorem about Two Tangent Circles
A Theory of Insurance Lapses
Atomium
A Tour of Second-Order Ordinary Differential Equations
A Triangle and an External Contact Triangle
A Triangle Formed by the Centers of Three Circles
A Triangle Formed by the Centers of Three Nine-Point Circles
A Triangle Inequality Involving the Altitudes, Semiperimeter, Inradius, and Circumradius
A Triangle Model of Criminality
Attainable Regions for Maleic Anhydride Manufacture in a Plug-Flow Reactor
Attractors of Iterated Affine Transform Systems
Atwood's Machine
A Two-State, Discrete-Time Markov Chain
Audio File Size Calculator
Aurifeuillean Factorization
Autocorrelation and Partial Autocorrelation Functions of AR(1) Process
Auto Lease Calculator
Auto Loan Calculator
Automata Generative Networks 1
Automata Generative Networks 2
Automatically Selecting Histogram Bins
Automatic Differentiation
Automatic Stabilizers in Krugman's Cross Diagram
Autoregressive Moving-Average Simulation (First Order)
Auto-Regressive Simulation (Second-Order)
A Variable Can Be Anything
A Variant of Kosnita's Theorem
Averaged Cellular Automaton Patterns
Averaged Gosper Curves
Average Rate of Change: Exploring More Functions
Averages of Totalistic Automaton States
Average Value of a Function
A Very Tough Puzzle
A Visualization of Logarithms
A Visual Proof of Girard's Theorem
A Visual Proof of Nicomachus's Theorem
A Visual Proof of the Cauchy-Schwarz Inequality in 2D
A Visual Proof of the Double-Angle Formula for Sine
A Word Problem about Boats
Axial Gradient
Axial Tilts of Planets
Axis of Revolution
Babylonian Numerals
Balanced Annihilation
Balanced Bracket Sequences
Balanced T-Coil Parameters
Balanced Ternary Notation
Ball Bouncing in a Potential Well
Ball Bouncing on a Moving Piston
Ball in a Box
Ball in V Groove Geometry
Ballistic Pendulum
Ballistic Trajectories
Ball on the Border of a Rectangle
Ball Rolling without Slipping inside a Vertical Cylinder
Balls and Sticks Geometrical Sculpture
Balls Rolling on a Sphere
Ball Valve
Band Spectrum in a Periodic Potential
Bankoff Circle
Banzhaf Power Index
Barlow's Wheel
Barrier Option Pricing within the Black-Scholes Model
Barrow's Inequality
Bars of a Hyperboloid
Bars of an Astroid
Bars with an L-Shaped Cross Section
Bascule (Seesaw) Bridge
Baseball Batting Metrics
Baseball: Graph of On-Base Percentage
Baseball: Graph of Slugging Percentage
Basic Angles
Basic Examples of Propositional Calculus
Basic FM Synthesis of Sounds
Basic Option Trading Strategies
Basic Statistics of Movable Points
Basic Structure, Plane Orientation, and Facetting in the Face-Centered and Body-Centered Cubic Crystal Structures
Basic Supply and Demand
Basic Views in Technical Drawing
Basis Instruments Contracts (BICs) in Baseball World Series Odds
Batch Distillation of a Non-Miscible Binary Mixture
Batch Rectification of a Binary Mixture at Total Reflux
Batch Rectification of an Acetone/Chloroform Binary Mixture
Batch Rectification of an Ethanol/Ethyl-Acetate Binary Mixture
Batch Rectification of an Ideal Benzene/Toluene Binary Mixture
Batch Rectification of a Partially Miscible Binary Mixture
Baum-Sweet Sequence
Bayes's Theorem and Inverse Probability
Bead on a Horizontally Rotating Hoop: Variable Initial Displacement and Tilted Axis
Bead on a Horizontally Rotating Hoop with Movable Axis
Bead on a Rotating Hoop
Bead on a Rotating Wire
Beat Chebyshev
Beat Frequency of Sound Waves
Beats Working
Beautiful Designs Made from the Knight's Tour
Beautiful Designs Made from Two Knights' Tours
Bed of Nails
Bell Curves
Bell Number Diagrams
Bell Wave Machine
Bending a Wire on a Cylinder
Bending of Light by a Star
Benford's Law and Data Spread
Benford's Law from Ratios of Random Numbers
Bennett's Octagon-to-Square Dissection
Beraha's Conjecture and Cyclic Graphs
Beraha's Conjecture, Wheels, and Cyclic Graphs
Bernoulli Inequality
Bernstein Polynomials
Bessel Function Lissajous Figures
Bessel Functions in the Complex Plane
Best Effort Global Warming Trajectories
Beta Cube
Beta Distribution
Between Ellipsoid and Spindle in Stereo
Between Sphere and Torus
Beverton and Holt's Yield per Recruit Model
Bézier Curve Approximation of an Arc
Bézier Curve by de Casteljau's Algorithm
Bézier Curve Flowers
Bézier Curves
Bhargava's Theorem
Bicycle Gear Ratios and Meters of Development
Bicycle Rides
Bifurcation Analysis of a Cubic Memristor Model
Bifurcation Diagram for a Simple Nonlinear Optical Fiber Ring Resonator
Bifurcation Diagram for the Gauss Map
Bifurcation Diagram for the Rössler Attractor
Bifurcation Diagram of the Hénon Map
Bifurcation Diagrams with Flow Fields
Bifurcation of Boundary Wavefronts for Some Graphs
Bifurcations in First-Order ODEs
Bifurcations of the Logistic Map
Big Circles on Small Tiles
Bilateral Accident Model
Bilinski's Vase
Bimetallic Strip
Bimodal Normal Distribution Mixtures
Binary-Coded Decimal Clock
Binary Counting Sequence
Binary de Bruijn 2D Hex Maps
Binary Election Sequences
Binary Gray Code
Binary Inspiral Gravitational Waves from a Post-Newtonian Expansion
Binary Numbers, Subsets and Compositions
Binary Operations with Cellular Automata
Binary Operators Satisfying Two Axioms
Binary Options: Pricing and Greeks
Binary Patterns of Integer Functions
Binary Run-Length Encoding
Binary Tree Enumeration
Binding Energies of Isotopes
Binet-Cauchy Identity
Bingo Card Generator
Binocular Disparity versus Depth (Visual Depth Perception 9)
Binocular Disparity (Visual Depth Perception 7)
Binomial Approximation to a Hypergeometric Random Variable
Binomial Approximation to a Poisson Random Variable
Binomial Distribution
Binomial Distributions Are Bernstein Vectors
Binomial Option Pricing Model
Binomial Theorem
Binomial Theorem (Step-by-Step)
Binomial Tree
Biodiversity in Spatial Rock-Paper-Scissors Games
Bioeconomics of a Discrete Ricker Model with Delayed Recruitment
Biphasic Exponential Decay and Growth
Birds on a Wire
Bisecting a Line Segment through the Orthocenter
Bisection Method
Bisection of Line Segment
Bisection of Segments by the Sides of a Triangle
Bisectors of Quadrilaterals
Bisectors of the Angles of the Orthic Triangle
Bit Count Classification of Binary Numbers
Bit Plane Effects
Bitwise Operations Mod n
Blackbody Radiation
Blackbody Spectrum
Black-Scholes Option Model
Blind Algorithm
Block and Tackle
Block on a Frictionless Inclined Plane
Blocks in Cellular Automaton Patterns
Blood Spatter Trigonometry
Blood Type Inheritance
Board Inversion
Boat and Rope Related Rates
Bochvar's Three-Valued Logic
Bohm Trajectories
Bohm Trajectories for a Type of Derivative Nonlinear Schrödinger Equation
Bohm Trajectories for Quantum Airy Waves in a Time-Dependent Linear Potential
Bohm Trajectories for Quantum Particles in a Time-Dependent Linear Potential
Bohm Trajectories for Quantum Particles in a Uniform Gravitational Field
Bohr's Orbits
Boiling Point of Water as a Function of Altitude
Boltzmann Gas
Bolzano's Continuous but Nowhere Differentiable Function
Bolzano's Function
Bolzano's Theorem
Bonding and Antibonding Molecular Orbitals
Bond Pricing
Book Reader
Boolean Algebra
Boolean Dissimilarities
Boolean NK Networks
Boolean Operations in Geometry
Boolean Primitives as Symmetric Boolean Functions
Boolean Symmetric Functions
Boole Differential Equation with Continued Fractions
Bootstrap Percolation Automaton
Bootstrapping Credit Default Swap Data
Bootstrapping to Compute Value-at-Risk Standard Errors
Boron Suboxide
Borromean Mesh
Borromean Ring Removal
Bose-Einstein Condensation (Free Boson Gas)
Bose-Einstein Condensation in a Harmonic Trap
Bose-Einstein, Fermi-Dirac, and Maxwell-Boltzmann Statistics
Bottema's Theorem
Bouncer
Bouncing Mass-Spring System
Boundary Conditions for a Semi-Infinite Potential Well
Bound States in a Square Potential Well
Bound States of a Finite Potential Well
Bound States of a Semi-Infinite Potential Well
Bound-State Spectra for Two Delta Function Potentials
Box Packing
Box Toppling Patterns
Brachistochrone Problem
Brachistochrones
Bracket with Cylindrical Hinge
Bradley's Two-Triangles-to-One Dissection
Bragg's Law
Brake Shoes
Braking a Car
Break Even
Breakthrough Curves for Adsorption with Longitudinal Diffusion Assuming Linear Equilibrium
Breit-Rabi Diagram
Bridges with Catenary Shaped Supports
Bridge Varieties
Brightness and Magnitude
Brillouin Zone of a 2D Square Lattice: Tight Binding Approximation
Brillouin Zone of a Single-Walled Carbon Nanotube
Brillouin Zone Sampling of a Periodic Chain with N Sites
Brillouin Zone Sampling of Chain with N Sites and Modified Periodic Boundary Conditions
Brinnel Hardness of Materials
Brodie's Pentagon-to-Square Dissection
Broken Heart
Broken Heart Tangram
Broken Heart Tangram (Variation)
Broken Stick Rule
Brouwer Fixed Point Theorem
Brownian Bridge
Brownian Motion in 2D and the Fokker-Planck Equation
Brownian Motion Path and Maximum Drawdown
Brute-Force String Matching
B-Spline Basis Functions
B-Spline Curve with Knots
B-Spline Surfaces
Bubble Chart Comparisons of Countries
Bubbles in a Simple Behavioral Finance Model
Buckyball in Icosahedron
Buckyball of Buckyballs
Buffon's Needle Problem
Building a Velocity Sensor
Building Cylinders from Congruent Triangles
Building Frame with Catenary Roof Beams
Build the Periodic Table
Bulls and Cows
Bullseye Illusion
Bump Map of Mars
Burgers Turbulence
Busschop's Hexagon-to-Square Dissection
Busy Beaver
Buttons in Manipulate
Buying Gas at Midnight
Buying Watermelons Intelligently
Byte Value Display (Maybe It Is EBCDIC?)
C3v Group Operations
Calabi-Yau Space
Calculating 24
Calculating Area, Perimeter, and Diagonal Length of a Rectangle
Calculating Integer Logarithms in Different Bases
Calculating the Distance between Two Inaccessible Points
Calculating the Path of a Point on a Sliding Ladder
Calculating Your Grade
Calculus-Free Derivatives of Sine and Cosine
Calculus Slalom
Calendar Calculator
Call Graphs of Fibonacci-Like Functions
Call Graphs of Nestedly Recursive Functions
Camera Animations on Circles, Spirals, Helices, and Conical Spirals
Canonical Polygons
Cantilever Bridge
Cantor Function
Cantor Set
Capacitors in Circuits
Capacity Planning for Short Life Cycle Products: The Newsvendor Model
Capillary Action
Capitalization Rate Probability
Captain Cook's Voyages
Carbon Dating
Carbon Dioxide Absorption in Water
Carbon Nanotubes
Cardinal Elements of an Aging Human Eye Model with a Gradient-Index Crystalline Lens
Carlyle Circle
Carnot Cycle on Ideal Gas
Carnot's Theorem
Carta Psicrométrica (Portuguese)
Cartesian Color Coordinate Spaces
Cartesian Coordinates Exercise
Car Traveling at Night
Cart with Pendulum
Cassegrain Telescope
Cassini Ovals and Other Curves
Catacaustics for Some Graphs
Catacaustics Generated by a Point Source
Catalan Number Trees
Catenary Stability for Variable-Length Cables
Catenary: The Hanging Chain
Cauchy Mean-Value Theorem
Cauchy-Schwarz Inequality for Integrals
Cauchy's Problem
Causal Interpretation of the Double-Slit Experiment in Quantum Theory
Causal Interpretation of the Free Quantum Particle
Causal Interpretation of the Quantum Harmonic Oscillator
Causal Interpretation of Two-Dimensional Time-Dependent Scattering
Causal Networks for the Wolfram 2,3 Turing Machine
Caustics on Spline Curves
Cavalieri's Principle
Cavity Quantum Electrodynamics with Bosons: Emission Spectra in the Strong and Weak Coupling Regimes
Cayley Graphs
Cayley Tables for Small Groups and Moufang Loops
Cellular Automata as Mappings
Cellular Automata Based on Permutation Groups
Cellular Automata Entropy
Cellular Automata Evaluation
Cellular Automata Model of an MPA Fishery
Cellular Automata on Trivalent Networks
Cellular Automata Ordered by Entropy
Cellular Automata with Global Control
Cellular Automata with Globally Determined Neighborhoods
Cellular Automata with Majority Rule
Cellular Automata without Redundant Information
Cellular Automata with Repetitive Initial Conditions
Cellular Automaton Browser
Cellular Automaton Carpets
Cellular Automaton Classification according to Attractor Type
Cellular Automaton Coil Representation
Cellular Automaton Compressibility
Cellular Automaton Explorer
Cellular Automaton Fluids
Cellular Automaton Fonts
Cellular-Automaton-Like Neural Network in a Toroidal Vector Field
Cellular Automaton Media Strategy
Cellular Automaton Model of Pine Savanna Dynamics in Response to Fire and Hurricanes
Cellular Automaton Rules on Dendrites
Cellular Automaton State Transition Diagrams
Cellular Shading
Celsius and Fahrenheit Thermometers
Center of an Equilateral Triangle Circumscribing a Given Parabola
Center of Mass of a Polygon
Center of Mass of n Points
Centipede3D
Central Cross Sections of Polyhedra
Central Limit Theorem for the Continuous Uniform Distribution
Centre d'un triangle équilatéral circonscrit ŕ une parabole (French)
Centroid Hexagon
Cepheid Variable Star Light Curve Analysis
Certainty Equivalent Wealth
Cesŕro Summation
Cesŕro Sums of Some Unit Sequences
Ceva's Theorem
Chain Rule
Chains of Amicable Numbers
Chains of Regular Polygons and Polyhedra
Chair-O-Plane
Champernowne Constant
Change in Inflection for a Basic Polynomial
Change of Basis in 2D
Changes in the Budget Line
Changing a Coefficient in Polynomials of Low Degree
Changing Proportions into Percents
Changing Sea Levels
Chaos and Order in the Damped Forced Pendulum in a Plane
Chaos Game
Chaos Game 2D/3D
Chaos in the Shift Map
Chaos While Sledding on a Bumpy Slope
Chaotic Attractor in Tumor Growth
Chaotic Dynamics of a Modulated Semiconductor Laser
Chaotic Itinerary but Regular Pattern
Chaotic Oscillation Circuit
Character Rotation Patterns
Charged Particle in a Constant Electric Field Produced by Two Plates
Charged Particle in a Uniform Magnetic Field
Charged Particle in Uniform Electric and Magnetic Fields
Charles's Law
Charting with BarChart
Chebyshev Polynomials
Chebyshev's Inequality
Chebyshev's Inequality and the Weak Law of Large Numbers
Chebyshev's Inequality and the Weak Law of Large Numbers for iid Two-Vectors
Checking Finite Difference Errors
Chemical Abundances and Properties
Chemical Compound Class Browser
Chemical Reaction and Diffusion in a Spherical Catalyst Pellet (the Nonisothermal Effectiveness Factor)
Chess
Chestnut Shell
Chinese Remainder Theorem
Chip Stack Game
Chirality of Substituted Methanes
Chiral Polyhedra
Chi-Squared Distribution and the Central Limit Theorem
Chladni Figures
Chooser Options
Choosing a Data Transformation with the Box-Whisker Plot
Chords in Equal Temperament and Just Intonation
Choreographic Motions of N Bodies
Christmas Stocking Identity
Chromatographic Reactions of Three Components
Chronological Interpretation of Mayan Numerals
CIE Chromaticity Diagram
Cipher Encoder
Circle Chord Envelope
Circle Covering by Arcs
Circle, Ellipse, Hyperbola, and Astroid
Circle Images
Circle Intersection Patterns
Circle Involute
Circle Involute 3D Art
Circle of Curvature
Circle of Disks
Circle-Point Midpoint Theorem
Circle Reflections
Circles and Ellipses
Circles of Apollonius
Circles on a Grid
Circles Packed in a Circle
Circles through the Orthocenter
Circles & Triangles
Circular and Elliptic Polarization of Light Waves
Circular Coverings
Circular Motion and Newton's First Law
Circular Rectangle
Circular Scissors Lift
Circular View of the Means of Two-Color Totalistic 2D Cellular Automata
Circumcenter of a Triangle
Circumcircle and Incircle of a Triangle
Circumcircles Intersecting at the First Fermat Point
Circumference of a Circle
Circumradii Perpendicular to the Sides of the Orthic Triangle
Circumscribing Polygons
Cities of Countries
Classical Approximations of Pi
Classical Barrier Crossing and Phase Space
Classical Motion and Phase Space for a Harmonic Oscillator
Classical Motion and Phase Space for a Morse Oscillator
Classical Particle in a Coulomb-Like Radial Potential
Classic Logistic Map
Classifying Quadrilaterals
Classifying the Complexity and Information of Cellular Automata
Clean Tile Problem
Clebsch-Gordan Coefficients
Click a Country
Clinking Wineglasses
Closed-Form Full Life Cycle Distribution
Closed Form Solutions for Spheroidal Functions
Closed Spline Ball
Cluster Analysis for 2D Points
Cluster Cellular Automata
Clustered Power-Law Networks
Cluster of 120 Spheres
Cluster of 195 Spheres
Cluster of 20 Icosahedra
Cluster of 30 Cubes
Cluster of 92 Spheres
Cluster of Spheres
CNOT and Toffoli Gates in Multi-Qubit Setting
Coalescent Gene Genealogies
Coalitions
Cobb-Douglas Production Functions
Cobb-Douglas Utility Function
Coconuts, Sailors, and a Monkey
Code 20 Cellular Automaton Calculates R110
Coevolving Complements
Coexistence of Qubit Effects
Coffee Mug to Donut
Cofunction Identities for Sine and Cosine
Cogwheel Drive
Coherent States of the Harmonic Oscillator
Coils, Knots, and Rosettes
Coincidences in Powers of Integers
Coin Flips
Collapsible Cup
Collatz Paths
Collatz Problem as a Cellular Automaton
Collatz Sequence Maximums
Collatz Sequence Paths
Colliding Cars
Colliding Galaxies
Collinearity of an Incenter and Two Circumcenters
Collinearity of an Orthocenter, the Incenter, and the Circumcenter
Collinearity of a Triangle's Circumcenter, Incenter, and the Contact Triangle's Orthocenter
Collinear Orthocenters
Collocation by Chi Square
Collocation by Symmetric Conditional Probability
Color Blindness
Colorcoded Country Comparison
Color Cube
Color Cube 2x2x2 Puzzle
Color Cube 3x3x3 Puzzle
Color Drill Down
Colored Klein Bottle
Colored Lights
Color Filters in Black and White Photography
Coloring Cycle Decompositions in Complete Graphs on a Prime Number of Vertices
Colorization of Cellular Automata
Colorizing an Image
Color Lambdoma
Color Relationships
Color Representations of Numbers
Color Triangles
Color versus Name
Color versus Wavelength
Color-Word Game
Color Your Pies
Column of Hexagons
Combination of Linear and Icosahedral Structures
Combination of Triangular Patches
Combinations of Sines in the Complex Plane
Combinatorics of Love from A Midsummer Night's Dream
Combining Blocks
Combining Colored Quarks
Combining Functions
Combining Quarks into Hadrons
Combining Sine Waves
Combining Two 3D Rotations
Comet Moving about the Sun
Comet Tails
Common-Factor Graph of Fibonacci Numbers
Common Font Families
Common Methods of Estimating the Area under a Curve
Common Subsequences of Words
Communities of Nations Bridged by Language Similarity
Commute p to the Left
Comparing Algorithms for the Traveling Salesman Problem
Comparing and Weighting Two Weibull Models
Comparing Binomial Generalized Linear Models
Comparing Data on Countries
Comparing Fractions
Comparing Fractions, Decimals, and Percents
Comparing Integers
Comparing Models for Two-Way Contingency Tables
Comparing Normal and Student's t-Distributions
Comparing Parametrizations
Comparing Properties of Chemical Elements
Comparing Real Mathematical Constants and Random Numbers
Comparing Rule 30 Pseudorandoms to a Uniform Distribution
Comparing Simple Crank/Slider and Scotch Yoke Mechanisms
Comparing Some Residuals for Generalized Linear Models
Comparing the Normal Ogive and Logistic Item Characteristic Curves
Comparing the Primes in Base 2
Comparing Two Isochronous Oscillators
Comparison of Factorials and Primorials
Comparison of Performance of Recycle Reactors with Plug Flow Reactors
Comparison of Real and Plug Flow Reactors for a First-Order Reaction
Comparison of Trigonometric Functions
Compass Tiling
Competing Bid-Rents under Monocentric Assumptions
Competition for Territory: The Levins Model for Two Species
Complement Tiling
Complete-Mixing Model for Gas Separation by Membranes
Completing the Square
Complex Addition
Complex and Real Planes of Discrete Fourier Transforms
Complex Exponential and Logarithm Functions
Complex Exponential Resonance
Complex Fibonacci Residues
Complex Functions Applied to a Square
Complexified Pseudosphere
Complex Mapping of Contours and Regions
Complex Multiplication
Complex Newton Iteration for a Cubic Polynomial
Complex Newton Map
Complex Number
Complex Number Game
Complex Numbers in Rectangular and Polar Form
Complex Numbers with Different Norms
Complex One-Time Pad
Complex Pendulum Oscillations at Complex Times
Complex Polynomials
Complex Power Plot
Complex Reaction with Segregation in a Semi-Continuous Reactor
Complex Roots of a Polynomial and Its Derivative
Complex Rotation of Minimal Surfaces
Complex Slide Rule
Complex Spherical Harmonics
Complex Views of Logistic Maps
Composing Functions Using Dynagraphs
Composing Rotations
Composition of Functions
Compound Interest Table
Compound of 5 Cubes
Compound of Five Octahedra
Compound of Two Icosahedra
Compounds of 5 and 10 Tetrahedra
Compounds of Tetrahedra
Compressed Constrained Cellular Automaton Initialization
Compressing Elementary Cellular Automaton Rules
Compton Scattering of Subatomic Particles
Computation of the Residue Curve Map
Computation of the Volume of a CSTR and a PFR
Concentrations of Multiple Reacting Species in a Semi-Infinite Region
Concentric Circles
Concentric Circles around a Circle
Concurrency Induced by a Cevian
Concurrency via Midpoints
Concurrent Line Segments Related to the Incircle
Concurrent Lines that Intersect on the Euler Line
Concyclic Circumcenters
Concyclic Points
Concyclic Points Associated with an Angle Bisector and an Excircle
Concyclic Points Derived from Midpoints of Altitudes
Concyclic Points from Reflections of the Circumcenter about the Altitudes
Concyclic Points Related to a Midpoint and the Incircle
Condensed Cellular Automaton States
Conditionally Reversible Elementary Cellular Automata
Conditional Probability
Cone Placed on a Sphere
Cone, Tent, and Cylinder
Confidence and Prediction Bands
Confidence Intervals: Confidence Level, Sample Size, and Margin of Error
Confidence Intervals, Confidence Levels, and Average Interval Length
Confidence Intervals for a Mean
Confidence Intervals for the Binomial Distribution
Confluent Hypergeometric Functions
Conformal Map of Exponential
Conformal Maps
Conformal Pixel Transformation
Congressional Apportionment Using General Divisor Methods
Congruential Color Sequences
Conic Pendulum
Conic Section
Conic Section Curves
Conic Sections: Equations and Graphs
Conic Sections: Move Foci
Conic Sections: Polar Equations
Conic Sections: The Double Cone
Conjecture de Beraha pour les cycles (French)
Conjecture de Beraha pour les roues (French)
Connected Astroids
Connected Components
Connecting the CDF and the PDF
Connecting Towns Using Kruskal's Algorithm
Connect the Dots
Consecutive Digits in the Expansion of Pi
Consecutive Exponential Decay
Consecutive Heads or Tails
Conservation of Energy with a Simple Pendulum
Conservative Byte Drift Machine
Conservative Hex Drift Machine
Constant-Addition Continuous Cellular Automaton
Constant Angle of Incidence Contours for an Air-SiO2-Si System
Constant Coordinate Curves for Elliptic Coordinates
Constant Coordinate Curves for Parabolic and Polar Coordinates
Constante d'Euler-Mascheroni (French)
Constant Elasticity of Substitution Production
Constant Risk Aversion Utility Functions
Constrained Optimization
Constructing a Cassegrain Telescope
Constructing and Manipulating Graphs
Constructing a Regular Pentagon with Ruler and Compass
Constructing A Simple Optical System
Constructing the Perfect Piston
Construction of a Bézier Curve
Construction of the Rate-Conversion-Temperature Chart from Kinetic Data
Constructive and Destructive Interference
Consumer and Producer Surplus
Consumer Demand
Contact Binary Star Envelopes
Contact Triangles of a Triangle
Continued Fraction Approximations
Continued Fraction Approximations of the Tangent Function
Continued Fractions
Continuous and Discrete Time Discounting
Continuous Cellular Automaton with Math Rules I
Continuous Cellular Automaton with Math Rules II
Continuous Cellular Automaton with Math Rules III
Continuous Exponential Growth
Continuous Functions Are Integrable
Contour Integral around a Simple Pole
Contour Integration
Contour Plots of the Zeta Function
Contours of Algebraic Surfaces
Contours of Constant Principal Angle in the Complex Dielectric Plane
Contrast Illusion
Controlling Chaos on the Logistic Map
Controlling Liquid Height within Two Tanks in Series Using a PID Controller
Convection between Two Vertical Walls
Convection-Diffusion in a Semi-Infinite Region
Convergence d'une série de Taylor vers la fonction développée (French)
Convergence of a Hyperpower Sequence
Convergence of a Monotonic Sequence
Convergence of a Recursive Sequence
Convergence of Minimization Methods
Convergence of Newton's Method for Approximating Square Roots
Convergence of the Empirical Distribution Function of a Sample
Conversion of Ethylbenzene to Styrene in the Presence of Steam
Convert an Image to Binary
Converting Fractions
Convex Golden Rhombic Polyhedra
Convex Hull
Convex Hull and Delaunay Triangulation
Convexification
Convolution Method in Hydrologic Computations
Convolution of Two Densities
Convolution Sum
Convolution with a Rectangular Pulse
Conway Sequence with Varying Initial Conditions
Conway's M(13) Puzzle
Cooke (Taylor) Triplet
Cookie Cutting
Cooling of a Rod with Initial Temperature Distribution
Cooperation in Heterogeneous Populations
Coordinates of a Point Relative to a Basis in 2D
Coordinate Sonification
Coordination in Ionic Compounds
Coordination of Insurance Policies
Copeland-Erdos and Champernowne Numbers for Small Bases
Copernicus's Theorem
Coprime Triples
Corner Cabinet with Rotating Shelves
Corner Reflector
Correction Trees
Correlated Binary Decision Rules
Correlated Gamma Variance Processes with Common Subordinator
Correlated Wiener Processes
Correlation and Covariance of Random Discrete Signals
Cosine and Sine Identities with Dihedral Transformations
Cosine of a Cubic Polynomial
Cosine Offset Curves
Cost and Revenue for Monopoly and Monopolistic Competition
Cotes Identity
Cotter Pin (Split Pin)
Countercurrent Extraction with Extract Reflux (the Janecke Diagram)
Countercurrent Leaching of Oil from Meal
Counting by Steps
Counting Exercise
Counting in Binary
Counting in Triangles
Counting Order Puzzle
Counting Paths through a Grid
Counting T-Shirts
Country Data and Benford's Law
Country Flag Quiz
Country Flags and Descriptions
Country Groups
Country Interior: Two Solutions to the Nonconvex Polygon Interior Problem
Country Ranking Maps
Country Statistics over Time
Coupled Cellular Automata: Symbiotic Patterns and Synchronization
Coupled Coils
Coupled Pendulum Oscillations
Coupling Cellular Automata with Substitution Systems
Coupon Collector Problem
Covering Maze
Covering the Plane with Polygons
Crane Model
Crankshaft-Driven Pendulum
Crazy Surfaces
Create a Face
Create Alternative Koch Snowflakes
Creating a GCD Grid
Creating and Manipulating Knots
Creation of a Magnetic Field by an Electric Current
Credit Risk
Crisscrossing a Polygon
Crookes Radiometer: A Comedy of Errors
Crossed Ladders Theorem
Cross-Flow Liquid-Liquid Extraction
Cross Product in Spherical Coordinates
Cross Product of Vectors
Cross Product of Vectors in the y-z Plane
Cross Recurrence Plot of Two Elementary Cellular Automata
Cross Sections of Graphs of Functions of Two Variables
Cross Sections of Regular Polyhedra
Cross's Theorem
Cryogenic Recovery of Acetone from Air
Crystal Art
Crystal Viewer
CSTR and PFR Holding-Time Ratio versus Conversion for Various Reaction Orders
Cube 5-Compound and Small Rhombicosidodecahedron
Cube and Hyperboloid
Cube and Octahedron Movement
Cube-in-a-Sphere Illusion
Cube Net
Cube Net or Not
Cube of Cubes
Cube of Cubic Koalas
Cube of Koalas
Cube Packing
Cube to Small Rhombicuboctahedron
Cubic Big Bang
Cubic Close Packing
Cubic Close Packing Cube
Cubic Close Packing versus Hexagonal Close Packing
Cubic Equation
Cubic Equation of State for the Compressibility Factor
Cubic Symmetry Types
Cubic Twist
Cumulation
Cumulative Cooling and Heating Degree Days
Cumulative Frequency Curves
Cupid's Arrow
Curlicue Fractal
Curling Structures Made of Square Pyramids
Curly Torus
Current-Voltage Characteristics of a Memristor
Curvas de nivel (Spanish)
Curvature and Torsion
Curvature of the Trefoil Knot
Curve Fitting
Cutting Polygons
Cycles from Permutations
Cycles in Random Sample Permutations
Cycles in Sums of Prime Factors
Cyclic Functions under Differentiation
Cyclic Quadrilaterals, Subtriangles, and Incenters
Cycloid and Archimedes's Spiral
Cycloid Curves
Cycloid from a Rolling Ellipse
Cylinder Area Paradox
Cylinder Hemisphere Cone
Cylinder Net
Cylinders on a Helix
Cylinders through Five Points
Cylindrical Cam
Cylindrical Coordinates
Cylindrical Segment Used as Shaft for a Switch
Cylindrical Waves Passing through a Double Slit
Damped 2D Lissajous Figures
Damped 3D Lissajous Figures
Damped Swing Pendulum
Damping in RLC Circuits
Dandelin Spheres for an Ellipse
Darboux and the Dog's Tail
D'Arcy Thompson's Affine Fish Transformations
Dart Practice
Data Compression Using Asymmetric Numeral Systems
Data Sampling and Interpolation
Data Smoothing
Day and Night World Clock
Deadweight Loss for a Monopoly
Deaf Island Puzzle Generator
Death Penalty Regressions
De Bruijn Graph Arcs
De Bruijn Initialization of 2D Automata
De Bruijn Sequences
De Bruijn Sequences Provide Compact Initial Conditions
Decagonal Clusters of Polyhedra
Decimal Division
Decimal Multiplication
Decimals as Fractions
Decisions Based on P-Values and Significance Levels
Decoding the Arecibo Message
Decomposing a Cuboctahedron into Three Rhombic Dodecahedra and Two Tetrahedra
Decomposing a Rhombic Dodecahedron into a Trapezoid-Rhombic Dodecahedron
Decomposing a Rhombic Solid into a Truncated Octahedron
Decomposition of 2,3 Turing Machine Evolutions
Decomposition of a Tetrahedron into Four Congruent Parts
Decomposition of a Vector in 2D
Decomposition of the Tetrahedron into Four Congruent Symmetrical Parts
Decorative Holiday Stars
Dedekind Cut
Deep Drawing of Metals
Definitions of Hill's Tetrahedra
Deformation Pattern in an Earthquake Source Region
Degenerate Eigenstates
Degenerate Energy States
Degree Distribution on a Random Network
Delannoy Number Carpet
Delannoy Path Exhaustion Diagonal
De Laval Nozzle
Delayed CA
Delay Logistic Equation
Delboeuf Illusion
Deltic Engine
Deltoidal Hexecontahedron
Deltoid Kite Tiling
Demanda, Oferta y Equilibrio (Spanish)
De Moivre's Theorem for Trig Identities
Dendrimer to Linear Polymer Transition
De Novo Growth Processes with Competing Mechanisms
Densest Tetrahedra Packing
Density
Density Estimation
Density Map for the 3n+1 Problem
Deploying and Steering a Parabolic Antenna
Derangement Diagrams
Derivation of Thermodynamic Derivatives Using Jacobians
Derivative as a Function
Derivatives: A Look at Graphs
Derivatives of a 2D Gaussian
Derivatives of Exponential Functions
Derivatives of Quintic Polynomials
Derivatives of Sine
Derivatives of the Sum of Two Sines
Derived Polygon
Deriving a Bilunabirotunda from Rhombic Solids
Descartes Signature Explorer
Descriptions of Univariate Data
Descriptive Reaction Kinetics
Designing a Car Body with Splines
Designing a Mercury Tumbling Machine
Designing End-Blown Flutes
Designing Optical Systems
Design of Alphabets with Fourier Transforms
Designs by Colorful Circles
Desk Lamp
Desynchronization Dynamics of Two Coupled Oscillators
Determinants Seen Geometrically
Deterministic Context Free (DOL) Systems
Device for Tube Extraction
Diagonal Planes of a Cube
Diagonals of Polyhedra
Dialectical Cellular Automata
Diamond Lattice
Diaphragm Shutter
Diauxic Growth of Bacteria on Two Substrates
Dice Probabilities
Dice Roller & Penny Flipper
Dichotomy Paradox Joke
Difference Formula for Cosine
Difference of Cubes
Difference of Squares
Difference Set
Differential of a Function
Differential of the Arc
Differentiation Microscope
Different Slants on Simple Geometric Figures
Diffraction Grating Intensities
Diffuse and Specular Reflection
Diffusion in One Dimension
Diffusion-Limited Aggregation
Diffusion-Limited Aggregation: A Real-Time Agent-Based Simulation
Diffusion-Limited Clogging
Diffusion-Reaction in a Catalytic Particle
Diffusion to a Spherical Reactor
Digital Tilt-Shift Photography
Digit Counts in Integers
Digit Frequencies in Digit Expansions
Digit Frequencies in the Columns of Cellular Automata
Digit Frequencies in the Copeland-Erdos Constant
Digit Mosaics
Digit Patterns in Integer Powers
Digit Patterns of Successive Numbers
Digit-Reversal Sequences
Digits in Powers of Integers for Various Bases
Digits of Hyperfactorial and Barnes G
Digits of Successive Fractions
Digits of the First 30 Mersenne Primes
Digit Sums
Digon Tiling of a Hosohedron
Dihedral Group n of Order 2n
Dim City
Dintzl's Law of Cosines Dissection
Dipole Antenna Radiation Pattern
Dipole Fields Are Complicated
Dirac Matrices in Higher Dimensions
Direct Control for Hex Machine
Directional Derivatives
Directional Derivatives and the Gradient
Directional Derivatives in 3D
Directivity Pattern of Line Arrays
Direct Rule Control for 2,2 Turing Machines
Direct Rule Control for 2,3 Turing Machines
Direct Rule Control for 3,2 Turing Machines
Direct Rule Control for 3,3 Turing Machines
Direct Sum of Rings
Dirichlet Characters
Dirichlet L-Functions and Their Zeros
Dirichlet L-Functions near the Critical Line
Dirichlet's Theorem
Disappearing Dot Illusion
Discontinuity
Discounted Present Value
Discrete Fourier Sine and Cosine Transforms
Discrete Fourier Transform of a Two-Tone Signal
Discrete Fourier Transform of Windowing Functions
Discrete Function Composition
Discrete Logistic Equation
Discrete Marginal Distributions
Discrete Number Theory Plots
Discrete Reaction-Diffusion
Discrete Spiral
Discrete Versions of Continuous Functions
Disk Rolling on a Horizontal Plane
Disk Sliding or Rolling in a Semicircular Well
Disparity, Convergence, and Depth (Visual Depth Perception 10)
Dispersion of Light through a Prism
Displacement along a Curve
Displacement versus Distance
Dissecting a Decagon into Two Pentagons and Two Pentagrams
Dissecting a Decagram {10/4} into Two Decagons
Dissecting a Dodecagram {12/2} into a Hexagram {6/2}
Dissecting a Dodecagram into a Triangle
Dissecting a Dodecagram into Two Hexagrams
Dissecting a Hexagon into Four Smaller Hexagons
Dissecting a Hexagram into Three Hexagrams
Dissecting an N-gram into Two Smaller N-grams
Dissecting an Octagon into Four Smaller Octagons
Dissecting a Pyramid into Four Equal Pieces
Dissecting a Regular Triangle into Three Similar Tiles
Dissecting a Rhombic Dodecahedron into an Octahedron and Two Tetrahedra
Dissecting a Rhombic Dodecahedron into Two Cubes
Dissecting a Rhombic Dodecahedron into Two Solids
Dissecting a Rhombohedron into Two Congruent Rhombohedra
Dissecting a Square Pyramid into 15 Pieces
Dissecting the Triangular Hebesphenorotunda (Johnson Solid 92)
Dissecting Two Rhombic Dodecahedra of the Second Kind into a Cube
Dissection Fallacy
Dissection of a Concave Rhombic Triacontahedron
Dissection of a Cube into a Cuboctahedron and an Octahedron
Dissection of a Cube into Five Polyhedra
Dissection of a Cube into Five Solids
Dissection of a Cube into Three Bilunabirotundas, a Dodecahedron, and a Smaller Cube
Dissection of a Cube into Two Kinds of Triacontahedra
Dissection of a Cube to a Quarter of a Rhombic Dodecahedron of the Second Kind
Dissection of an Oblique into a Right Prism
Dissection of a Pentagonal Antiprism into a Metabidiminished Icosahedron
Dissection of a Prismatoid into Prisms
Dissection of a Rhombic 210-hedron into a Combination of Icosahedron and Truncated Dodecahedron
Dissection of a Rhombic Dodecahedron of the Second Kind into a Rectangular Solid
Dissection of a Rhombohedron into a Right Prism with the Same Base
Dissection of a Special Rectangular Solid to a Cube
Dissection of a Triangle to Its Mirror Image I
Dissection of a Triangle to Its Mirror Image II
Dissection of a Truncated Octahedron into Hexagonal Skew Prisms and Parallelepipeds
Dissection of Four Rhombic Dodecahedra into a Truncated Tetrahedron and a Tetrahedron
Dissection of Hill's Tetrahedron of Type 1
Dissection of the Oblate Golden Rhombohedron
Dissection of Three Bilunabirotundas and a Dodecahedron into Rhombic Solids
Dissection of Three Polyhedra into Two
Dissection of Three Rhombic Solids into an Icosahedron, a Dodecahedron, and an Icosidodecahedron
Dissection of Two Rhombic Dodecahedra of the Second Kind
Dissection of Two Rhombic Solids into an Icosidodecahedron and a Rhomb-Icosi-Dodecahedron
Dissection of Two Stellated Dodecahedra
Dissection of Two Truncated Octahedra into a Cube
Distance and Average Velocity for Piecewise Trajectory
Distance and Polygons
Distance between Two Points
Distance Distributions in Finite Uniformly Random Point Processes
Distance from a Point to a Line
Distance Functions
Distances from the Centroid
Distance Transforms
Distillate and Still Compositions for Batch Rectification of a Binary Mixture at a Very Large Reflux Ratio
Distillation Column with an Intermediate Reboiler
Distillation Column with Liquid Side Stream
Distillation Column with Two Feeds
Distillation Lines for a Mixture of Chloroform, Acetone, and Methanol at 1 atm
Distillation Lines versus Residue Curves for a Mixture of Oxygen, Argon, and Nitrogen at 1 atm
Distillation of Substances with Widely Differing Heats of Vaporization: The Peters Transform
Distorted Cube
Distribution of Palindromes in Base b
Distribution of Primes
Distribution of Returns from Merton's Jump Diffusion Model
Distribution of Spore Custom Content
Distribution of the Last Digit of the Primes
Distributions of Leading Digits
Distributions of Order Statistics
Divergence from the Mandelbrot Set
Divide Triangles
Dividing a Right Triangle by the Altitude to the Hypotenuse
Dividing a Triangle by Lines Parallel to Two Sides
Dividing Fractions
Dividing Integers
Dividing Mixed Numbers
Dividing Segments by Origami
Divisibility Graph
Divisibility Networks
Division of an Angle Bisector by the Incenter
Division of Polynomials up to Degree Six
Division of the Opposite Side by an Angle Bisector
Divisors of a Number
DNA Screening Incidentalome
Dodecahedral Cluster of RTs and RHs
Dodecahedron and Bilunabirotunda
Dodecahedron Made of 20 Blocks
Dodecahedron Measures
Dodecahedron to Small Rhombicosidodecahedron
Dollar Duration
Domain and Range of Inverse Trigonometric Functions
Domany-Kinzel Stochastic Cellular Automata
Dome Frames Made of Arches
Dome Made of Cylindrical Segments
Doped Silicon Semiconductors
Doppler Effect
d-Orbitals in an Octahedral Field
Dot Product
Dots and Boxes
Double Delta Function Scattering Potential
Double Helix
Double Integral for Volume
Double Pendulum
Double P-Slide Dissection
Double Refraction by a Uniaxial Crystal
Doublets
Doubling a Distance with a Compass
Doubling Cellular Automata
Draining of Two Connected Tanks
Drawbar/Harmonic Waveform Viewer
Drawing a Colored Polygon
Drawing a Line on a Digital Display
Drawing an Ellipse Using Circles and Lines
Drawing an Oval with a String and Three Nails
Drawing a Triangle
Drawing a Wire
Draw Polygons on Grids
Dressed Multi-Particle Electron Wave Functions
Drilled Truncated Octahedron
Drilling a Hexagonal Hole
Drilling a Square Hole
Driven Damped Oscillator
Driven Damped Oscillator with Resonance Effect
Driving a Piston
Drop Shadows
Droste Effect
Dual Billiards
Duals by Rotating the Edges of Polyhedra
Duals of Platonic Solids
Duck Cut
Dudeney's Cross-to-Isosceles-Right-Triangle Dissection
Duffing Oscillator
Duff-Porter Alpha-Compositing Operators
Dust with Line Shadows in Every Direction
Dutch Drawbridge
Dwarf Planets
Dynagraphs
Dynamical Network Design for Controlling Virus Spread
Dynamic Approximation of Static Quantities (Visual Depth Perception 14)
Dynamic Behavior of a Bioreactor
Dynamic Behavior of a Heated Stirred Tank
Dynamic Behavior of an Addition Polymerization in a Batch Reactor
Dynamic Behavior of a Nonisothermal Chemical System
Dynamic Behavior of a Reactive Distillation Column: Mixture of Four Components
Dynamic Behavior of a Simple Canonical System
Dynamic Damping
Dynamic Proximity Networks
Dynamic Simulation of a Gravity-Flow Tank
Dynamics of a Binary Distillation Column
Dynamics of a Continuous Distillation of a Mixture of Benzene, Toluene, and p-Xylene
Dynamics of a Cube-Shaped Mass-Spring Network
Dynamics of an Elementary Cellular Automaton
Dynamics of Coupled Pendulums
Dynamics of Triangle Iterates
Dynamic Step Distance Transforms
Earthquake Focal Mechanism
Earth's Second Moon
Eccentric Cylinder and Tappet Drive
Echoes
Economic Trade-Offs in the Operation of a Single Reactor
Ecosystem Dynamics
Ecouter la convergence (French)
E-curve and F-curve for the Tanks-in-Series Model of MacMullin and Weber
Edge Enhancement on Multiple Images
Edit Distance
Effect of a Perturbation on the Stable Points of a Dynamical System
Effect of Detail on Visual Perception
Effects of Bin Width and Height in a Histogram
Efficacy of Degenerate Keyboard Maps
Efficient Single Limit Liability Insurance
Egg Colored by Gradients
Egg Colored Using Polynomials
Egg Colored Using the Riemann Zeta Function
Egg Colored with Sinusoids
Eggnigmatica
Eggnigmatica II
Egg Shape Construction
Egg-Shaped Orbits of Non-Uniform Oscillators
Eggstravaganza
Eggs using Spherical Coordinates
Egyptian Fractions
Egyptian Multiplication
Ehrenstein Contour Illusion
Eigenfunctions for a Linear Potential Field
Eigenfunctions of the Helmholtz Equation in a Right Triangle
Eigenstates for Pöschl-Teller Potentials
Eigenvalue Plots of Certain Tridiagonal Matrices
Eigenvalue Problem for 2×2 Hermitian Matrices
Eigenvalues, Curvature, and Quadratic Forms
Eigenvalues of Cellular Automaton Matrices
Eigenvalues of Random Symmetric Matrices
Eigenvectors by Hand
Eigenvectors in 2D
Eight Dodecahedra
Eight Icosahedra
Eight Queens Puzzle
Eight Spheres
Einstein's Box
Einstein's Most Excellent Proof
Einstein Solid
Eisenstein Integer
Eisenstein Snowflakes
Elapsed Time
Elastic Collision of Two Spheres
Elastic Collisions of Two Spheres
Elasticity of Shear
Elasticity, Total Revenue, and the Linear Demand Curve
Electrical Impedance of Biological Tissues
Electric Current
Electric Dipole Potential
Electric Field Generated by Two Point Charges
Electric Fields for Three Point Charges
Electric Potential of a Metallic Cylinder Surrounded by a Dielectric in an External Static Electric Field
Electromagnetic Energy Density and Poynting Vector of a Relativistic Oscillator
Electromagnetic Ring Toss
Electromagnetic Wave
Electron Configuration Browser
Electronic Band Structure of a Single-Walled Carbon Nanotube by the Zone-Folding Method
Electronic Structure of a Single-Walled Carbon Nanotube in Tight-Binding Wannier Representation
Electrostatic Bonding Forces between Atoms
Elementary CA Map Composition Diagram
Elementary CA Mapping Composition Fan
Elementary Cellular Automaton Egg
Elementary Cellular Automaton Process Visualization
Elementary Cellular Automaton Rules by Gray Code
Elementary Cellular Automaton with Network
Elementary Nearest Ambidextrous Automata
Elementary Processes in Protein Folding
Elementary Propagation of a Wave
Elementary Transcendental Functions and Their Inverses
Element Density Comparisons
Elements propres interactifs (French)
Elevator System
Ellipse and Friends
Ellipses and Hyperbolas with the Same Focal Points
Ellipsograph or Reciprocator
Ellipsoid
Elliptical Drive
Elliptical Spheres
Elliptic Chords
Elliptic Cylindrical Coordinates
Elliptic Gears
Elliptic Hyperboloid of One Sheet
Elliptic Hyperboloid of Two Sheets
Elliptic Paraboloid
Embeddings of Graphs in a Torus and in a Moebius Strip
Embroidery I: Honeycomb Lattice
Embryonic Cleavage
Emoticon
Emotion Tiling
Empirical Characteristic Function
Empty Focus Approximation to Kepler's Second Law
Emulating Land Use Evolution with a Cellular Automaton
Encryption with Irrational Numbers and the One-Time Pad
End Behavior of Polynomial Functions
Endothermic Reaction in a Heated Batch Reactor
Energy and Position Relationships in Simple Harmonic Motion
Energy and Power of Signals
Energy Balance Model with Greenhouse Parameterization
Energy Density of a Magnetic Dipole
Energy Density of an Electrostatic Dipole
Energy Density of a Particle Moving at Uniform Speed
Energy Flow in a Negative-Index Material
Energy Levels in a Half-Infinite Linear Well
Energy Levels of a Morse Oscillator
Energy Levels of a Quantum Harmonic Oscillator in Second Quantization Formalism
Energy of a Slingshot: David and Goliath
Energy Spectrum for a Finite Potential Well
Enigmatic Comet Holmes
Enlargement in Two Dimensions
Enneper Surface
Entangled Torus Coils
Entanglement between a Two-Level System and a Quantum Harmonic Oscillator
Enthalpy-Concentration Plot for a Benzene-Toluene Binary Mixture
Enthalpy versus Composition Diagram for n-Hexane/n-Octane at 101 kPa
Entropy of a Message Using Random Variables
Enumerating Pythagorean Triangles
Enumerating the Directed Graphs
Enumerating the Rationals
Envelope of Two Families of Plane Curves
Enzymatic Reaction in a Batch Reactor
Enzymatic Reactor with Product Inhibition
Epicycloid
Epitrochoids
Equal Areas
Equal but Opposite Percentage Changes Do Not Cancel
Equal Chords
Equally Distant Chords
Equally Sized Mixed Flow Reactors in Series versus Plug Flow Reactor
Equation of a Line Game
Equation of a Line in Vector Form 2D
Equation of Continuity
Equations for Valentines
EquationTrekker
Equatorial Telescope Mounts
Equidecomposition of a Rhombohedron, Hexagonal Prism, and a Rhombic Dodecahedron
Equilateral Triangles in 3D with Integer Coordinates
Equilateral Triangles That Form a Star
Equilateral Triangle to Square
Equilibrium of a Rigid Bar
Equilibrium Water Activity of Binary Dry Mixtures
Equivalence Classes Modulo m
Equivalence of Three Catalan Number Interpretations
Equivalence of Two Fibonacci Number Interpretations
Equivalent Isothermal Time at a Reference Temperature as a Function of Time
Equivalent Products of Integers
Eratosthenes's Measure of the Earth's Circumference
Erlang B and Telephony
Error-Diffusion Dither Patterns
Estimating Conditional Expectations with Monte Carlo Simulation and Least Squares Regression
Estimating Insurance Premiums Using Exceedance Data and the Method of Moments
Estimating Loss Functions Using Exceedance Data and the Method of Moments
Estimating Percentages
Estimating Perimeter and Area of Simple Polygons
Estimating the Local Mean Function
Estimating the Side Length of a Square from Its Area
Estimating the Surface Area and Volume of a Rectangular Prism
Estimating the Thermal Properties of Foods from Their Moisture Contents
Euclid, Book 3, Proposition 22
Euclidean Algorithm Steps
Euclid's Proof of the Pythagorean Theorem
Euler Angles for Space Shuttle
Euler Line
Euler Product for the Zeta Function
Euler's Distribution Theorem
Euler's Generating Function for the Partition Numbers
Euler's Method for the Exponential Function
Euler's Polyhedral Formula
Euler's Theorem for Pedal Triangles
Euler's Triangle Formula
Euler Zigzag Numbers
European Option Greeks
Europe Vote Exponent
Evaluate Hot Pizza
Even and Odd Functions
Evens and Odds
Event-by-Event Simulation of Double-Slit Experiments with Single Photons
Event-by-Event Simulation of the Mach-Zehnder Interferometer
Everything about Gell-Mann Matrices (Part 1): Unary Operations
Everything about Gell-Mann Matrices (Part 2): Binary Operations
Evidentiary Uncertainty
Evolutes of Some Basic Curves
Evolutionary Multiobjective Optimization
Evolution of a Gaussian Wave Packet
Evolution of Rotating Substitution Systems in 3D
Exact and Variational Energies for the Hydrogen Molecular Ion and the Helium Hydride Ion
Exact Recurrent Computations with Integers
Exact Values for Sine and Cosine
Exact Values of Trigonometric Functions for Given Angles
Examples of 1D Three-Color Totalistic Cellular Automata
Examples of Fourier Series
Excavator Operator
Exchange Networks
Excising Edges from Cubic Graphs
Exit Times of Brownian Motion in 3D
Expandable Flat Ring
Expanded Fermi Solution for Risk Assessment
Expanding a Set of Possibilities
Expanding Surface Structure
Expanding Tetrahedral Network
Expansion of a Polynomial
Expectation Maximization for Gaussian Mixture Distributions
Expectations for Roulette
Expected Motion in 2x2 Symmetric Games Played by Reinforcement Learners
Expected Returns of the Dow Industrials, Beta Model
Expected Returns of the Dow Industrials, Fama-French Model
Expected Utility: Optimal Asset Investment
Expected Utility: Optimal Insurance
Expert System for Identifying Pedigree Dogs
Explaining Real Estate Price Bubbles
Exploration of Fluid Meandering by Statistical Simulation
Exploration versus Consumption
Exploring a Logo by Grignani
Exploring Cylindrical Coordinates
Exploring Measures of Association
Exploring Minimal Models of the Complexity of Security Prices
Exploring Polar Graphs
Exploring Skewness in Box Plots
Exploring Social Choice Theory
Exploring Spherical Coordinates
Exploring the Black-Scholes Formula
Exploring the Crease Length Problem
Exploring the Mercedes-Benz Logo
Exploring the Reyflex Textilien Logo
Exploring the Tails of the Normal Distribution
Exploring the Tate Logo
Exploring Unicode Space
Exploring with Inset
Exponential Allowance Riddle
Exponential Congruences
Exponential Decay
Exponential Generating Functions
Exponents
Expression Machine
Expression Trees for Integrals
Extendable Circular Table
Extendable Fence
Extendable Frame
Extendable Shelter
Extendable Tower
Extend Dodecahedron Edges
Extended Bead-Sort
Extended Euclidean Algorithm
Extend Icosahedron Edges
Extraction of Acetic Acid from Water Using Isopropyl Ether
Extremal Trigonometric Polynomials
Extreme Value Theorem
Eyeball and Two Muscles
Eye Parameters (Visual Depth Perception 6)
Factorial
Factorial Digits
Factorial Number Base
Factoring an Integer
Factoring Binomials
Factoring Gaussian Integers
Factoring Polynomials over Various Rings
Factoring Real Numbers
Factoring the Sphenic Numbers
Factor Trees
Fagnano's Problem
Failure Probabilities from Quality Control Charts
Falling Mass Attached to a Rigid Pulley
Falling Stick
False Logarithm Series
Families of Figure-Eight Curves
Families of Solutions for ODEs
Family of Curves, Tangents, and Intuition
Faraday Cage
Faraday Disk Dynamo
Farey Sequence
Farkas's Lemma in Two Dimensions
Farris Wheels
Fastening Mechanism Using a Rocker
Fault-Free Tilings
Feasible Regions for Product Compositions in Ternary Mixtures
Fed-Batch Fermentation
Feinberg-Horn-Jackson Graph
Fender Tone Circuit
Fenómeno de Difusión-Reacción en una partícula catalítica (Spanish)
Fermat Point
Fermat's 4n+1 Theorem and the n Queens Problem
Fermat's Little Theorem
Fermat's Magic Cube
Fermat's Theorem on Stationary Points
Fermi-Dirac Distributions for Free Electrons in Metals
Ferrers Diagrams
Feuerbach's Theorem
Feynman's Relativistic Electrodynamics Paradox
Feynman's Wobbling Plate
Fibonacci Determinants
Fibonacci Mountain Matra Meru
Fibonacci Numbers and the Golden Ratio
Fibonacci Numbers Count Domino Tilings
Fibonacci Rabbits
Fibonacci Residues are Periodic
Fibonacci Tree
Fifteen Great Circles on a Sphere
Fifty-Nine Icosahedra
Filling a Container Defined by a Curve
Filling a Cube Seven Parts at a Time
Filling the Gaps in Mendeleev's Periodic Table
Filtering a White-Noise Sequence
Filtering Using Common Value
Find a Formula for Pi
Find a Model for a Set of Propositions
Finding an Inverse
Finding a Pattern (Linear Equations)
Finding a Tangent Line to a Parabola
Finding Probabilities for Intervals of a Normal Distribution
Finding Roots Using Binary or Interpolation Search
Finding Segments of an Image by Filtering Its Distance Transform
Finding the Greatest Common Divisor of Two Numbers by Factoring
Finding the Least Common Multiple of Two Numbers by Factoring
Finding the Linear Function Rule
Fingerprints of Electronic Files: Performance Tests
Fingerprints of Electronic Files: Precision Tests
Finite Difference Schemes of One Variable
Finite Element Solution of 1D Poisson Equation with Galerkin Spectral Decomposition
Finite Field Tables
Finite Limit at a Finite Point
Finite Limit at Infinity
Finite-State, Discrete-Time Markov Chains
Finsler-Hadwiger Theorem
Fireworks
First Droz-Farny Circle
First-Order Logic Test
Fisher Discriminant Analysis
Fish-in-Pond Simulation
Fit, Interpolation, or Polynomial Interpolation in Uncertain Calculus
Fitness Model for Gene Frequency in Populations
Fitting a Curve to Five Points
Fitting an Elephant
Fitting Lifetime Data to a Weibull Model
Fitting Noisy Data
Fitting Primes to a Linear Model
Five Circles and the Golden Ratio
Five Definitions of Strain for Large Deformations
Five-Gear Manual Transmission
Five-Neighbor 2D Totalistic Cellular Automata
Five-Neighbor Two-Color Outer Totalistic 2D Cellular Automaton
Five Points Determine a Conic Section
Five-Squares-to-One Dissection
Fixation and Distraction (Visual Depth Perception 5)
Fixed Point Elementary Rules
Fixed Point of a 2x2 Linear System of ODEs
Fixing a Fixed Point
Flash, Bang, Thwack
Flash Calculations Using the Peng Robinson Equation of State
Flash Distillation Cascade for a Benzene-Toluene Mixture
Flash Distillation Cascade for an Acetone-Chloroform Mixture
Flash Distillation Cascade for an Ethyl Acetate-Ethanol Mixture
Flash Vaporization of a Heptane-Octane Mixture
Fleeting Random Question Marks
Flett's Theorem
Flexible Galton Board
Flight through a Star Cluster
Flipping between Two Elementary Cellular Automaton Rules
Floating Disk Stereogram
Floating Sphere
Floor, Ceiling, and Round
Flor con la función seno (Spanish)
Flower and Seed Spirals
Flower Around a Limit Point
Flower-like Parametric Plots
Flower-Like Polar Plots
Flow from a Tank at Constant Height
Flow in a Rectangular Channel with Double Orthogonal Collocation
Flow in a Vertical Channel with Walls at Different Temperatures
Flow of a Vector Field in 2D
Flow of Liquid through a Hole
Fluid Rotating in a Cylinder
Fluids in the Critical Region
Flux
Flying Saucer Builder
Flying to the Moon
Focal Attraction with Rotation
Foldable Frame
Foldable Frame on a Telescopic Lift
Foldable Panel
Folding and Unfolding Polar Zonohedra
Folding and Unfolding Polyhedral Nets
Folding an Equilateral Triangle in a Square
Folding an Octahedron from Squares
Folding a Triangular Net into a Boat
Folding Hexagons
Folding Ladder
Following Chaotic Reflections
Fonction de répartition (French)
Font Polygon
Forced Convection for Internal Laminar Flow
Forced Oscillations
Forced Oscillator with Damping
Forced Pendulums with Damping
Forces Acting on a Ladder
Forces Exerted on an Immersed Object
Forces on a Pendulum
Force to Overcome Vacuum Pull
Ford Circles
Forest Fire Simulation Using a Randomized Hexagonal Automaton
Forest of Points
Forming the Efficient Frontier When Returns Are Non-Normal
Formula Generator
Formulas and Structures for Some Simple Molecules
Forward Kinematics
Forward Problem Solved with GeoDestination
Foucault's Pendulum
Four-Bar Linkage
Four Collinear Points Related to the Altitudes
Four-Coloring Real Maps
Four-Color Maps
Four-Color Outer Median Cellular Automata in 1D
Four-Color Outer Median Cellular Automata on Graphs
Four-Color Outer Median Cellular Automata on the Grid
Four Concurrent Lines in a Cyclic Quadrilateral
Four Concyclic Points
Four Dividable Rhombohedra
Four Icosahedra around a Tetrahedron
Fourier Coefficients of a Square Pulse
Fourier Series of Simple Functions
Fourier Sound Synthesis
Fourier Synthesis for Selected Waveforms
Fourier Transform of Radially Symmetric Potential Functions
Fourier Transform Pairs
Fourier Transforms of Cellular Automaton Images
Fourier View of an Outer Totalistic Three-Color Cellular Automaton
Four Kinds of Rhombic Polyhedra
Four-Piece Dissection of Hill's Tetrahedron of Type 2 into a Triangular Prism
Four-Piece Dissection of Juel's Pyramid to a Prism
Four Polyhedra, Each with an Unfolding that Overlaps Itself
Four Space Curves
Fractal Blades
Fractal Creation with Iterated Function Systems
Fractal Curves
Fractal Octahedra I
Fractal Polygon Tiler
Fractal Reflection of Circle Configurations
Fractal Right Triangle
Fractal Robot Arm
Fractals Generated by Multiple Reflections of Circles
Fractal Tetrahedron
Fractal Trees
Fractional Derivative of a Power
Fractional Derivative of Sine
Fractional Fourier Transform
Fractional Graphs and Flowers
Fractional Parts of Powers of Numbers
Fractional Parts of Powers of Rationals
Fraction Blocks
Fractions
Fractions in Different Bases
Fraunhofer Diffraction at One Slit
Free Convection past an Isothermal Vertical Plate
Free Fall on the Solar System Planets and the Moon
Free Rotation of a Rigid Body: Poinsot Constructions
Freese's Two-Octagons-into-One Dissection
Free Surface next to a Wet Wall
Frenet Frame
Frequencies of Character Pairs
Frequency Chord Projection
Frequency Distribution of the Logistic Map
Frequency Response of an LCR Circuit
Frequency Spectrum of a Noisy Signal
Fresnel Diffraction at an Edge
Fresnel Equations
Frictional Pressure Drop in a Pipe
Frog Jumping Problem
From Circle to Sine and Cosine Curves with Angle in Degrees
From Quadrophony to Octophony
From Vector to Line
From Vector to Plane
Fuel Break
Fuhrmann's Theorem
Fully Random, Five-Rule Interactive Cellular Automata (ICA)
Function Explorer 2D
Function Explorer 3D
Function Identification Game
Functions of Matrices
Functions Related to Primes
Functions with Vertical, Horizontal, and Oblique Asymptotes
Function Transformations
Fundamental Law of Fractions
Fundamental Theorem of Arithmetic
Fund Drawdown Simulation
Fun with Koch Snowflakes
Future Value Calculator
Gabriel's Horn
Gains from Trade
Galactohedra
Galileo's Experiment at the Leaning Tower of Pisa
Galileo's Paradox
Galton Board
Galvanometer as a DC Multimeter
Game Clock
Game of Dice
Game of Dice 2
Game of Life Configurations
Game of Life in 3D Layers
Gamma Correction
Gammes pythagoriciennes (French)
Gantt Charts and Network Diagrams
Gaps between Consecutive Primes
Gaps between Successive Primes
Gaps in GPS Data
Garbage Collection by Ants
Garden of Eden States
Gas Absorption with a Rapid Chemical Reaction
Gas Absorption with Chemical Reactions
Gateshead Millennium Bridge
Gauss Code Loops
Gaussian Approximations to 1s Slater-Type Orbitals
Gaussian Brackets
Gaussian Divisors of an Integer
Gaussian Filtering for Blurring
Gaussian Laser Modes
Gaussian Matrix Kernels
Gaussian Primes
Gaussian Quadrature
Gaussian Smoothing
Gaussian Wave Packet in a Linear Potential Field
Gaussiennes, effet des paramčtres (French)
Gauss Map
Gauss Map and Curvature
Gauss's Circle Problem
Gauss's Line
GCD Cube
Gear Shifting
Gematria (Letter Sums)
Genealogy Graphs from XML
General Additive Cellular Automata
General Assembly Resolution Viewer
General Divisor Methods
Generalized Arrhenius Function
Generalized Central Limit Theorem
Generalized Coons Surface
Generalized Digit Parity
Generalized Digit Parity for Integer Sequences
Generalized Hyperbolic Distribution
Generalized Kaprekar Routine
Generalized Logistic (Verhulst) Isothermal Microbial Growth
Generalized Mobile Automata
Generalized Unsöld Theorem for Hydrogenic Functions
Generalized Waterman Polyhedra
Generating a Cardioid III: Circles Passing through a Point
Generating a Cardioid II: Reflecting in Tangents
Generating a Cardioid I: One Circle Rolling around Another
Generating a Cardioid IV: Feet of Perpendiculars
Generating a Cardioid VII: Joining Points on a Circle
Generating a Cardioid VI: Reflected Rays
Generating a Cardioid V: Rolling a Hoop
Generating a Lemniscate I: Envelope of Circles
Generating a Lemniscate III: Using a Linkage
Generating a Lemniscate II: Rotating Line
Generating a Lemniscate IV: Rotating a Line
Generating Conics by Newton's Method
Generating Entangled Qubits
Generating Functions
Generating Functions and Rodrigues's Formulas for Special Functions Used in Quantum Mechanics
Generating Lissajous Figures
Generating Patterns Similar to Peruvian Textiles
Generating Realistic Baseball Line Scores
Generator of Pythagorean Triples
Generic Random Walk and Maximal Entropy Random Walk
Genetic Drift of Allele Frequency
Geneva Drive (Maltese Cross)
Geodesate
Geodesic Balls in the Nil-Space
Geodesic Cone in Nil-Geometry
Geometrical Doll
Geometric and Continued Fraction Expansion of the Golden Ratio
Geometric Difference between a Finite Difference and a Differential
Geometric Elements of a Circle
Geometric Illustration of Machin-Like Formulas
Geometric Mean Constructions
Geometric Mean Filtering for Blurring
Geometric Mean of Continued Simple Fraction Terms
Geometric Multiplication
Geometric Proof of the Tetrahedral Number Formula
Geometric Series
Geometric Square Root Construction
Geometry of Logarithms
Geometry of the Pyramid of Khufu
Geosynchronous Orbit
Gerling's 12-Piece Dissection of an Irregular Tetrahedron into Its Mirror Image
Gershgorin Circles
Getting e from Randomness
Gibbs Phenomena for 1D Fourier Series
Gilbreath's Conjecture
Gingerbreadman Trajectories
Girder Bridge Model
GL(2,p) and GL(3,3) Acting on Points
Glass Cut
Global and Local Errors in Runge-Kutta Methods
Global and Local Sequence Alignment Algorithms
Global Extrema on an Interval
Global Recurrence Plot of Cellular Automaton Dynamic
Gödelization
Golay Code
Goldbach Comet
Goldbach Conjecture
Golden Octet Truss
Golden Ratio by the Fibonacci Sequence
Golden Ratio Dividers
Golden Section
Golden Spiral
Goldilocks and the 3 Similar Triangles
Golf Player Handicap Calculation
Gomory's Theorem
Goodness of Fit for Random Subsets
Goodstein Function in Terms of Fast-Growing Function Hierarchies
GPS Simulator
Gram-Schmidt Process in Two Dimensions
Granger-Orr Running Variance Test
Graph and Contour Plots of Functions of Two Variables
Graph Automorphisms
Graph Browser
Graph Detangling
Graph Embedding Trajectories
Graphene Brillouin Zone and Electronic Energy Dispersion
Graphical Illustration of Bivariate Constrained Optimization
Graphical Linear Programming for Three Variables
Graphical Linear Programming for Two Variables
Graphical Method for Evaluating the Performance of N Tanks in Series
Graphical Modulo-4 Image Encryption
Graphical Representation of Geometric Series
Graphing Derivatives
Graphing Functions with Four Parameters
Graphing Systems of Inequalities
Graph of Inequalities
Graphs and Their Complements
Graph Searching: Breadth First and Depth First
Graphs from Zero-One Matrices
Graphs of Partitions
Graphs of Random Permutations
Graphs of Successive Digits of Pi, e, and Phi
Graphs of Taylor Polynomials
Graphs of the Beta Function
Graphs of the Successive Digits of Rational Numbers
Grating
Grating 3D
Gravitational Lensing by a Point Mass
Gravitational Lensing of a Point Source
Gravitational Potential of a Cuboid
Gravitational Redshift
Gravitational Wave Polarization and Test Particles
Gravitation versus Curved Spacetime
Gray Color Meditation
Gray Indexed Minimum Change Permutation
Great Circles on Mercator's Chart
Great Rhombic Triacontahedron Sculpture
Greedy Algorithms for a Minimum Spanning Tree
Greek Numerals
Greenberg-Hastings Model
Green's Functions for Diffusion
Gregory Series
Grids of Lines
Grouping Country Data
Grouping Images by Color
Group of Symmetries of the Square
Grover's Quantum Search Algorithm
Grow an Alexander Horned Sphere
Grow Dahlias
Growth Model for the Blue-Green Alga Anabaena catenula
Guess Who?
Guilloché Outline Rules
Guilloché Patterns
Guitar Chord Diagram
Guitar Chord Maker
Guitar Chords
Guitar Tuner
Gyroelongated Dipyramid
Gyroscope
Gyroscopic Joint
Haar Function Interval Points
Haar Functions
Haberdasher's Problem
Hackers Language
Hacker's Symbol: The Glider in the Game of Life
Hadwiger-Finsler Inequality
Half-Distance Rules with High Resolution
Half-Distance Rules with Low Resolution
Half-Life of a Radioactive Element
Half-Tone Screens
Halving a Triangle
Hamiltonian Tours on Polyhedra
Hammer Time
Handling Molecular Pathways in Systems Biology
Hand Model
Hand Moving on a Moebius Strip
Hanegraaf's Dissection of a Truncated Octahedron into a Hexagonal Prism
Hanegraaf's Rectangular Block of Size 2×1×1 to a Cube Dissection
Hanging a Hollow Sphere with a Leaf Spring
Hangman Word Game for a Computer Player
Hangman Word Game for a Human Player
Harmonic Oscillation
Harmonic Oscillator Eigenfunctions
Harmonic Oscillator in a Half-space with a Moving Wall
Harmonic Oscillator Wavefunctions
Hasse Diagram of Power Sets
Head Motion of the Wolfram 2,3 Turing Machine
Head-to-Toe Vector Addition
Health-Wealth Tradeoffs
Hearing Cellular Automata
Heart Disease Risk
Heat Balance in Freezing and Thawing Food
Heat Capacity of Solids in the Debye Approximation
Heat Conduction in a 1D Harmonic Chain with 3D Vibrations
Heat Conduction in Some Standard Solids
Heat Diffusion in a Semi-Infinite Region
Heating of Bodies in a Limited Volume of Well-Stirred Fluid
Heat Transfer in a Heat Exchanger
Hecks Boards
Hedging the Black-Scholes Call Option
Hedging the European Put Option
Hegemonic Zones of Influence
Heilbronn Triangles in the Unit Square
Heilbronn Volumes
Helical Spring
Helical Spring between Two Cylinders
Helical Staircase
Helices
Helices Made of Icosahedra and Rhombohedra
Helices of Tetrahedra
Helicopter Tilt Control
Helix Made of Golden Rhombohedra
Hello Block World
Helmholtz Coil
Helmholtz-Coil Fields
Herd Immunity for Smallpox
Hering Illusion
Hermite Lissajous Figures
Hertzian Contact Stress
Hexagonal Close Packing and Triangular Orthobicupola
Hexagonal Outer Totalistic Cellular Automata
Hexagonal Pattern Made of Tetrahedra
Hexagonal Prism Totalistic Cellular Automaton
Hexagonal Two-Color Four-Neighbor Totalistic Cellular Automata
Hexagons and the Golden Ratio
H-Fractal
Hidden Variables in Quantum Mechanics
Hierarchical Zonotopal Spaces
Highlighting Patterns in Cellular Automata
High-Pressure Chemical Equilibrium
High-Pressure Vapor-Liquid Equilibrium Data for a Binary Mixture: the Poynting Factor Correction
High Pressure Vapor-Liquid Equilibrium Data of a Binary Mixture of Chloroform and Acetone
Hilbert and Moore 3D Fractal Curves
Hilbert and Moore Fractal Curves
Hill Cipher Encryption and Decryption
Hill-Climbing Algorithm
Hindmarsh-Rose Neuron Model
Hinged and Translational Dissection of Two Squares to One
Hinged Dissection of an Equilateral Triangle into a Hexagram
Hinged Dissection of an Equilateral Triangle into Two Hexagons
Hinged Dissection of Equilateral Triangles
Hinged Dissection of Four Hexagons into One
Hinged Dissection of Squares
Hinged Dissections: From Equilateral Triangle to Square
Hinged Dissections: From Three Squares to One
Hinged Dissections of a Hexagram and a Hexagon
Hinged Lining for a Tunnel
Hinged Octahedron
Hinged Tripod for Holding a Vertical Pole
Hippias Quadratrix
Histograms of Heating and Cooling Degree Days
Histogram Threshold by Max Contour Contrast
Hjelmslev's Theorem
Hoberman Cube
Hoberman Rings Attached in a Plane
Hoberman Sphere (Cuboctahedron)
Hoberman Sphere (Octahedron)
Hoehn's Theorem
Hölder Approximations to Popular Means
Hofstadter's MU Riddle
Hole-Free Magic 45-Ominoes
Hole in a Cylinder
Holes in a Spikey
Holes in the Faces of a Dodecahedron
Holiday Greenery with Stars
Holistic Theorem
Hooke's Law
Hopf Bifurcation in the Brusselator
Horizontal and Vertical Line Tests
Horn Growth in 2D
Horosphere Packings of the (3, 3, 6) Coxeter Honeycomb in Three-Dimensional Hyperbolic Space
House Profits in Negative Edge Games
How Area Changes with Diameter
How Does the Vertex Location of a Parabola Change?
How Far Can One See from a Height?
How Fast Can You Run?
How Loess Works
How Long Does It Take a Society to Learn a New Term?
How Many Gallons in My Pool?
How Many Rectangles Are There?
How Many Triangles Are There?
How Old Would You Be on Another Planet (or Pluto)?
How Rare Is that U.S. Coin?
How Receiver Operating Characteristic Curves Work
How Tall Is That Building?
How Text Rotation Works in Mathematica
How the Altitude Divides the Hypotenuse
How the Proton and Neutron Got Their Masses
How the Superposition of the Periodic Pulsations of +1 and -1 Generates Values of the Mertens Function
How the Zeros of the Zeta Function Predict the Distribution of Primes
How to Avoid a Collision
How to Torpedo a Ship
How Zippers Work
HSV Colors
HSV Loci in the RGB Color Space
Hue, Saturation, and Brightness Image Controls
Hue-Saturation-Brightness (HSB) Explorer
Hunger's Law of Cosines Dissection
Hunveyor-Surveyor Field Work Simulations
Hurricane Risk by State
Huygens's Principle
Hydrogen Atom Radial Functions
Hydrogenic Radial Functions via Supersymmetry
Hydrogen Orbitals
Hyperbolic Distribution
Hyperbolic Paraboloid Embracing an Elliptic Paraboloid
Hyperbolic Quadrilateral
Hyperbolic Triangle
Hyperbolization of a Dodecahedron
Hyperboloid as a Ruled Surface
Hyperlabyrinth Chaos
Hypothesis Tests about a Population Mean
Hysteresis Modeled with Cellular Automata
I-Beam Support
I Ching 3D
Icosahedra along the Edges of Icosahedra
Icosahedral Group Polyhedra
Icosahedron Ball
Icosahedron Fractal
Icosahedron to Small Rhombicosidodecahedron
Icosahedron Towers
Icosidodecahedron and the Compound of Five Octahedra
Ideal Gas Law
Ideal Gas Law Solver
Idealized Belousov-Zhabotinsky Reaction
Idealized Cam-Follower Mechanism
Idealized Closed Loop for Processes in Three Continuous Stirred-Tank Reactors
Idealized Controller
Idealized Galton Board
Idealized Gear Mechanism
Idealized Hoppering
Identifying Trinomials That Are Perfect Squares
IFS: Rotation, Translation, and Scaling
Ikeda Attractor
Ikeda Delay Differential Equation
Illustrating Cosecant with the Unit Circle
Illustrating Cosine with the Unit Circle
Illustrating Cosine with the Unit Circle II
Illustrating Cotangent with the Unit Circle
Illustrating Secant with the Unit Circle
Illustrating Sine with the Unit Circle
Illustrating Tangent with the Unit Circle
Illustrating the Law of Large Numbers
Illustrating the Locker Problem
Illustrating Trigonometric Curves with the Unit Circle
ImageAdjust Parameters
Image Blurring
Image Compression via the Fourier Transform
Image Compression via the Singular Value Decomposition
Image Jitter Filter
Image Padding
Image Processing of Calligraphy
Image Quality Adjustment
Image Restoration for Degraded Images
Image Sharpening
Image Warping
Implicit Function Game
Implied Volatility in Merton's Jump Diffusion Model
Implied Volatility in the Variance Gamma Model
Implosion Mechanism for Plutonium Bomb
Impossible Frames
Improper Integrals
Impulse Acting on a Pendulum
Incipient Growth Processes with Competing Mechanisms
Incircles and a Cevian
Incircles in Right Triangles
Income and Substitution Effects
Incomplete Gamma Function with Continued Fractions
Indestructible Perpendicular Lines
Individual Insurance Decisions under HR 3560 and HR 3962
Induced EMF through a Wire
Inelastic Collisions of Two Spheres
Inertia
Inertial Coordinate Systems
Inertia versus Gravity in 3D
Infinite Limit at a Finite Point
Infinite Limit at Infinity
Infinite Number of Cubes inside a Cube
Infinite Number of Squares inside a Square
Infinite Series Explorer
Inflating a Koch Snowflake
Inflation-Adjusted Yield
Influence of the Relative Phase in the de Broglie-Bohm Theory
Injective, Surjective, Bijective
Inscribed and Central Angles in a Circle
Inscribed Angles in a Circle Subtended by the Same Chord
Inscribed Angles That Intercept the Same Arc
Inscribing Four Circles in a Triangle
Inset Plot Magnifier
Inside the Rhombic Hexecontahedron
Inside World of the Torus
Insolvency Setoff
Inspecting the Rosetta Stone
Instability of Laplace's Equation
Instantaneous Rate of Change
Instantaneous Rate of Change: Exploring More Functions
Instantaneous Rate of Change: Exploring More Functions with the First and Second Derivatives
Instantly Periodic General Continued Fraction Representations of Square Roots
Insulin Molecule
Insurance Disclosures
Insurer Assessments with Tax Credits
Insurer Ruin
Integer Exponentiation Mappings
Integer Polynomial
Integer Rotation Patterns
Integers
Integer-Sided Triangles
Integers Relatively Prime to the First n Primes
Integer Torus Maps
Integer Torus Maps Applied to Letterforms
Integer Trig Hunt
Integer Trigonometric Patterns
Integer Value Spatial Distance
Integral Grid Triangles with Consecutive Side Lengths
Integral Mean Value Theorem
Integrals of Nested Elementary Functions
Integrals over Dirac Delta Function Representations
Integrating a Vector Field along a Curve
Integrating Odd Powers of Sine and Cosine by Substitution
Integration by Riemann Sums
Integration Is a Sum
Integration using Hermite Reduction
Intensity Distribution for Multiple Slit Diffraction
Intensity of Segregation versus Time in a Semi-Continuous Reactor
Interacting Vortices in a Disk
Interaction at a Distance: The Radiation Model
Interactive Curve Fitting
Interactive Maze
Interactive Rings Tessellation
Interactive Spline
Interest Rate Swap
Interference of Waves from Double Slit (Young's Experiment)
Interference Patterns
Interior and Exterior Angles of Equilateral Figures
Interior Angles of a Triangle
Interleaving Theorems for the Rayleigh-Ritz Method in Quantum Mechanics
Interlocking Disks
Interlocking Rings
Interlocking Tori
Intermediate Value Theorem
Interpolating a Set of Data
Interpolating B-Spline Curves with Boundary Conditions
Interpolating Polynomial
Interpolating the Hilbert Curve with a B-Spline
Interpolating the Hilbert Curve with a B-Spline to Create a Surface
Interpolating Vertical Segments
Interpreting a Square Grid as a Hexagonal Grid
Intersecting 3D Triangles
Intersecting a Rotating Cone with a Plane
Intersecting Chords Theorem
Intersecting Circles
Intersecting Circles and a Right Angle
Intersecting Cylinders
Intersecting Line Segments
Intersecting Lines in All Possible Ways
Intersecting Lines Using Slope-Intercept Form
Intersecting Secants Theorem
Intersection of a Line and a Circle
Intersection of an Altitude and a Line through the Incenter
Intersection of Two Circles
Intersection of Two Cylinders
Interval Interpolating Polynomial
Interval Notation
Intrinsically Defined Curves
Intrinsically Knotted Graphs
Introduction to Taylor-Maclaurin Series
Introduction to the Doppler Effect
Invaders from the Computational Universe
Inverse Composition Rule
Inverse Fourier Sound
Inverse Hilbert Matrix Coefficients Mod m
Inverse Kinematics
Inverse Square Laws
Inverse Transformation of s-Reflection Coefficient between Oblique and Normal Incidence
Inversion of an Array of Cubes
Inversive Geometry I: Basic Properties of Inversion
Inversive Geometry III: Inverting an Ellipse
Inversive Geometry II: The Peaucellier Inversor Mechanism
Inversive Geometry IV: Inverting a Point with a Compass in Three Steps
Inversive Geometry V: Circle through Three Points
Inversive Geometry VI: Inverting a Polar Grid
Inversive Geometry VII: Spider Eyes
Inverted Pendulum Controls
Investigating Limits Numerically
Investment Leverage Effect
Investment Returns
Inviscid Burgers's Equation
Irradiation Illusion
Irrational Digits Walk
Irreducible Gaussian Fractions
Irregular Tilings
Irreptiles
Is a Point Inside or Outside a Regular Polygon?
Ising Model
Isobaric Vapor Liquid Equilibrium Computations
Isobaric Vapor-Liquid Equilibrium Data of an Immiscible Mixture
Isobaric Vapor-Liquid Equilibrium Data of a Partially Miscible Mixture
Isolated RGB Channels of Images
Iso-Optic Curve of a Regular Polygon
Iso-Optic Curve of the Ellipse
Iso-Optic Plane of the Regular Tetrahedron
Isoperimetric Inequality for Polygons
Isosceles Trapezoid Area
Isothermal Reactor with Complex Reactions
Isothermal Yield of Allyl Chloride versus Conversion of Propylene
Isotope Browser
Isotope Decay
Is the Given Function Injective?
Is the Golden Ratio Really the Most Beautiful Proportion?
Item Characteristic Curves
Iterated Cyclic Averaging of Points
Iterated Games
Iterated Maps Using Complex, Dual, and Split-Complex Numbers
Iterated Matrix Operations in 3D
Iterated Nonlinear Map
Iterated Random Rotation 3D
Iterates, Cycles, and Bifurcations of the Logistic Map
Iterates of the Logistic Map
Iterates of the Logistic Map in 3D
Iterating Linear Functions
Iterating Rational Functions
Iteratively Reflected n-gons
Iterative Polygon Simplification
Jacobi Polynomials in an Orthogonal Collocation Method
Jamie's Function
Jazz Voicings
Jefferson National Expansion Memorial
Jensen's Disks
Jingle Bell Surface
Johnson's Theorem
John von Neumann's First Pseudorandom Number Generator
Joint Density of Bivariate Gaussian Random Variables
Joint Density of Trivariate Gaussian Random Variables
Josephus Problem
Joule's Experiment
Jug Problem
Julia Sets and the Mandelbrot Set
Junk Food Molecules
Kaiser Window Transform
Kakutani's Solution of the Dirichlet Problem
Kaleidoscope
Kaleidoscopic Polygons
KAM Tori Reforming
Kanizsa Polygon Illusion
Kaprekar Routine
Karhunen-Loeve Directions
Karhunen-Loeve Directions and Regression
k-Cayley Trees
Kelland's Gnomon-to-Square Dissection
Kelly Portfolio Analysis
Kempe's Angle Adder
Kempe's Angle Trisector
Kempe's Multiplicator
Kempe's Translator
Kempe's Universality Theorem: An Example
Kenmotu Circle
Keplerian Orbital Elements
Kepler's Mysterium Cosmographicum
Kepler's Second Law
Kermit's What-Happens-Next-Machine
Kernel Density Estimation
Keyboard and Composer
Keynesian Cross Diagram
Keypad Security Timer
Kinematics of a Moving Point
Kinematics of Traffic Congestion
Kinetics of Chemical Reaction with an Intermediate Product
Kite Tiling and a Dodecahedron
Kleene's Three-Valued Logic
Klein-Nishina Formula for Photon-Electron Scattering
Klein Paradox
k-Nearest Neighbor (kNN) Classifier
Kneser Arrays
Kneser Graphs
Knights and Knaves Puzzle Generator
Knights, Knaves, and Normals Puzzle Generator
Knitting Hyperbolic Pants
Knopp's Osgood Curve Construction
Knots
Kolkata Paise Restaurant (KPR) Problem
Kosnita's Theorem
Kremser Shortcut Method for Absorption and Stripping
Krypto, or 24-Game
Kürschák's Dodecagon
Labeled Polygon Triangulations
Labeled Trees of 10 Vertices Using Prüfer Sequences
Labeling the Integer Heptagon
Ladder Operators for the Harmonic Oscillator
Ladybug Counting Game
Lagrange Points
Lambdoma Matrix
Lamé's Ellipsoid and Mohr's Circles (Part 1)
Lamé's Ellipsoid and Mohr's Circles (Part 2: Parallels)
Lamé's Ellipsoid and Mohr's Circles (Part 3: Meridians)
Lamé's Ellipsoid and Mohr's Circles (Part 4: Spheres)
Laminar Flow between Two Eccentric Tubes
Laminar Flow of a Power-Law Fluid in a Horizontal Pipe
Land Use Regulation and Municipal Utility
Land Use with Contract
Laplace's Equation on a Circle
Laplace's Equation on a Square
Laplacian of Gaussian Filtering
Large Number Names
Largest Isosceles Triangle Inscribed in a Circle
Largest Triangle Inscribed in a Circle
Laser Diffraction Pattern
Laser Lineshape and Frequency Fluctuations
Laser Sword
Last Minute Valentine's Graphic
Latent Heats of Fusion and Vaporization
Latitude and Longitude of a Point on a Sphere
Lattice Circles
Lattice Effects in Cellular Automata
Lattice Effects in Cellular Automata-Comparisons
Lattice Method Multiplication Grids
Lattice Multiplication
Lattice of Factors
Lattice Path Interpretations
Lattice Points on a Circle
Lattice Points under a Hyperbola
Lattice Random Walk in 2D
Lattice Random Walk in 3D
Lattice Rule
Lattices in Perspective
Latticework Design by Runs in Automaton States
Launching a Rocket
Lava Lamp
Laws of Exponents
Laws of Reflection
Lawsuit Settlement Calculator
LCM, GCD, and MOD
Leads in Coin Tossing
Learning Newton's Method
Learning Place Value by Counting Squares
Learning the Alphabet
Learning the Arabic Alphabet
Learn Musical Notes
Learn the Game of Go
Least Common Multiple and Greatest Common Divisor
Least Squares
Least Squares Criteria for the Least Squares Regression Line
Least Squares Estimate Using a Monte Carlo Method
Least-Squares Estimation of an Ellipse
Lebesgue 3D Curves
Lebesgue Integration
Left-Right-Rotating Spiral Array
LEG CA
Legendre Lissajous Figures
Leibniz Criterion for Alternating Series
Lemniscate Linkage
Lemniscate Plotting Device
Lemon Batteries
Length of a Coiled Rope
Length of Line Illusion
Lengths of Permutation Cycles
Length Units of Measurement
Length, Width, and Height of an Impossible Object
Lens Doublets
Lens Flares
Lensmaker's Equation
Lester's Theorem
Letterform Comparison
Letter Highlighting in Text
Letter Products
Level Curves
Levels of a Mathematica Expression
Level Surfaces for a Polynomial
Leverage Ratios
Lévy Measures
Levy's Conjecture
Lewis Carroll's Bilateral Diagram
Lewis Carroll's Diagram and Categorical Syllogisms
L'Hospital's Rule for 0/0 Forms
Life Expectancy in the U.S. Population
Life Insurance Pricing
Life Transitions
Lift Table
Light Beams through Multiple Polarizers
Light Clocks and Time Dilation
Light Cone Causality
Lighthouses
Light on an Impossible Object
Light Polarization and Stokes Parameters
Light Ray in a Prism
Light Ray Passing through a Transparent Plate
Light Rays in a Lens
Light Refraction with a Waterdrop
Lights On/Lights Out Game and Solver
Lights Out Puzzle
Likelihood-Based Goodness of Fit in Two-Way Contingency Tables
Lily Pond
Limited Diagonalization
Limited Mandelbar Sets
Limit Laws
Limit of a Family of Curves
Limit of a Sequence
Limit of the Sequence a^(1/n)
Limit of the Sum of Two Sequences
Limits of a Rational Function of Two Variables
Limits of Sequences
Limits of Tree Branching
Lindgren's Dissection of an Octagram into an Octagon
Lindgren's Dissection of a Pentagram into a Decagon
Lindgren's Dissection of One Decagram {10/3} into Two Pentagrams {5/2}
Lindgren's Dissection of One Octagram {8/3} into Two
Lindgren's Dissection of One Octagram into Two
Lindgren's Hinged Dissection of Quadrilaterals
Lindgren's Square-to-Dodecagon Dissection
Lindgren's Two-Triangles-to-One Dissection
Lindgren's Two-Triangles-to-One Hinged Dissection
Line Angle Histogram
Linear and Quadratic Curve Fitting Practice
Linear Congruential Generators
Linear Congruential Sequences
Linear Dependence between Two Bernoulli Random Variables
Linear Equations: Row and Column View
Linear First-Order Differential Equation
Linear Function Game
Linear Interpolation between a Square and a Circle
Linear Interpolation between Two Images
Linear Sweep Voltammetry: Infinite Series Approximation
Line Art
Linear Transformation Given by Images of Basis Vectors
Linear Transformations of a Polygon
Linear Transformation with Given Eigenvectors
Lined Tunnels
Line Jaggedness Visualization with the Mandelbrot-Weierstrass Function
Line Plots, Histograms, and Stem-and-Leaf Plots
Line Segments through the Vertices and the Circumcenter of an Acute Triangle
Lines of Force for Two Point Charges
Lines on a Cubic Surface
Lines Parallel to the Sides of a Triangle
Lines: Point-Slope
Lines: Slope-Intercept
Lines through Points in the Poincaré Disk
Lines: Two Points
Line through Two Points
Linkage for the Cissoid
Linkage Inverting Figure-Eight Curves
Linked Electric and Magnetic Fields for Source-Free Maxwell Equations
Linked Sliders
Linked Torus Knots
Linking Two Rings
Lipschitz Continuity
Liquid Chromatography Simulation
Liquid Chromatography with Linear Adsorption Equilibrium and Plate Model
Liquid in a Rotating Cylinder
Liquid-Liquid Equilibrium Diagram for a Chloroform-Water-Acetic-Acid Mixture at 25°C and 1 atm
Liquids in Containers of Different Shape
Lissajous Array
Lissajous Figures
Listing's Law
Local Behavior of a Polynomial near a Root
Local Digit Values
Local Growth in an Array of Disks
Local Markov Chains for Elementary Cellular Automata
Locate a Point on a Map of the World
Locate Important Real Numbers
Location of Complex Roots of a Real Quadratic
Location Theory - 3D Bid Rent Curves Become Urban Form
Location Theory - Is the Bid Rent Curve Linear?
Location Theory - Land Use Determination
Loculus of Archimedes
Logarithmic Integral on the Critical Line
Logarithmic Mean Temperature of a Heat Exchanger
Logarithmic Spiral
Logic on the Number Line
Logic with Complex Numbers
Logic with Letters
Logic with Logicians
Logistic Equation
Logistic Map: Onset of Chaos
Logistic Sigmoid Market Model
Logit and Probit Models for O-Ring Failure Data
Logo Text Rotation
Log Survival Ratio as a Function of Time
London Eye
Long Addition
Long Division
Longitudinal and Transverse Waves
Longitudinal Oscillations and Resonance of a Four-Spring, Three-Mass System
Long Multiplication
Long-Run Average Total Cost
Look and Say Sequence Steps
Look and Say Sequence Substrings
Looking at a Cube
Looped Curves
Loops around a Cuboctahedron
Lorentz Transformation for a Rotating Square
Lorenz Attractor
Lorenz Curves and the Gini Coefficient
Lotus Effect
Lower Excited States of Helium Atom
Low-Temperature Heat Capacity of Hydrogen Molecules
Loxodrome
Loyd's Two-Greek-Crosses-to-a-Square Dissection
Lubricated and Frictional Squeezing Flow
Lucas Circles
Lucas-Gauss Theorem
Lucky Numbers
LU Decomposition
Lukasiewicz's Three-Valued Logic
Luna Landers on the Moon
Lunar Calendar Maker
Lunar Libration
Lunar Libration of Longitude
Macaulay Duration
Macaulay's Triangle and Trapezoid Hinged Dissection
Machine Language Pattern for lsmod Program
Mackey-Glass Equation
Macroeconomic Effects of Interest Rate Cuts
Magic Cube
Magic Cube of Order 5
Magic Multiplications with Digits
Magic Number Bidding
Magic Numbers in the Nuclear Shell Model
Magic Square
Magic Squares and Designs
Magnetic Braking
Magnetic Dipole in a Uniform Magnetic Field
Magnetic Flux through a Loop of Wire
Magnetic Hysteresis
Magnetic n-gon Billiards
Magnetic Resonance and Bloch Equations
Magnified Views of the Mandelbrot Set
Magnifying Glass
Major Multinational Languages
Make Your Own Solar System
Making Change with U.S. Currency
Making Necklaces
Making Patterns
Mamikon's Proof of the Pythagorean Theorem
Mandelbrot's Binomial Measure Multifractal
Mandelbrot Set Doodle
Mandelbrot Set Print
Manipulable Complete Graph
Many Sine Curves
Mapping a Convergent Sequence by a Continuous Function
Mapping an Interval by a Cubic Polynomial
Mapping an Interval to Another
Mapping a Square by Complex Functions
Mapping Lines and Circles onto the Riemann Sphere
Mapping Rectangles by the Elementary Transcendental Functions
Maps of a Complex Variable
Marden's Theorem
Marginal Utility Budget Line
Markov Chain Monte Carlo Simulation Using the Metropolis Algorithm
Markov Volatility Random Walks
Mark-Recapture Investigation
Mascheroni's Construction of the Center of a Circle
Mass Balance in a Single-Stage Evaporator
Mass Balance of Binary Mixtures
Mass Transfer in Binary Star Systems
Mass Units of Measurement
Mathematica 7's Discrete Distributions
Mathematica Characters
Mathematics of Tsunamis
Mathematics Subject Classification 2000
Mathematics Subject Classification Graphs
Math Songs
Matrix Multiplication
Matrix Norm and Spectral Norm
Matrix Rank of Arrays in Coprimality Tensor
Matrix Transformation
Matrix Transformations: "F"
Maurer Rose Chords
Maurer Roses
Maximizing the Area of a Rectangle with Fixed Perimeter
Maximizing the Area of Some Geometric Figures of Fixed Perimeter
Maximizing the Viewing Angle of a Painting
Maximizing the Volume and Surface Area of Geometric Solids Inscribed in a Sphere
Maximum Absolute Column Sum Norm
Maximum Absolute Row Sum Norm
Maximum Area Field with a Corner Wall
Maximum Fruit Salad
Maximum Likelihood Estimation
Maximum Likelihood Estimation for Coin Tosses
Maximum Likelihood Estimators for Binary Outcomes
Maximum Likelihood Estimators with Normally Distributed Error
Maximum Rectangle Inscribed in a Circle
Maximum Size of Involute Gear Teeth
MAX Puzzle
Maxwell's Demon
Maxwell's Displacement Current
Mayan Pyramid
Mazes on Polyhedra Nets
McCabe & Thiele Graphical Method
McCabe-Thiele Graphical Method for a Non-Ideal Binary Mixture
McNugget Problem and Frobenius Numbers
Mean and Single Prediction Bands for a Nonlinear Model
Mean and Standard Deviation of a Distribution
Mean Distance of Agents in Diffusion-Limited Aggregation
Mean, Median, and Standard Deviation for Random Values
Mean, Median, Mode
Mean-Reverting Random Walks
Mean Value Theorem
Measures of Network Centrality
Measures of Node Prominence on a Network
Measuring Angles with a Protractor
Measuring Entangled Qubits
Measuring Gravity at the Surface of the Earth
Measuring the Speed of Light with Marshmallows
Mechanical Involute Gears
Mechanical Sensitivity of Soft Testing Machines
Mechanical Work
Medial Division of Triangles
Median Split
Median Theorem for Polygons
Members of International Organizations
Memristor Based Chaotic System
Menelaus' Theorem
Mercury's Perihelion Precession
Mercury Transits
Merger Guidelines
Merging Schools of Fish
Mersenne Twister and Friends
Mertens Accumulation of Möbius Values
Mertens Conjecture
Merton's Jump Diffusion Model
Meshing Involute Gear Teeth
Meshless Approximation
Metamorphic Soccer Ball
Metamorphosis of a Cube
Method of Image Charges: Point Charge inside a Planar Capacitor
Method of Images in Electrostatics
Methods of Social Choice
Método de McCabe-Thiele (Spanish)
Michaelis-Menten Enzyme Kinetics and the Steady-State Approximation
Michael Jackson's Anti-Gravity Illusion
MIDI Frequency Calculator
MIDI Keyboard Limits
MIDI PPQN Duration Calculator
Milk Centrifugation to Cream and Skim
Miller Indices for a Simple Cubic Lattice
Minimal and Maximal Surfaces Generated by the Holomorphic Function log(z)
Minimal Crossings for Complete Graphs
Minimal Disjunctive Normal Form
Minimal Model of Simulating Prices of Financial Securities Using an Iterated Finite Automaton
Minimizing the Rosenbrock Function
Minimizing the Surface Area of a Cylinder with a Fixed Volume
Minimizing the Time for a Dog to Fetch a Ball in Water
Minkowski Spacetime
Minkowski's Question Mark Function
Miquel's Pentagram Theorem
Miquel's Theorem
Mirror Flip Transpose
Mitering a Closed 3D Path
Miter Joint and Fold Joint
MIU Explorer
Miura Map Folding and Unfolding
Mixed and Improper Fractions
Mixed Base Gray Codes
Mixed Boundary-Value Problem for the One-Dimensional Heat Equation
Mixed Radix Number Representations
Mixing and Segregation in Chemical Reactors (CSTR versus PFR)
Mixing Colors with Blend
Mixing Entropy for Three Letters
Mixtilinear Incircles
Modeling a Staircase
Modeling Light Curves
Modeling Return Distributions
Model Modal Modulo 18
Model of a Firefly Swarm
Model of Shell Growth
Modificaciones de la función cuadrátrica (Spanish)
Modificaciones de la función cúbica (Spanish)
Modified Logistic Isothermal Microbial Growth Ratio
Modifying Overall Image Brightness
Modify Simple Life
Mod Mod Zoom
Modulaciones AM (Spanish)
Modular Addition, Multiplication, and Exponentiation
Modular Arithmetic
Modular Arithmetic Tables
Modular Interference Diagrams
Modular Inverses
Modular Music
Moduli Space of Shapes of a Tetrahedron with Faces of Equal Area
Modulus Counting
Möbius Collar
Möbius Monorail
Möbius Mu Function Walk
Möbius Net Corner
Möbius Star
Möbius Strip as a Half-Twisted Square Torus
Möbius Strip Made from a Rotating Bar
Möbius Transforms
Mohr's Circle
Moiré Pattern of Two Equally Spaced Circular Ring Patterns
Moiré Pattern of Two Fresnel Zone Plates
Moiré Pattern of Two Straight Line Patterns
Moiré Patterns
Molarity of Aqueous Salt Solutions
Molecular Dynamics of Lennard-Jones Particles Using the Velocity Verlet Algorithm
Molecular Motion in Solids, Liquids, and Gases
Molecular Orbitals for First- and Second-Row Diatomic Molecules
Molecular Pie Charts
Molecules Distributed between Two Compartments
Moment of Inertia
Mona Lisa and the Golden Rectangle
Money Supply Process
Mong
Monodromy of z^n + b z + 1 = 0
Monoliths in Cellular Automaton 146
Monopolist's Profit Maximization
Monopoly and Natural Monopoly
Monopoly Profit and Loss
Monte Carlo Estimate for Pi
Monte Carlo Methods to Estimate Area
Monte Carlo Simulation of Line-of-Sight Distance Measurements
Monte Carlo Valuation of an Option
Monty Hall Paradox
Monty Hall Problem
Moon Lander
Moon Landing Simulation
Moral Hazard
Moral Hazard and Least-Cost Contracts: Impact of Changes in Agent Preferences
Moral Hazard and Least-Cost Contracts: Impact of Changes in Conditional Probabilities
Morley's Theorem
Morphological Processing
Mortal Fibonacci Rabbits
Mortgage Calculator
Most Common Angles
Motion Blur
Motion Grouping
Motion in a Central Field
Motion in a Gaussian Potential
Motion in Two Dimensions with Constant Acceleration
Motion of a Disk Hanging from a Spring on an Inclined Plane
Motion of a Dumbbell Acted on by Constant Forces
Motion of an n-Point Dynamical System
Motion of a Planet around a Star
Motion of a Rolling Disk Driven by a Rod and Spring
Motion of a Simple Pendulum with Damping
Motion of a System Consisting of a Rolling Cart, Pendulum, and Spring
Motion of the Moon: Eclipses
Motion of the Moon: Phases
Motion of Two Masses Connected by Springs
Motion Parallax versus Depth, 2D (Visual Depth Perception 12)
Motion Parallax versus Depth, 3D (Visual Depth Perception 13)
Motion, Pursuit, Fixate, and Distractor (Visual Depth Perception 2)
Motion/Pursuit Law in 1D (Visual Depth Perception 1)
Motion/Pursuit Law in 2D (Visual Depth Perception 3)
Motion/Pursuit Law on Invariant Circles (Visual Depth Perception 4)
Motion/Pursuit Ratio and Depth in 1D (Visual Depth Perception 17)
Motion/Pursuit Ratio and Depth in 2D (Visual Depth Perception 18)
Motions of a Hexapod Robot
Motion Traced on the Torus and Klein Bottle
Motzkin Number Diagrams
Movable Concentric Color Rings
Movement between Three Populations in a Closed System
Moving a Couch around a Corner
Moving Eyes
Moving Pearl Chains in 3D
Moving Stick Figures
Moving the Vertices of the Great Rhombicuboctahedron
Müller-Lyer Illusion
Multicolor Neighbor-Dependent Substitution Systems
Multicolor Random Array of Squares
Multicomponent Distillation: The Case of Pentane/Hexane/Heptane System
Multicomponent Simple Batch Distillation
Multidimensional Scaling
Multilanguage Word Lengths
Multilingual Palindromes
Multi-Particle Evolution Map
Multiple Derivatives
Multiple Ferris Wheels
Multiple Reflections of a Rotating Triangle
Multiple Reflections of a Superball
Multiple Slit Diffraction Pattern
Multiple Steady States in Continuous Culture with Substrate Inhibition
Multiples with Only the Digits 0 and 1
Multiple Tubular Reactors with Interstage Cooling
Multiplication as Jumping
Multiplication Bingo
Multiplication by Adding Logs
Multiplication Flash Cards
Multiplication of Fractions
Multiplication of Integers
Multiplication Table
Multiplication Table Mod m
Multiplying a Double-Digit Number by a Single-Digit Number
Multiplying a Monomial and a Linear Polynomial
Multiplying Binomials
Multiplying Complex Numbers
Multiplying Fractions
Multiplying Mixed Numbers
Multiplying Probabilities
Multiplying with Lines
Multipoint Padé Approximants
Multipole Fields
Multipole Shapes
Multiseed 2D Cellular Automata
Multiseed Averaged 2D Cellular Automata
Multiset Partitions
Multistage Operation with Cold-Shot Cooling
Multiterm Nestedly Recursive Functions
Multivariable Epsilon-Delta Limit Definitions
Munching Square Thresholds
Munker-White Illusion
Musical Instruments and Colored Surfaces
Musical Interval Recognition
Musical Lines and Curves
Musical Logo Widget
Music from the Game of Life
Music from the Rationals
Nagel's Theorem
Named Binary Cartesian 2D String Rewriting Systems
Named Colors
Nanotube Builder
Napier's Bones
Napier's Inequality (I)
Napier's Inequality (II)
Napoleon's and van Aubel's Theorems
Napoleon's Theorem
Narayana Numbers
Nash Equilibria in 3×3 Games
Nash Equilibria with Continuous Strategies
Natural Convection between Two Vertical Plates
Natural Logarithm Approximated by Continued Fractions
Natural Modes of Vibrations for Some Simple Systems
n-Digitaddition
Nearest Center Orbit Twins
Nearest Hexagonal Positions
Nearest Named RGBColor
Nearest Neighbor Graph Connections
Nearest Neighbor Networks
Necker Cube: A Variation
Negative Pedal Curves of an Ellipse
Negligible Senescence Scenario
Neighbor-Dependent Substitution System
Neighbor-Dependent Substitution Systems of Two Kinds
Neighborhood Ranges
Nested Gears
Nested Image Rotations
Nested Polygon Disks
Nested Square Root Representation of the Golden Ratio
Nested Structure of the Continued Fraction of Liouville's Constant
Nested Trigonometric Function Plots
Net Lease Economics
Net Present Value
Nets of Polyhedra
Network Centrality Using Eigenvectors
Networks for Basic Number Theoretic Functions
Neural Impulses: The Action Potential in Action
Neuronal Bursting
Neutrino Oscillations
Newton-Cotes Quadrature
Newton-Cotes Quadrature Formulas
Newton Flows of Polynomials
Newtonian Reflecting Telescope
Newton's Color Wheel
Newton's Cradle
Newton's Ellipse
Newton-Simpson's Formula for the Volume of a Prismatoid
Newton's Integrability Proof
Newton's Law of Cooling
Newton's Method
Newton's Method on a Mesh of Initial Guesses
Newton's Rotating Bucket Experiment
Newton's Second Law
n-Flakes
NFL Quarterback Passer Rating
n-gon Polynomials
Nilpotent Matrices in Jordan Decompositions
Nine-Point Circle
Nine-Point Cubic
n+l Rule for Atomic Electron Configurations
Nodal Domains of Spherical Harmonics
Nodal Surfaces of Degenerate States
No-Four-In-Plane Problem
Noise Retrieval from Averaged Sequences
Nomogram of p and s Reflectances for an Ambient-Film-Substrate System
Non-Crossing Rule for Energy Curves in Diatomic Molecules
Nonlinear Coupled Oscillator
Nonlinearized Motion of a Simple Pendulum
Nonlinear Warping
Nonlinear Wave Equation Explorer
Nonlinear Wave Equations
Nonlinear Wave Resonances
Nonlocal Cellular Automata
Non-Renewable Resource Economics
Nonsense Sentence Generator
Non-Sinusoidal Oscillation: A Balloon on a String
Nonsteady-State Heat Conduction in a Cylinder
Non-Terminating Reversal-Addition
Nordstrand's Weird Surface
Normal Approximation to a Binomial Random Variable
Normal Approximation to a Poisson Random Variable
Normal Distribution with Continued Fractions
Normal Forces Can Do Work
Normalizing Vectors
Normal Lines to a Parabola
Normal Modes of a Beaded String
Normal Modes of a Circular Drum Head
Normal Modes of a Double Pendulum
Normal Mode Vibrations of Buckminsterfullerene (C60)
Normal Oscillatory Modes of Rotating Orthotropic Disks
Normal Sections and Gauss Map for the Hyperbolic Paraboloid
Normed Line
Nosquare Puzzle
No Supply Curve in a Monopolistic Market
Nowhere-Neat Squaring the Square
Nowhere-Neat Tilings
Nowhere-Neat Tilings of the Plane
Nowhere-Neat Tilings of the Plane, Part 2
n-Player Chessboards
NPV and its Contributions
nth Roots of a Complex Number
Nuclear Liquid-Drop Model Applied to Radioactive Decay Modes
Nuclear Magnetic Resonance Spectrum of Ethanol
Null Distribution of the Correlation Coefficient
Numbered Cube
Number Guessing with Binary Numbers
Number Guessing with Ternary Numbers
Numbering the Vertices of Polyhedra
Number Lines
Number Line Solutions to Absolute Value Equations and Inequalities
Number of Digits in Base k Expansion of Fractions
Number of Finite Groups
Number of Normals to a Curve
Number of Repeating Digits in Base b Expansion of Fractions
Number of Squares in a Square
Number of Ways to Complete a True/False Test
Numbers in Various Bases
Number Systems in 3D
Number Systems Using a Complex Base
Number-Theoretic Construction of Digital Circles
Number Theory Tables
Numerical Approximation of the Fourier Transform by the Fast Fourier Transform (FFT) Algorithm
Numerical Evaluation of Some Definite Integrals
Numerical Flowers
Numerical Instability in the Gram-Schmidt Algorithm
Numerical Integration: Romberg's Method
Numerical Integration using Rectangles, the Trapezoidal Rule, or Simpson's Rule
Numerical Inversion of the Laplace Transform: The Fourier Series Approximation
Numerical Inversion of the Laplace Transform: The Zakian Method
Numerical Iso-Optic Curve of the Ellipse
Numerical Methods for Differential Equations
Numerical Solution of the Advection Partial Differential Equation: Finite Differences, Fixed Step Methods
Numerical Solution of the Dispersion Equation for a First-Order Reaction
Numerical Solution of the Falkner-Skan Equation for Various Wedge Angles
Numerov Solutions for Single- and Double-Well Potentials
Nuremberg Scissors
Nutation of a Symmetric Top
Nutts Puzzle
Observing Magnetic Fields with Iron Filings
Obtuse-Angle Shadowing Networks and Distance-Based Interpolation
Occurrence versus Claims-Made Insurance Policies
Octahedral Ring
Octahedral Vase
Octahedron Fractal
Octahedron Frame Made of Rhombic Dodecahedra
Octahedron in a Sphere
Octahedron Stars
Octahedron to Small Rhombicuboctahedron
Octonion Algebra Multiplication Tables
Octonions and the Fano Plane Mnemonic
Odd-4 Graph, Fano Planes, and the Coxeter Graph
Odd Disk Overlap
Odd Number Theorem
Odd or Even
Offspring of Adam and Eve
Ohm's Law
Olbers's Paradox
Oldham Coupling
Oloid
Olympic Medal Times in the Men's 100 Meter
Olympic Pole Vaulting
One- and Two-Word Palindromes
One-Dimensional Fractional Brownian Motion
One Interpretation of a Catalan Number Recurrence Relation
One-Sided Polyominoes
One-Term Nestedly Recursive Functions
One to Zero Cellular Automaton Function
Onion Polyhedra
On the Areas of Triangles Associated with an Excircle
Opacity Explorer 2D
Opacity Explorer 3D
Op-Art
Openable Cosine Dome
Openable Zonohedral Dome
Open a Cube
Open Circuit Voltage Coupled into a Semi-Infinite Transmission Line by an Exponential, Plane Waveform
Opening a Door
Opening an Umbrella
Opening a Sphere
Operating an AC Three-Phase Induction Motor
Operation of a Rectifying Column
Operation of a Stripping Column
Opposite Angles of a Quadrilateral in a Circle
Optical Matrix Elements of Single-Walled Carbon Nanotubes for Longitudinal Polarization of Light
Optical Model of the Human Eye
Optical Properties of Graphene
Optical Pumping: Population Dynamics
Optical Pumping: Visualization of Steady State Populations and Polarizations
Optical Rogue Waves (Rogons)
Optical Rotation and Ellipticity
Optics: Law of Reflection
Optimal Bin Packing with Random Lengths
Optimal Consumption Paths
Optimal Economic Strategy for the Operation of a Batch Reactor with Dead Time
Optimal Induction of Foreign Protein Synthesis
Optimality of Greedy Change-Making
Optimal Operation of a Batch Reactor with Dead Time
Optimal Patch Exploitation
Optimal Queue of Cars
Optimal Rectangles for Circles
Optimal Setup of Two Continuous Stirred-Tank Reactors (CSTRs) in Series
Optimal Transport Scheduling
Optimization of Cobb-Douglas Function
Optimization of the Rosenbrock Banana Function
Optimize the Length of the Crease of a Folded Piece of Paper
Optimum Temperature for Lactose Hydrolysis in Batch Reactors
Option Prices in Merton's Jump Diffusion Model
Option Prices in the Variance Gamma Model
Option Prices under the Fractional Black-Scholes Model
Options Board Using Black-Scholes Prices
Options: Time Value
Orbison's Illusion
Orbital Elements
Orbital Speed and Period of a Satellite
Orbital Wobble
Orbit Diagram of the Hénon Map
Orbit Diagram of the Logistic Map
Orbit Diagram of Two Coupled Logistic Maps
Orbit Diagram of Two Coupled Logistic Maps with Delay
Orbits and Stabilizers of Groups Acting on Colorings of 4x4 Chessboards
Orbits around and through a Sphere
Orbits around a Spinning Black Hole
Orbits around Schwarzschild Black Holes
Orbits of a Triple Pendulum
Orbits of Martin's Map
Orbits of the Chirikov Standard Map
Orbits of the Gingerbreadman Map
Orbits of the Tent Function's Iterates
Orbit Spectral Density in Olfactory Signal Processing
Orchard-Planting Problem
Ordered Dither Patterns
Order of Operations Tree
Ordinary Regression and Orthogonal Regression in the Plane
Oriented Run-Length Compression
Oriented Triangle Angles
Orienting a Compass
Origami Bases
Origami Flanged Pots
Ornamental Holiday Decoration
Ornamental Snails
Orthocentre d'un triangle inscrit dans une hyperbole (French)
Orthogonal Circle Inversion
Orthogonality of Two Functions with Weighted Inner Products
Orthogonal Projection of a Rectangular Solid
Orthogonal Views of Named RGB Colors
Orthonormality of Standing Waves
Oscillation of Two Masses Connected by Springs
Oscillations of a Pogo Stick
Osculating Circles
Osculating Circles 3D
Osmotic Pressure
Outer-Totalist Hexagons Between Past and Future
Outer Totalistic 2D Cellular Automata with Memory
Outer-Totalistic 3-Color Cellular Automaton 50010
Outgoing Waves Generated by a Pulsating Source
Overhanging Cards
Overlapping Go Games
Overlapping Intervals
Overlapping Lattices of Figures
Overlapping Rings
Overtones
Oxidation of Naphthalene in a Tubular Flow Reactor
Oxygen Transport by Hemoglobin and Myoglobin
Packed Bed Reactor Design for a Gas Phase Catalytic Reaction
Packing Squares with Side 1/n
Packing Squares with Triangles
Packing the Slothouber-Graatsma Cube
Padé Approximants
p-Adic Continued Fractions
Pairwise Axes Rotations in Factor Analysis
Pairwise Tangent Circles Centered at the Vertices of a Triangle
Palindrome Primes
Pandigital Olympic Circles Puzzle
Pantograph Linkage
Paper Folding
Paperfolding Dragon Curve
Paper Size Series A
Pappus Chain
Pappus's Hexagon Theorem
Parabolas from Rays
Parabolic Cylindrical Coordinates
Parabolic Envelope from Congruent Triangles
Parabolic Function Game
Parabolic Projectile Motion: Shooting a Harmless Tranquilizer Dart at a Falling Monkey
Parabolic Trough Solar Concentrator
Parallax
Parallel Lines: Angles with a Transversal
Parallel Lines Optical Illusion
Parallelogram Slide
Parallel-Plate Capacitors
Parameterized Offset Surfaces
Parameterizing Mathews versus Eldridge
Parameters for Plotting a Cubic Polynomial
Parameters for Plotting a Quartic
Parametric Movement of a Rubber Band
Parametric Necklace
Parametric Representations of Four Surfaces
Parametric Trace
Parametrized Cosine Surfaces
Parametrized Families of Circles
Paraxial Approximation and the Mirror
Parity Entangled Links
Parity Entangled Rings
Parity Recurrence in Thue-Morse Sequence
Partial Derivatives in 3D
Partial Diversion of a Stream through a Slot
Partial Fraction Decomposition
Partial Fractions in the Set of Rationals
Partially Filled Capacitor
Particle-in-a-Box Spectra for Delta-Function Perturbation
Particle in an Infinite Spherical Well
Particle Moving around Two Extreme Black Holes
Particle Moving on a Rotating Disk
Particles Thrown from a Rotating Bar
Particle System: Fountain
Partitioning a List
Partitions of Integers
Pascal-like Triangles Made from a Game
Pascal-like Triangles Mod k
Pascal's Mystic Hexagon
Pascal's Principle
Pascal's Tetrahedron Mod k
Pascal's Triangle
Pascal's Triangle and the Binomial Theorem
Pascal's Triangle Mod k
Pascal's Triangle Row Digits
Passing a Cube through a Cube of the Same Size
Passive Sonar
Patch Reentry of Elementary Cellular Automata
Path Integrals on Causal Sets
Path of a Projectile in a Rotating Frame
Path Rewriting Cellular Automata
Paths of Two-Dimensional Oscillators
Paths Through a Cubic Lattice
Pattern Generator with Fourier Transforms
Pattern Recognition Primer
Patterns for Functions on the Gaussian Integers
Patterns from All Possible Elementary Cellular Automata
Patterns from a Simple Math Function I
Patterns from Elliptic Integrals
Patterns from Math Rules Using Complex Numbers
Patterns from the Mean of Two-Color Totalistic 2D Cellular Automata
Patterns in Partitions of Integers
Pauli Spin Matrices
Payoff Gradients in Two-Player Games
Pay the Points?
Peano Curve Explorer
Peano Curve in 3D
Pearl Necklace
Peaucellier's and Hart's Inversor
Peaucellier's Inversor
Pedal Curves
Pedal Triangles of Isogonal Conjugates
Pendule simple (French)
Pendulum on a Moving Support
Pendulum with an Oscillating Support
Pendulum with Rotating Pivot
Pendulum with Three Magnets
Penetration of Potential in a Semi-Infinite Region
Pen Falling Off a Finger
Penrose Diagram
Penrose Tiles
Penrose Triangle
Pentagonal Art Template
Pentagonal Pyramid
Pentagonal Tiling
Pentagonal Tiling around a Decagon
Pentagonal Tiling around a Pentagon
Pentagon Spirals
Pentagon Tilings
Pentaprism
Percentage Change
Percentiles of Certain Probability Distributions
Percent of a Number
Percents as Fractions or Decimals
Percolation on a Square Grid
Perfect Number Generator
Perfect Riffle Shuffling
Perfect Tilings
Performance of a Batch Reactive Distillation System versus a Plug Flow Reactor
Performance of a Plug Flow Reactor with Recycle
Perigal's Three-Squares-to-One Dissection
Period and Frequency
Periodic Boundary
Periodic Boundary Condition
Periodic Box-Ball System
Periodic Heat Kernel
Periodicity of Euler Numbers in Modular Arithmetic
Periodic Table in 3D
Permafrost
Permutation Notations
Permutations
Permutations, Derangements, and Other Forbidden Position Problems Using Non-Attacking Rooks
Permutations in Lexicographic Order
Permutation Tree
Permuted Projections of Nested Arrays
Permuted Tabulation of Vanishing Determinants
Perpendicular Lines Generated by a Quadrilateral
Perpendiculars from a Point on the Line between the Endpoints of the Angle Bisectors
Perpendiculars from the Midpoints of the Orthic Triangle
Perpendiculars to a Chord
Perpetuum Mobile I
Perpetuum Mobile II
Perpetuum Mobile III: Drinking Duck
Person Walking Away from Spotlight
Perspective Drawing of Billiards Table
Perturbation Theory Applied to the Quantum Harmonic Oscillator
Per Unit Tax
Peter de Jong Attractors
P&F Charts
Phase Matching of SHG in Nonlinear Optics
Phase Mismatch in SHG in Nonlinear Optics
Phase of a Complex Polynomial
Phase Percentage and Carbon Content Estimation in Hypoeutectoid Steels by Image Processing
Phase Plane Plot of the Van der Pol Differential Equation
Phase Plots of Trigonometric Functions
Phase Portrait and Field Directions of Two-Dimensional Linear Systems of ODEs
Phases of Planets
Phase Space of an Intermittently Driven Oscillator
Phase Space Trajectories of a 1D Anharmonic Oscillator
Phase Transitions in 2D Cellular Automata
Phase Transitions in 2D Cellular Automata--A 3D View
Phase Transitions in Cellular Automata
Phase Transitions in ECA Rule 184
Phasor Model for RC Filter Electronic Circuit
Phone Numbers by Fibonacci Methods
Phone Numbers to Words
Photographic Orton Effect
Photon Number Distributions
Phyllotaxis Spirals
Phyllotaxis Spirals in 3D
Piano Keys
Pick's Theorem
Pictorial Bar Charts
Picture Frame
Picture Puzzle
PID Control of a Tank Level
Pi Digits Bar Chart
Pi Digits Pie
Piecewise Functions
Pie Charts
Piezoelectricity in Barium Titanate
Pi-Like Ratios for Circular Lakes
Piling of Bars along Dodecahedron Axes
Pinch Composition versus Reflux Ratio for a Benzene-Ethylenediamine Mixture at 1 atm
Pine Cone
Pinwheel Tiling
Pipe Flange
Pipeline Expansion Loop
Piston Linkage
Pixelization of a Font
Place Value
Placing Objects at Predetermined Sets of Points
Planarity
Planar Three-Body Problem
Planar Three-Connected Trivalent Network Mobile Automaton
Planar Trivalent Network Growth Using Two Rewrite Rules
Plane Cubic Curves
Plane Nets for Pyramids
Plane Sections of a Cone
Plane Seismic Waves
Planes, Solutions, and Gaussian Elimination of a 3×3 Linear System
Platonic Solids
Platonic Spline Radiolari
Plato's Shadow Problem
Playing and Solving Peg Solitaire
Playing Cards
Playing with Stellations of the Icosahedron
Plot a Quadratic Polynomial
Plot a Quintic Polynomial
Plot, Log Plot, and Beyond
Plot of a Geometric Sequence and Its Partial Sums
Plots of Compositions of Bitwise Operations
Plots of Functions and Their Iterates
Plots of Legendre Polynomials
Plots of the Fermi-Dirac Distribution
Plotting a Parabola
Plotting Closed Curves with a Four Bar Linkage
Plotting Price Divergences
Plotting Pythagorean Triples
Poggendorff Illusion
Poincaré Hyperbolic Disk
Point Horopter
Point in Triangle
Points around an Ellipse
Points on a 3D Spiral
Points on a Spiral
Points Symmetric to the Orthocenter with Respect to the Sides of a Triangle
Poisson Distribution
Poisson Equation on a Circular Membrane
Poisson Spot
Poker
Polar and Rectangular Coordinates
Polar Angles and Points on the Unit Circle
Polar Area Sweep
Polar Coordinates
Polar Graph with Rectangular Graph
Polarization of an Optical Wave through Polarizers and Wave Plates
Polarization State Generator
Polarized Atoms Visualized by Multipole Moments
Polarizer-Compensator Combination as a Controlled Polarization Filter
Polar Plot of a Year's Hourly Temperatures
Polar Plots of Legendre Polynomials
Polar Prime Spiral
Polar Trace
Polar Zonohedra
Polar Zonohedron around an Icosahedron
Polar Zonohedron Built on a Dodecahedron
Pólya Plots
Pólya's Mechanical Model for the Fermat Point
Pólya's Space-Filling Curve
Pólya's Urn
Pólya Vector Fields and Complex Integration along Closed Curves
Polycube Snakes
Polyform Explorer
Polygonal Holes in Polygonal Bars
Polygonal Numbers
Polygon Diagonals
Polygons Arranged in a Circle
Polygons at the Vertices of Polyhedra
Polygon Spirals
Polygon Triangulations
Polyhedra Assembled from 20 Parts
Polyhedra Copied to the Vertices of the Same Kind of Polyhedron
Polyhedral Koalas
Polyhedrane
Polyhedra, Spheres, and Cylinders
Polyhedra with Prismatic Symmetry
Polyhedron Dual
Polyline Simplification
Polynomial and Derivative
Polynomial Fits of Random Walks
Polynomial Fitter
Polynomial Graph Generator
Polynomial Long Division
Polynomial Long Multiplication
Polynomial Roots
Polynomial Roots in the Complex Plane
Polynomials and Derivatives
Polyrhythyms
Ponchon-Savarit Graphical Equilibrium Stage Method
Ponchon-Savarit Graphical Method at Total Reflux
Pool Shot
Pool Shot 3D
Population Cube
Population Donut Chart
Population Genetics: Hardy-Weinberg Equilibrium with Two Loci
Population Selector
Populations of Cities
Position-Dependent Opacity
Positive-Contact Claw Clutch
Positive Frobenius Numbers of Three Arguments
Positive Integer Explorer
Positive Wigner Function Transformation by Quantum Operations
Post-Event Bonding
Potential and Kinetic Energies of a Falling Object
Potential Energy of Objects from Daily Life
Potential Field of Two Charges
Potential Flow over a NACA Four-Digit Airfoil
Potter's Wheel
Power Content of Frequency Modulation and Phase Modulation
Power Efficiency of Amplitude Modulation
Power in Weighted Voting Systems
Power Law Tails in Log Normal Data
PowerMod Is Eventually Periodic
Power of a Test about a Binomial Parameter
Power Series Interval of Convergence
Powers in Modular Arithmetic
Powers of Complex Numbers
Powers of Complex Points
Powers of Numbers
Power Spectrum of Cellular Automaton Dynamics
Power Spectrum of the Logistic Map
Power versus Work
pq-System Explorer
Practicing Aircraft Holding Pattern Entries
Pratt Certificates of Primality
Precession of the Equinoxes
Precision Error
Predator-Prey Dynamics with Type-Two Functional Response
Predator-Prey Ecosystem: A Real-Time Agent-Based Simulation
Predator-Prey Equations
Predictive Scores and Ultimate Test Passage
Preferred Values of Collatz Paths
Prefixes of Multiples
Premium Assistance Calculator for HR 3590 and HR 3962
Premium Ratios with Capital Costs Included
P-Representation of Laser Light
Present Value Calculator
Pressure
Pressure-Volume-Temperature Diagram
Price Competition
Price Controls
Price-Yield Curve
Pricing a Bermudan Option with the Longstaff-Schwartz Monte Carlo Method
Pricing a Diamond
Pricing a European-Style Arithmetic Asian Option: Comparing Bootstrapping and Simulation Approaches
Pricing Power Options in the Black-Scholes Model
Primality Bit Spiral
Primality Formal System Explorer
Prime Beat
Prime Factorization Table
Prime Gaps
Prime-Generating Cellular Automaton
Prime-Generating Recurrence
Prime Number Races
Prime Numbers in Various Bases
Prime Plaid
Prime Walk
Principal Axes of Inertia
Principal Components
Principal Stresses in Compacted Cohesive Powders
Principle of the Lever
Probabilistic Dartboard
Probabilité pour que X soit dans un intervalle (French)
Probabilities of Combinations
Probabilities of Matching Colors
Probabilities of Permutations
Probability Densities, Expectation Values, and Uncertainties for Gaussian Wavepackets
Probability Density in an Infinite Square Well
Probability Distribution for the kth Greatest of a Sequence of n Random Numbers
Probability Logic Test
Probability of Being Sick After Having Tested Positive for a Disease (Bayes's Rule)
Probability of Long Leads
Probing Macroscopic Quantum States with Gravitational-Wave Detectors
Probit and Logit Models with Normal Errors
Problem 11340 of The American Mathematical Monthly
Problems in Peg Solitaire
Problems on Circles I: Dissecting a Circle
Problems on Circles II: Halving a Set of Points
Problems on Circles III: Apollonius's Problem
Problems on Circles IV: Circles Tangent to Two Others with a Given Radius
Problems on Circles V: External and Internal Common Tangents
Problems on Circles VII: Circles of a Given Size Tangent to Other Two
Problems on Circles VIII: A Spiral of Circles
Problems on Circles VI: Poncelet Transversals
Problems on Circles XI: Perpendicular Tangents to an Ellipse
Problems on Circles X: Tangent Circles Generate Ellipses
Processing Various Parts of an Image Differently
Product Distribution Curves for Laminar, Plug, and Mixed Flows
Production of Cis2-Butene and Cis2-Hexene by Cis2-Pentene Disproportionation
Product of Two Gaussians
Product of Two Levi-Civita Tensors
Product of Two Levi-Civita Tensors with Contractions
Product of Two Primes
Products Determined by a Point on the Circumcircle
Products of Diagonal Lengths in the Regular Polygon
Products of Lines
Products of Segments of Altitudes
Profit Maximization in Perfect Competition
Projectile Motion
Projectile with Air Drag
Projecting a Lattice of Points
Projecting a Triangle
Projecting Images from a Point Cloud
Projecting Points on Spheres to Lower Dimensions
Projection of Polytopes into Three Dimensions
Projection of Three Orthogonal Planes
Projections, an Altitude, and the Orthocenter
Projections of a 4D Hilbert Curve
Projections of Midpoints onto Circumradii
Projections of the 16-Cell
Projections of the Four-Cube
Projective Planes of Low Order
Proof by Induction
Proof of Proposition 28, Book XI, Euclid's Elements
Proofs in Multiway Systems
Proof without Words: 1+2+...+(n-1)=n Choose 2
Propagating Waves
Propagation of a Plane Electromagnetic Wave
Propagation of Reflected and Refracted Waves at an Interface
Properties of a Simple Random Walk with Boundaries
Properties of Chemical Elements
Properties of Fresnel's Interface Reflection Coefficients
Properties of the Elements
Property Coinsurance
Proportional-Integral Control of Batch Rectification Process
Proportional Quantities
Proportional Temperature Control
Proportional Triangle Nesting
Proportions of a Box
Proposition 29, Book XI, Euclid's Elements
Proposition 30, Book XI, Euclid's Elements
Proposition 3, Book XII, Euclid's Elements
Proposition 7, Book XII, Euclid's Elements
Propositional Logic Puzzle Generator
Propositional Logic Test
Propp Circles
Prospecting for Complexity in k=3, r=1 Cellular Automata
Protein Alignment Wheel
Protein Dot Plot
Protein Synthesis Induction
Prototype of a Virtual 2D Orrery
Protractor Practice
Prüfer Codes of Labeled Trees
Pseudocolor
Pseudoprime
Pseudorandom Password Generator
Psychrometric Chart
Ptolemy's Theorem
Pulsars
Pulsating Starfish
Pulse Width Modulation Principle
Pump Jack
Punched Sphere
Puppet
Purchasing Power Calculator
Pure Tones
Pure Tones with Sample Rate
Pursuit Curves
Pyramid Builder
Pythagoras Tree
Pythagorean Analogs for Similar Triangles
Pythagorean Dissection
Pythagorean Graphs
Pythagorean Means
Pythagorean, Meantone, and Equal Temperament Musical Scales
Pythagorean Theorem
Pythagorean Theorem 3D
Pythagorean Triangles with Consecutive Values
Pythagorean Triangles with the Same Area
Pythagorean Triples
Pythagorean Triples Star
q-Bézier Curves
q-Cosine and q-Sine Functions over the Extended Complex q-Plane
q-Pascal Triangle
QR Decomposition
Q-Representation of Number States
q-Trigonometric Functions
Quadratic Equation
Quadratic Julia Sets
Quadratic Practice
Quadratic Residues
Quadratic Surfaces
Quadrilateral Slide
Quantized Solutions of the 1D Schrödinger Equation for a Harmonic Oscillator
Quantum Circuit Implementing Grover's Search Algorithm
Quantum Computational Basis Vectors
Quantum Computer Search Algorithms
Quantum Fourier Transform Circuit
Quantum Logic Gates: Roots, Exponents, and Eigensystems
Quantum-Mechanical Particle in an Equilateral Triangle
Quantum Particles in an Infinite Square Potential Well
Quantum Random Walk
Quantum Well Explorer
Quasi-Rent in Open Access Fisheries
Quaternion Julia Set
Qubits on the Poincaré (Bloch) Sphere
Quick Return Mechanism
Quicksort versus Selection Sort
Quotients and Remainders Wheel
Qutrit States as Probability Vectors
Racing Disks
Radial Engine
Radial Gradient
Radially Polarized Laser Beam Intensity Distributions
Radial Metric
Radial Temperature and Concentration Gradients in a Packed Bed Reactor
Radiation by an Accelerated Charge
Radiation Pattern for Double-Couple Earthquake Sources
Radioactive Decay as a Probability Distribution
Radioactive Decay in the Causal Interpretation of Quantum Theory
Radioactive Decay of Five Elements: Time Dependence of Remaining Mass
Radio Propagation and Multipath with Diversity Antennas
Radius-1/2 Cellular Automata
Radius and Temperature of Main Sequence Stars
Ragged Automaton Disk
Rainbow String Figures
Raindrop Art
Ramsey(3,3,3)>16
Ramsey(3,3) = 6
Random 2D Nearest Neighbor Networks
Random 3D Nearest Neighbor Networks
Random Acyclic Networks
Random Binary Differences
Random Branching Process
Random Branching Process in 3D
Random Chord Paradox
Random Circles with Power-Law Sizes
Randomly Filling an Array
Randomly Reducing Objects to Spheres
Randomness in the Digits of Pi
Randomness in the Digits of Square Roots
Random Number Exploration
Random Number Generation
Random Partitioning of a List
Random Permutations of a Given Length
Random Points on a Sphere
Random Spheres with Power-Law Sizes
Random Spline Polygon
Random Subdivision of a Square
Random Tree Aggregation
Random Values from Distributions
Random Walk and Diffusion of Many Independent Particles: An Agent-Based Simulation
Random Word
Random Word Generator
Range-Zero Binary Cellular Automata
Rankine Vortex: A Simple Hurricane Model
Rank Plots for Countries
Rare Event Analysis by Flow Cytometry
Raster Bubble
Rasterization of a Data Plot
Rate of Change of the Distance between Two Point Masses
Ratio-Based Modified Logistic Isothermal Microbial Growth
Ratio Match
Rational Function Graphs
Rational Functions of Small Degree
Rational Harmonics
Rational Number Explorer
Rational Points on an Elliptic Curve
Rational Triangles with Area Less than 102
Ratio Notation
RATS Sequences
Ray Absorber
Ray Diagrams for Lenses
Ray Diagrams for Microscope and Telescope
Ray Diagrams for Spherical Mirrors
Ray Tracing With Lenses
Reaction Force on a Lever
Reactive Collision Dynamics in Two Dimensions
Reactive Distillation Computations Including Heat Effects
Reactive Scattering on a LEPS Surface
Reactor Steady States
Readable Mess between First and Last Letters
Reading Hertz's Own Dipole Theory
Real Elliptic Curves
Real Options
Real Roots of Sparse Polynomials
Real-Time Simulation of the Game of Life
Rearranging Color Channels in an Image
Rearranging the Alternating Harmonic Series
Receiver Operating Characteristic Curves and Uncertainty of Measurement
Reciprocals of Diagonal Lengths in the Regular Polygon of Unit Side
Reconstruction of the Great Wall
Recovering the Purchase Price of Hybrid Vehicles in Fuel Savings
Rectangles: Perimeter and Area
Rectangular Prisms
Rectangular Pulse and Its Fourier Transform
Rectangular Screw Thread
Rectangular Slice Twist
Rectifying and Stripping Profiles for the Reactive Cascade
Recurrence Network Measures for the Logistic Map
Recurrence Plot of Mathematical Functions and Constants
Recurrence Table Attractors
Recurring States in Cellular Automaton Evolution
Recursion in the Ackermann Function
Recursive Exercises I
Recursive Exercises II: A paradox
Recursive Exercises III: Fire Patterns
Recursive Exercises IV: Rep-Tiles
Recursive Exercises IX: Gothic Windows
Recursive Exercises VIII: Kites and Darts
Recursive Exercises VII: Stacks of Cubes
Recursive Exercises VI: Tori
Recursive Exercises V: Nested Triangles
Recursive Extended Euclidean Algorithm
Recursively Defined Partial Tilings of the Plane
Red Giant Stars and the Death of the Sun
Reduce, Replicate, Rebuild
Redundancy in Written Language
Reflecting a Lattice of Points
Reflecting a Point through the Midpoints of a Triangle's Sides
Reflecting in Intersecting Lines
Reflecting in Intersecting Planes
Reflecting in Parallel Lines
Reflecting in Parallel Planes
Reflection and Refraction on a Flat Surface between Two Different Materials
Reflection and Transmission of Acoustic Waves
Reflection from a Freehand Polynomial Mirror
Reflection from a Polynomial Mirror
Reflection in a Circle
Reflection in Two Dimensions
Reflection Matrix in 2D
Reflection of Parallel Rays by a Sphere
Reflection of Parallel Rays by a Two-Dimensional Body of Nearly Maximal Resistance
Reflections in a Mirrored Corner
Reflections in an Elliptical Region
Reflections of a Point in the Midpoints of a Triangle's Sides
Reflective Properties of Conic Sections
Reflux Policy for a Batch Distillation Operation with a Constant Distillate Composition
Refraction from a Submerged Light Source
Refraction through Parallel Faces
Refrigeration Cycle Coefficient of Performance
Refrigerator Poetry
Regelation
Regions of Feasible Distillate and Bottom Product Composition for an Ideal Ternary Mixture at 1 atm
Register Machines
Regular Cone Nets
Regular k-ary Trees
Regular n-Gons as Polygons and Stars
Regular Paperfolding Sequence
Regular Polygon Rolling on a Catenary
Regular Polygons
Regular Right Prism between Two Points
Regular Tilings
Reinsurance Cut-Through
Related Rates: A Boat Approaching a Dock
Related Rates Clock
Related Rates: Triangle Angle and Area
Relating a Rhombic Triacontahedron and a Rhombic Dodecahedron
Relating the Rhombic Triacontahedron to the Rhombic Enneacontahedron
Relating the Rhombic Triacontahedron to the Rhombic Icosahedron
Relation of Radius, Surface Area, and Volume of a Sphere
Relations and Functions between Small Sets
Relations and Graphs
Relations between Some Triangles Associated with Excircles
Relationship between Sine and Tangent
Relationship of Sine and Cosine to the Unit Circle
Relative Circular Motion
Relatively Prime Numbers and Zeta(2)
Relative Motion in a Subway Station
Relative Primality
Relativistic Energy Levels for Hydrogen Atom
Relativistic Headlight Effect
Relativistic Mass
Relativistic Time Dilation in Muon Decay
Relativistic Traffic
Relaxation Method for Electrostatic Problems
Relaxation Methods for Solving the Laplace Equation
Relaxation of a Maxwell Element
Reliability Distributions
Relief-Shaded Elevation Map
Repeating Continued Fractions
Replicating Patterns in 2D Binary Cellular Automata with the Parity Rule
Representation of Diffraction by Vector Summation
Representation of Qubit States by Probability Vectors
Representations by a Sum of Squares
Representations of the Dirac Deltafunction
Representing an Integer with an Elementary Automaton
Representing Elementary Cellular Automaton Rules
Rep-Tile
Residence Time Distribution for Continuous Stirred-Tank Reactors in Series Using the First Four Moments
Residence Time Distribution Function of a Stirred-Tank Cascade (E-Curve)
Residence Time Distribution of Single Continuous Stirred-Tank Reactors Subject to a Square Pulse Input
Residue Curve and Bubble Temperature Distribution for a Ternary Mixture
Residue Curve and Bubble Temperature Distribution for Methyl Tert-Butyl Ether Chemistry at 8 atm
Residue Curve and Temperature Distribution for Ethyl Acetate Chemistry
Residue Curve Computation for Methyl Tert-Butyl Ether (MTBE) Chemistry
Residue Curve Map for a Benzene-Toluene-p-Xylene Mixture
Residue Curve Map for a Mixture of Three Alcohols; Nature of the Singular Points
Residue Curve Map for a Mixture of Water, Ethanol, and Ethylene Glycol
Residue Curve Map for a Non-Ideal Ternary Mixture with a Distillation Boundary
Residue Curve Map for Methyl Acetate and Isopropyl Acetate Chemistry at 1 atm
Residue Curve Map Presenting Four Distillation Regions
Residue Curves at High Total Pressures
Residue Curves for Different Activity Coefficient Models
Resistors in Parallel
Resistors in Series
Resizable Number Theory Tables
Resizing Images
Resonance Lineshapes of a Driven Damped Harmonic Oscillator
Response of a Jacketed Exothermic Continuous Stirred-Tank Reactor to a Perturbation
Response of a Reactor with Axial Dispersion to a Pulse Input Tracer (E-Curve)
Restricted Non-Cooperative Game Theory
Restricted Three-Body Problem
Restricted Three-Body Problem in 3D
Resultant of a Vector
Retiree Stop-Loss Reinsurance
Retrocausality: A Toy Model
Retrograde Motion
Retrograde Motion Illusion for an Electric Fan
Reuleaux-Like Polygon Shafts
Revenue and Elasticity
Reverberations in Acoustic Layers
Reversal-Addition Related Systems
Reversal-Addition Systems
Reverse Collatz Paths
Reversed-Out Letterforms
Reverse Engineering GPA Distributions from Honors Data
Reverse-Then-Add Sequences
Reversible Multiplex of Totalistic Cellular Automata
Reversible Two-Color Cellular Automaton Rules
Revolutionary Cup
RF Attenuator T or Pi Values in 5% Resistors
RGB and CMYK Colors
RGBColor Gradients
RGB Color Space
RGB Explorer
RGB Gamma Controls
RGB Match Up
Rheological Behavior of FENE Dumbbell Suspension under Elongational Flow
Rhombi at the Incenter of a Triangle
Rhombic 210-hedron
Rhombic Dodecahedra Totalistic Cellular Automaton
Rhombic Dodecahedron 5-Compound
Rhombic Dodecahedron and Polar Zonohedron
Rhombic Dodecahedron Made of Four Rhombohedra
Rhombic Drive for Speed Governor
Rhombic Drive for Stirling Engine
Rhombic Enneacontahedron with 30 Icosahedra
Rhombic Hexecontahedron Frame
Rhombic Icosahedron Helices
Rhombic Spirallohedra
Rhombic Triacontahedron Built on a Cube
Rhombic Triacontahedron Measures
Rhombic Triacosiohedron
Rhombicuboctahedron Star
Rhombicuboctahedron Star on a Sphere
Rhombohedra
Rhombohedron with Variable Faces
Rhyme Finder
Riccati Differential Equation with Continued Fractions
Richards Growth Curve
Ridged Pip Die
Riemann Hypothesis
Riemann's Example of a Continuous but Nowhere Differentiable Function
Riemann Sums
Riemann Sums: A Simple Illustration
Riemann Sums for Functions of Two Variables
Riemann Surfaces of Inverses of Sums of Two Trigonometric Functions
Riemann Surfaces of Inverse Trigonometric Functions
Riemann Surface Transition
Riemann versus Lebesgue
Riemann Zeta Function
Riemann Zeta Function near the Critical Line
Riesz's Rising Sun Lemma
Righty Tighty; Lefty Loosy
Rigidity Testing of Frameworks
Rigorous Steady-State Simulation of a Multicomponent Distillation Column
Rimless Wheel Locomotion
Ring Fractal
Ring of 2n Rhombohedra
Ring of 90 Rhombohedra
Ring of Hoberman Rings
Ring of Ten Icosahedra
Rings of 20 Icosahedra
Rings of 5 and 10 Rhombic Triacontahedra
Rings of Dodecahedra
Rings of Five and Ten Polyhedra
Ring Structures Made of Square Pyramids and Matching Tetrahedra
Risk Aversion, Load, and Optimal Insurance
Risk, Ownership, and Control
Risk Premiums
Roads and Wheels
Robinson Tiling
RoboSnarf
Robustness of the Longstaff-Schwartz LSM Method of Pricing American Derivatives
Rocket Fire
Rock-Scissors-Paper
Roget's Palisade
Rogue Ocean Waves
Roller Bearing
Roller Coasters
Roller Crushing Machine
Rolling a Circle on a Parabola
Rolling a Coin around a Coin
Rolling a Cube on a Chessboard
Rolling a Disk on a Curve
Rolling Ball inside a Cylinder
Rolling Bridge
Rolling Circles
Rolling Circles with Points and Lines
Rolling Cylinder with a Hole
Rolling Disk
Rolling Mill
Rolling Polygons
Rolling Polygons in a Circle
Rolling Up a Sheet of Graphene
Rolling Wheel with Spoke
Roman Numeral Graph
Roman Numerals
Romeo and Juliet
Roofing a Cube to Produce a Dodecahedron
Root Finder
Root Plot of a Polynomial Class
Root Routes
Roots of Complex Numbers
Roots of the Derivatives of a Certain Real Polynomial in the Complex Plane
Rosettes and Baskets Made of Tori
Rotary to Linear Motion
Rotate and Fold Back
Rotated and Scaled Text in Graphics
Rotating a Hypercube
Rotating a Lattice of Points
Rotating a Mask over a Tone Scale
Rotating by Powers of i
Rotating Plane with Intersecting Sphere and Circle
Rotating Polygon Illusion
Rotating Regular Polygonal Rings
Rotating Space Station
Rotating Substitution Systems
Rotating Text
Rotating the Faces of the Icosahedron
Rotating the Sides of a Right Regular Prism
Rotation about Carbon-Carbon Bonds
Rotational Symmetries of Colored Platonic Solids
Rotational Symmetries of Platonic Solids
Rotational-Vibrational Spectrum of a Diatomic Molecule
Rotation and Translation of a Rigid Body
Rotation in Two Dimensions
Rotation Matrix Entries
Rotation of Feynman Diagrams around an Electron-Photon Vertex
Rotation Point Mandala
Rotation with Zero Angular Momentum (The Square Cat)
Roulette: A Comfortable Ride on an n-gon Bicycle
Roulette Plotter Device
Rounding Large Integers
Rounding Rectangle Corners with the RoundingRadius Option
Routh's Theorem
Rovibronic Infrared Spectrum of a Rigid Diatomic Rotor
RSA Encryption and Decryption
Rubik's Cube
Rudin-Shapiro Curve
Rule 54
Rule Case Fractions in the Wolfram 2,3 Turing Machine
Rule Case Transition Diagrams for the Wolfram 2,3 Turing Machine
Rule Cycling
Rule of 72
Rules for Logarithms
Rule Shaped Zonohedron
Runge's Phenomenon
Running Circles around Circles
Running in the Rain
Running Objects along a Path
Russell's Thought Experiment in Special Relativity
Rutherford Scattering
Saddle Points and Inflection Points
Safety Pin
Sagitta, Apothem, and Chord
Sailing against the Wind
Sailplane Altimeter Tools to Show Flight Levels
Sale Price Calculator
Sales Tax and Discounts
Salinon
Same but Different
Sample Size Calculation - Means
Sample Size Calculation - Proportions
Samples of Random Graphs
Sample versus Theoretical Distribution
Sampling a Digital Signal
Sampling Distribution of the Sample Mean
Sampling of Boolean Functions
Sampling Theorem
Sandpile Model
Sangaku for Incircles
Sarrus Linkage
Saturn's Disappearing Rings
Saturn's Rings and the Cantor Set
Saunders Digit Graphics
Saving for Retirement
Scalar Feynman Diagrams and Symanzik Polynomials
Scalar Retarded Potential of a Point Charge Moving on a Circle with Constant Velocity
Scalenohedron
Scalings of a GI/G/1 Queue Realization
Scanning Electrochemical Microscopy: Diffusion on a Microdisk Electrode
Scanning Electrochemical Microscopy: Feedback Approach Curve
Scanning Electrochemical Microscopy: Fitting Rg with Negative Feedback Curve
Scattering by a Square-Well Potential
Scattering of a Gaussian Wave Packet on an Infinite Wall
Scattering over a Square Potential Well
Scattering over Potential Step
Schinzel's Theorem
Schoenberg Plane-Filling Curve
Schröder Number Diagrams
Schröder Rectangulations
Schrödinger Equation for a Dirac Bubble Potential
Schrödinger Equation for a One-Dimensional Delta Function Potential
Schwarzschild Space-Time Embedding Diagram
Scientific Notation
Scissors Lift with Constant Diameter
Scrap Metal Price of Coins
Screw with Sharp V-Thread
Sea Level
Seasons
Secant and Tangent Lines
Secant Approximations
Secant Intersection with Two Internally Tangent Circles
Secant Root Finding Method
Second Droz-Farny Circle
Second Hardy-Littlewood Conjecture
Second-Order Partial Derivatives
Sections of the 16-Cell
Sections of the Four-Cube
SectorChart
SectorChart Applied to GDP
Sedimentation of Four or Six Spheres in a Newtonian Fluid at Low Reynolds Numbers
Segmenting a Medical Image
Segment Intersection
Seismicity of Germany
Seismoscope
Selected Reflections in the Sides of a Regular Polygon
Selectivity for Adsorption of Ethylene Relative to Ethane in NaX Zeolite at 20 °C
Self-Avoiding Random Walks
Self-Avoiding Random Walks in 3D
Self-Delimiting Binary Representation of a Ternary List
Self-Intersecting Polygons
Self-Intersections in a Polygon
Selfish Binary Halfrange Automaton Cycles 2D
Selfish Elementary Cellular Automata
Self-Similarity in Random Walk
Semenov's Algorithm for Solving Systems of Nonlinear Equations
Semi-Rotation Automaton Plot
Sendov's Conjecture
Sensitive Dependence in Iterated Maps
Sensitivity, Specificity, and Incidence
Sensitivity to Initial Conditions in Chaos
Sensitivity To Peturbation in Elementary Cellular Automata
Separation of a Reactive Quaternary Mixture in a Distillation Column
Separation of a Reactive Ternary Mixture: Olefin Metathesis
Separation of a Reactive Ternary Mixture with a Distillation Column
Separation of Topological Singularities
Sequence Alignment of Words
Sequential Substitution Systems
Series: A Few Examples
Series Expansion of 1/(1-x)
Series Expansions of Powers of Trigonometric Functions
Series of Spheres at the Vertices of Polyhedra
Series RLC Circuits
Series: Steps on a Number Line
Serra's Torqued Ellipse IV
Serrated Cylinder
Set of Nash Equilibria in 2x2 Mixed Extended Games
Seven Cylinders
Seven-Letter Anagrams with High-Frequency Letters
Seven-Letter Anagrams with High-Frequency Letters, Part 2
Seven-Segment Display
Shadows of a 3D Shape
Shakespearean Networks
Shakespearean Play Summaries
Shape Descender
Shapes Produced by Rotating an Astroid
Shaping a Road and Finding the Corresponding Wheel
Shaping a Tubular B-Spline Surface
Shaping of Polar Zonohedron
Shell Space: Flare, Verm, and Spire
Shell Space: The 'Snugness' Condition
Shepard Tones
Shifted Elementary Cellular Automata
Shifts in the Demand Curve
Ship Sailing over the Horizon
Shooting a Bullet from a Car
Shooting Method
Short Computation Patterns I
Short Computation Patterns II
Short Computation Patterns III
Shortening the 2008 Beijing Olympic Torch Tour
Shortest Distance between a Point and a Line in 2D
Shortest Path between Two Points on a Sphere
Shortest Time Problem
Short-Run Cost Curves
Short-Run Production and Cost Curves
Short Simple Continued Fractions
Short-Time Expansion of Quantum Amplitudes
Sicherman Dice
Side Capacity Model of Stagnancy in a Continuous Stirred-Tank Reactor
Side Lengths Opposite Unequal Angles
Sierpinski 3D Arrowhead Curve
Sierpinski Carpet
Sierpinski Sieve
Sieve of Eratosthenes
Signal Detection Theory
Signed Area of a Polygon
Signed Determinant Terms
Signs in Space
Similar Right Triangles
Similar Triangles Determined by Miquel Circles and the Circumcircle
Simple Bar Chart
Simple Batch Distillation of a Partially Miscible Binary Mixture
Simple Best-Fit Line
Simple Chaotic Motion of Quantum Particles According to the Causal Interpretation of Quantum Theory
Simple City Builder
Simple Contrast Adjustment
Simple Distillation Residue Curves for a Mixture of Chloroform and Acetone at 760 mmHg
Simple Distillation Residue Curves for a Mixture of Ethanol and Isopropanol at 750 mmHg
Simple Distillation Residue Curves for a Mixture of Pentane and Dichloromethane at 750 mmHg
Simple Harmonic Motion
Simple Harmonic Motion of a Spring
Simple Inequalities in the Unit Cube
Simple Initial Conditions in Code 20
Simple Line and Point Spiral Flower
Simple Phone Dialer
Simple Rational Functions
Simple Reaction with Segregation in a Batch Reactor
Simple Simulation of Tides
Simple Sphereman
Simple Spline Curves
Simple Spring Mass Damping
Simple Stair Calculator
Simple Three-Letter Cube
Simple versus Compound Interest
Simplifying Some Algebraic Expressions Using Mathematica
Simson's Theorem
Simulated Epidemics in Random Networks of Varying Density and Size
Simulated Quantum Computer Algorithm for Database Searching
Simulating a Catastrophe Insurer
Simulating a Normal Process from Sums of Uniform Distributions
Simulating Asset Prices with a GARCH(1,1) Model
Simulating a Three-Armed Pendulum
Simulating Flickering Fire with Noisy Cellular Automaton
Simulating Harmonographs
Simulating Temperature versus Time Relationships in the Thermal Preservation of Foods
Simulating the 2008 U.S. Presidential Election
Simulating the IRR
Simulating the Neutral Current of an AC System
Simulating the Power Factor of an AC System
Simulation of a Simple Gas Pressure Model
Sin/Cos Function Game
Sine and Cosine Graph Generator
Sine, Cosine, Tangent and the Unit Circle
Sine Oval and Nested Trig Functions
Sine Waves for Musical Scales
Single Color Images
Single-Slit Diffraction
Single-Slit Diffraction Pattern
Single-Slit Diffraction Using Feynman's Method
Single Solute Batch Two-Phase Extraction
Singular Value Decomposition
sin(t)=cos(pi/2-t)
Sinusoidal Bellows
Sinusoidal Spirals
SIR Epidemic Dynamics
Six Basic Arithmetic Functions
Six-Box Game of Chance
Six Circles Theorem
Six Decagons and Six Great Circles of a Sphere
Six Hexagons
Six Incircles in an Equilateral Triangle
Six Interwoven Loops
Six-Piece Dissection of a Tetrahedron into Its Mirror Image
Six-Piece Dissection of Hill's Tetrahedron of Type 3 into a Triangular Prism
Six-Sphere Coordinates
Six Square Pyramids
Skewed Cone
Skew Lines
Skolemization
Sleepless Knights Puzzle
Slicing a Cylinder
Slicing an Indented Sphere
Slicing a Solid of Revolution
Slicing a Torus
Slider Puzzle
Slide Rule
Sliding the Roots of Cubics
Sliding the Roots of Quadratics
Sliding Tiles
Slope and Equations of Lines through Points
Slope Fields
Slurry Reactor with Declining Catalyst Activity
Small-Digit Addition
Small Lattice Ellipses
Small Set Partitions
Small Signal (AC) Bipolar Parameters from SPICE Parameters
Small Signed Integer Addition 3D
Small World Networks
Smiley Changer
Snake-Arm Robot
Snell's Law of Refraction (Wave Fronts)
Snowflake Growth
Snowflake-Like 3D Growth
Snowflake-Like Patterns
Sobel Edge Detection Filter
Social Golfer Problem
Social Networking
SOHCAHTOA Triangle
Soil Temperature
Solar and Lunar Eclipses
Solar Neighborhood
Solar Panel of NASA's Phoenix Mars Lander
Solar System Mandalas
Solid Angles on a Sphere
Solid Cylinder Rolling in a Turnable Hollow Cylinder
Solid of Revolution
Solids of Known Cross Section
Solids of Revolution
Solids Whose Cross Sections Have the Same Shape
Solid Wood Texture
Solitary Wave Solution to the Nonlinear Schrödinger Equation
Solitons from the Korteweg-de Vries Equation
Soliton Trajectories According to Bohmian Quantum Mechanics
Soliton Trajectories for the Kadomtsev-Petviashvili Equation
Soliton Trajectories of the Modified Korteweg-de Vries Equation (mKdV)
Sol LeWitt (Virtual) Wall Drawing #146
Solow Growth Model
Solution of One-Dimensional Stefan Problem with Orthogonal Collocation
Solution of Quadratic Equations
Solution of Two Equations in Two Unknowns
Solutions of Fermat's Equation
Solutions of the Finite Difference Discretized Laplace Equation
Solve Quadratic Equations with Integer Coefficients
Solving a Linear System with Uncertain Coefficients
Solving Hard Traveling Salesman Problems
Solving Matrix Systems with Real, Interval, or Uncertain Elements
Solving Multiplication Equations
Solving Oblique Triangles
Solving Simple Equations
Solving Systems of Transcendental Equations
Solving the Cable Equation
Solving the Subprime Loan Problem
Some Integer Functions Involving Division
Some Polar Plots
Sonar
Sonar and Radio Navigation Techniques
Sorted Evolutions of Cellular Automata
Sorting Networks
Sorting the Values of Two Number Theoretic Functions
Sound Examples
Sounds from Amplitude and Frequency
Sounds of Waveforms
Sources, Saddle Points, and Sinks in Vector Fields
Spacecraft Flying in Formation
Space Curve Tubes
Space-Filling Polyhedra
Space-Filling Polyhedra Based on a Truncated Octahedron
Space Filling with a Prismatoid
Space Filling with a Rhombic Dodecahedron of the Second Kind
Space Filling with Rhombic Dodecahedra and Cubic Close Packing
Space Filling with Trapezoid-Rhombic Dodecahedra and Hexagonal Close Packing
Space Filling with Two Kinds of Polyhedra
Space Filling with Two Kinds of Triacontahedra
Space-Quantization of Angular Momentum
Spacetime Diagram
Spanning Tree of Points on Sphere
Sparse Triangle Tiling
Spectral Realizations of Polyhedral Skeleta
Spectral Series of the Hydrogen Atom
Spectra of Substitution Systems
Spectra of the D-Lines of Alkali Vapors
Specular and Diffuse Reflection
Specularity in Computer Graphics
Speed, Motion, Pursuit, and Depth (Visual Depth Perception 15)
Speed of a Point on the Surface of the Earth
Speed of Sound
Speed, Pursuit, M/P Ratio, and Depth (Visual Depth Perception 16)
Speed Regulation for Visualization of Elementary Cellular Automata
Sphere and Torus Made of Segments
Sphere-of-Influence Graphs
Sphere Packing
Spheres on a Cylinder
Spheres on a Sphere
Spheres on a Truncated Icosahedron
Sphere with Tunnel Brachistochrone
Spherical Coordinates
Spherical Harmonic on Constant Latitude or Longitude
Spherical Harmonics
Spherical Pendulum
Spherical Plain Bearing
Spherical Product of Two Superformulas
Spherical Seismic Waves
Spherical Shell in a Magnetic Field
Spherical Trigonometry on a Gnomonic Projection
Spherical Yin-Yang
Sphericon
Spheroidal Foci
Spieker Center Triangle Construction
Spikey and Dodecahedron
Spikey Diamond
Spikey Gear
Spinning Double Yin-Yang
Spinning Ice Skater
Spinning Out Sine and Cosine
Spinthariscope
Spin-Weighted Spherical Harmonics
Spiral Explorer
Spiral Formations from Iterated Exponentiation
Spiral or Not?
Spirals!
Spirals of Squares
Spirals of Trapezoids
Spire Star with Rhombic Pyramids
Spirograph
Spirographic Orbit of a Star in Extended Mass Distribution
Spirolateral
Spline Sketchpad
Splitting a Cube
Spoked Wheel
Spontaneous Oscillations in Yeast Chemostat Cultures
Sports Seasons Based on Score Distributions
Spring Pendulum
Spring Return Button
Sputnik 1 Orbiting the Earth
Square Helmholtz Coils
Square-Hole Drill in Three Dimensions
Square Koch Fractal Curves
Square Koch Fractal Surface
Square Matrix Permutations
Square of a Sum
Square of the Sum of Two Numbers
Square Passing through Four Points
Square Pyramid and Its Right Triangles
Square Roots with Newton's Method
Square Shuffle Puzzle
Square Spiral for a Given Number of Divisors
Squaring a Binomial
Squaring a Number
Squaring Numbers
Squeeze Theorem
Stabilizers and Fixes of the Action of D4 on the Two-Colorings of a 2 by 2 Chessboard
Stabilizing a Mast
Stable Density Function
Stable Distribution Computed with the Zolotarev Integral
Stable Distribution Function
Stable Equilibria
Stable Lévy Process
Stable Marriages
Stackelberg Equilibrium Set in 2x2 Mixed Extended Games
Stacking Cannonballs
Stacking Dominoes to the Limit
Stacking Squares on a Circle
Stacks of Reflecting Plates
Stage Illumination Problem
Stairs from Cube
Stairstep Interpretation of Catalan Numbers
Stalactites
Stamping Out Shaped Disks
Standard American and European Options
Standard and Generalized Versions of the Monty Hall Problem
Standard Normal Distribution Areas
Standing Waves and Chords
Star Dipyramid
Star of David Theorem
Star Polygons
Starr Rose Mesh
Start-Up of a Plate Gas Absorption Unit
State Emphasis for 2,3 Turing Machine 596440
State of Light Polarization
State Population Growth
State Sequences in the Wolfram 2,3 Turing Machine
States from All Possible Elementary Cellular Automata
State Transition Diagrams for Modular Multiplication
State Transition Diagrams for Modular Powers
State Transition Graphs for Integer Torus Maps
Static Equilibrium and Triangle of Forces
Static Friction
Static Longitudinal Stability of Fixed Wing Aircraft
Stationary Precession of a Spinning Top
Statistical Mechanics of Money
Statistical Thermodynamics of Ideal Gases
Statistics Associated with Normal Samples
Steady-State Heat Conduction in a Cylinder
Steady-State Heat Transfer through an Insulated Wall
Steady-State Temperature Distribution in Conducting Square
Steam Distillation of a Mixture of Hydrocarbons
Steam Locomotive Drive
Stefan-Boltzmann Law
Steffen's Flexible Polyhedron
Steiner Chain Inversion
Steiner Chain of Circles
Stella Octangula
Stellar Luminosity
Stellar Nucleosynthesis
Stengel's Theorem
Step and Impulse Response of a Second-Order System
Step Response of a Manometer
Step Response with a P Controller
Stepwise Trisection of a Square
Stereographic Projection
Stereo View without Glasses
Stern-Brocot Tree
Stewart's Theorem
Stirling Numbers of the First Kind
Stirling Numbers of the Second Kind
Stirling's Triangles
Stochastic Resonance
Stochastic Rock-Paper-Scissors Population Dynamics
Stochastic Simulation of Test Results with Guessing
Stock Market Returns by Party
Stock Price Envelopes
Stock Price Probability with Stable Distributions
Stock Price Simulation Using Stable Random Variables
Stokes Boundary Layer
Stokes Parameters for Superposition of Two Slightly Noncollinear Polarized Beams
Stokes Parameters for the Superposition of Two Slightly Noncollinear Light Beams with Orthogonal Polarizations
Stolarsky Approximations to Popular Means
Straight Cylinder Engine
Straightening Up an Image
Strange Attractor of Gumowski-Mira
Strang's Strange Figures
Stratification as a Device for Variance Reduction
Streamlines for Laminar Flow Past a Rotating Solid Cylinder
Streptohedron and Trapezohedron
Stress Distribution in a Circular Plate with Concentrated Radial Loadings
Stress Propagation in a Truss Bridge
String Matching Automaton
Stripping a Hydrocarbon from Oil Using Live Steam
Stripping and Rectifying Cascades for Isopropyl Acetate Chemistry
Strips of Equal Width on a Sphere Have Equal Surface Areas
Stroboscopic Photograph of a Bouncing Ball
Strong RSA Cryptosystem
Structure of Binary Numbers
Student's t-Distribution
Study of a Four-Stage Batch Distillation Column
Study of Successive First-Order Reversible Reactions
Study of the Dynamic Behavior of the Lorentz System
Study of the Dynamic Behavior of the Rossler System
Styles of Faulting
Styling Line Caps and Line Joins
Stylized Pie and Bar Charts for Fast Food Nutrition
Subrange Data View
Subrogation
Substituted Pyrimidines and Purines
Substitution System Defined by Splitting Each Cell into Four
Substitution System Defined by Splitting Each Cell into Nine
Substitution Systems for Elements of Two Kinds
Substitution Systems in Two Dimensions
Subtraction Bingo with Negatives
Subtraction Bingo with No Negatives
Subtriangles Formed by Concurrent Lines Parallel to the Sides of a Triangle
Successes and Failures in a Run of Bernoulli Trials
Successive Averages of the Vertices of a Polygon
Successive Derivatives of a Polynomial
Successive Differences and Accumulations of the Jacobi Symbol
Successive Differences of Sequences
Successive Finite Difference Triangle
Successive Truncations of a Polygon
Successive Truncations of a Polygon, Stacked in 3D
Sudoku Game
Sugarscape: Agent-Based Modeling
Summation by Parts
Summing an Arithmetic Progression
Sum of a Geometric Series
Sum of an Arithmetic Progression
Sum of a Telescoping Series
Sum of a Telescoping Series (II)
Sum of Exterior Angles of a Polygon
Sum of Odd Numbers
Sum of Reciprocals of Triangular Numbers
Sum of the Alternating Harmonic Series (I)
Sum of the Alternating Harmonic Series (II)
Sum of Three Triangular Numbers
Sums of Consecutive Integers
Sums of Generalized Cantor Sets
Sums of Sine Waves with Several Step Sizes (Sawtooth or Square Approximations)
Sums of Squares of Segments Created by a Pedal Triangle
Sunny Random Rays
Sunshine Logo
Superellipse Shapes
Superellipsoid Shapes
Superimposed Gaussians
Super Moiré Patterns
Superposition of Quantum Harmonic Oscillator Eigenstates: Expectation Values and Uncertainties
Superposition of Sound Waves
Superposition of Standing Waves
Superposition of Two Sine Waves with Sum and Beats
Superposition of Waves
Superquadrics
Supersymmetric Partner Bessel Functions
Supersymmetry for the Square-Well Potential
Supplemental Jurisdiction
Supply and Demand
Supremum of an Increasing Bounded Sequence
Surface Area Increase by Size Reduction
Surface Area of a Solid of Revolution
Surface Kinetics with Pore Diffusion Resistance
Surface Morphing
Surface Property Editor
Surfaces and Gradients
Surfaces of Constant Tisserand Parameter
Surface Tension: Walking on Water
Surface with Branch Cuts
Surplus Production in Logistic Growth
Surplus Production Models and Equilibrium Harvest
Surveyor-Hunveyor Tripod
Survival of the Quasispecies
Swapping Qubit States
Swapping Rule and Initial Condition
Swash Plate Drive
Sweet Heart
Swimming in Curved Space
Swim, Swim and Walk, or Walk?
Swinging Double Slider-Crank
Swirl Arcs
Swirl Points
Swirls
Swirly
Swiss Compass
Sydler's Dissection of a Hill Tetrahedron into an Isosceles Triangular Prism
Sydler's Dissection of a Pyramid into a Prism
Symbolon Hemisphere
Symbol Rotation Patterns
Symmetrical Spline Curves
Symmetric Chaos in the Complex Plane
Symmetric Compositions of Rhombic Dodecahedra of the Second Kind
Symmetric Decompositions of a Rhombic Triacontahedron
Symmetries of a Prism
Symmetries of a Space-Filling Polyhedron with 54 Faces
Symmetrizing Positive Random Variables
Synchronization of Coupled Phase Oscillators
Synchronizing Pendulum Clocks
Synchrotron Radiation
Synergistic Enzyme Hydrolysis of Cellulose in Biomass
Synthetic Division (Ruffini's Rule)
Synthetic Legal Precedent Structures: Feature Distance
Synthetic Legal Precedent Structures: Lévy Flight
System Bandwidth for Cascaded Amplifiers
System of Pendulums: A Realization of the Sine-Gordon Model
Szegö Curve
Table Legs Made of a Single Bent Bar
Table of Dirichlet Characters
Table of Nuclides
Table of Primes
Tag Systems
Tail Areas under Chi-Squared Distributions
Tail Conditional Expectations
Tail Rotor of a Helicopter
Takagi Curve
Taking Apart a Cube
Tangent Chord Angle
Tangent Circle and Arbelos
Tangent Circles
Tangent Circles and Parallel Diameters
Tangent Circles to Two Circles
Tangent Lines to a Conic Section
Tangent Lines to a Parabola
Tangent Lines to Exponential and Logarithmic Functions through the Origin
Tangent Line Using Many Different Limit Configurations
Tangent ODE Chase
Tangent Planes on a 3D Graph
Tangent Planes to Quadratic Surfaces
Tangent Points on a Semicircle
Tangents to a Circle
Tangents to an Excircle
Tangents to the Circumcircle at the Vertices
Tangents to the Incircle from a Point on the Circumcircle
Tangent to a Curve
Tangent to a Surface
Tangram
Tangram (Variation)
Tank Equilibria
TARP Toxic (Illiquid) Assets Pricing Model
Tautochrone Problem
Taxicab Geometry
Taxicab Voronoi Diagram by a Cellular Automaton
Tax Rates and Tax Revenue
Taylor Expansions with Noninteger Number of Terms
Taylor Polynomials
Taylor Polynomials Approximated by Interpolations
Taylor Series
Taylor's Hinged Dissection of One Parallelogram into Another
Telescopic Tower
Television Monitor Based on a Cathode Ray Tube
Temperature Control of a Batch Fermentor
Temperature Dependence of Dehydrogenation of Ethyl Benzene to Styrene
Temperature-Dependent Rotational Energy Spectrum
Temperature Propagation in a Thin Steel Rod
Temperature Scales
Tempestuous Teapot
Templator Model of Chemical Self-Replication
Ten Billion Barrel Puzzle
Tension in a Hanging Chain
Ten Year Calendar
Terminal Wealth Optimization with Power and Log Utility
Term Structure of Interest Rates
Test Your Spatial Visualization Abilities
Tetrads
Tetrahedra in a Hexagonal 3D Net
Tetrahedral Animated Ring
Tetrahedral Spine
Tetrahedron to Cuboctahedron
Tetrahedron Volume
Tetraviews of Elementary Functions
Tetrix Viewpoints
Text Positioning in Three Dimensions with Nested Transformations
Texture Perception Patterns
Thales' Theorem
The 12 Coin Weighing Problem
The 12x12 Queens Problem
The 2001 CSO Mortality Tables
The 2008 Convention Speeches
The 2-(64,8,1) Design
The 26-Sided Unilluminable Room
The 29 Tetrachords and Other Set Classes
The 3n+1 Problem
The 4 Puzzle
The 57-Cell
The 6×6 Points Connection Problem
The Acceleration of a Two-Block System
The Action of the Modular Group on the Fundamental Domain
The Aeolipile
The Aging Spring
The Alcubierre Warp Drive
The Alexander Polynomial of a Knot
The Alpha and Beta Components of the Hodgkin-Huxley Model
The Ancestral Genetic Code Cube
The Andromeda Paradox
The Arbelos
The Arcsine Transformation of a Binomial Random Variable
The Area between Two Circles
The Area of a Triangle as Half a Rectangle
The Area of a Triangle inside the Incircle
The Area of a Triangle, its Circumradius, and the Perimeter of its Orthic Triangle
The Area of a Triangle Using a Determinant
The Area of Part of an Isosceles Triangle
The Area of the Contact Triangle
The Area of the Excentral Triangle
The Area of the Incentral Triangle
The Area of the Pedal Triangle of the Centroid
The Argument Principle in Complex Analysis
The Arithmetic-Geometric Mean Inequality (I)
The Arithmetic-Geometric Mean Inequality (II)
The Arithmetic-Geometric Mean Inequality (III)
The Arithmetic-Logarithmic-Geometric Mean Inequality
The Bakos Compound
The Banach-Tarski Paradox
The Base-Phi Number System
The Basics of Two-Dimensional Vectors
The Begonia Point
The Belousov-Zhabotinsky Reaction
The Best-Fit DDA Algorithm
The Binomial Inequality
The Birthday Problem
The Birthday Problem and Some Generalizations
The Bivariate Normal and Conditional Distributions
The Bivariate Normal Distribution
The Black-Scholes European Call Option Formula Corrected Using the Gram-Charlier Expansion
The Block Pushing Game
The Bomb Problem
The Boundary of Periodic Iterated Function Systems
The Bouniakowsky Conjecture
The Bricard 6R-Mechanism
The Broken Stick in n Dimensions
The Buchberger Gröbner Basis Algorithm
The Butterfly Theorem
The Cake Icing Puzzle
The Cantor Sequence with Bits
The Carpets Theorem
The Casimir Effect
The Cauchy-Schwarz Inequality for Vectors in the Plane
The Causal Interpretation of Quantum Tunneling through a Square Barrier and Well
The Causal Interpretation of the Nonlinear Schrödinger Equation: An Analytic Example
The Causal Interpretation of the Stern-Gerlach Experiment
The Causal Interpretation of the Triangular Quantum Billiard
The Celestial Sphere
The Celestial Two-Body Problem
The Center and Radius of the Nine-Point Circle
The Central Limit Theorem
The Chinese Rings Puzzle
The Chi-Squared Distribution
The Chordal Theorem
The Chow-Ruskey Order 5 Venn Diagram
The Circle Covering Puzzle
The Circles of Descartes
The Circumcircle of the Orthocenter and Two Vertices and a Parallelogram
The Circumcircles of Four Triangles
The Coase Theorem
The Coffee Cooling Problem
The Colorit Puzzle
The Commutative Property of Multiplication
The Completion of Pappus' Chain of Circles
The Complex Exponential
The Complex Unit Circle
The Contact Triangle and Two Other Subtriangles
The Continuous Uniform Distribution
The Convergence Behavior of a One-Parameter Family
The Conway Circle
The Cram Game
The Cyclotron
The Definition of the Derivative
The Deltafunction as the Limit of Some Special Functions
The de Moivre-Laplace Theorem in Probability Theory
The Derivative and the Integral as Infinite Matrices
The Desargues Configuration
The Determinant Using Traces
The Difference between European Option Prices in the Black-Scholes and NIG Models Computed with the DFT
The Discrete Preisach Model of Hysteresis
The Disk Method
The Distance from the Circumcenter to an Excenter
The Distances to the Incenter and Excenters from the Circumcenter
The Distance to the Orthocenter
The Distance to the Orthocenter from a Vertex
The Distributive Property
The Divergence (Gauss) Theorem
The Divisor Sigma Function
The Doppler Effect
The Dow Jones Industrial Average Since 1928
The Drake Equation
The Duty to Settle
The Edgeworth Box
The Effect of Holding Period on Real Estate Investment Return
The Effect of the Spherical Harmonic Gravitational Potential on Satellite Orbits
The Effects of Some Simple Compositions on the Graph of a Function
The Efficient Dual-Limit Liability Insurance Contract
The Eigenvectors of a Random Graph
The Electroscope
The Envelope Theorem: Numerical Examples
The Equation of Time for a Sundial
The Erdös-Mordell Inequality
The Esscher Transform of the Densities of a Symmetric NIG Lévy Process
The Euclidean Algorithm
The Euclidean Algorithm and Simple Continued Fractions
The Eutrigon Theorem
The Excentral Triangle and a Related Hexagon
The Expanding Universe
The Exponential Distribution
The Extouch and Intouch Triangles of a Triangle
The Eyeball Theorem
The Eye-Pentagon Construction
The Final Number of Bacterial Cells
The First Lemoine Circle
The Fix of an Element of a Group Acting on a Two-Colored Figure
The Four-Number Game
The Four-Runner Problem
The F-Ratio Distribution
The Frequentist Intuition behind Assigning Probabilities
The Fuhrmann Circle
The Fundamental Theorem of Calculus
The Fundamental Theorem of Finite Abelian Groups
The Fundamental Theorem of Linear Programming
The Future Solar System
The Gambler's Ruin
The Game of Nim
The Game of Toads and Frogs
The Gamma Distribution
The Generalized Weierstrass-Riemann Functions
The Geometry of Integrating a Power around the Origin
The Geometry of Lagrange Multipliers
The Gergonne Point
The Goldman-Hodgkin-Katz Equation
The Gompertz Sigmoid Function and Its Derivative
The Gordon-Schaefer Model
The Greatest Fenced Area along a Barn
The Group of Rotations of the Cube
The Hagge Circle
The Hamming(7,4) Code
The Hazard of Multiple Comparisons
The Hazards of Propping Up: Bubbles and Chaos
The Headlight Effect
The Higgs Particle
The Hilbert Hotel
The Houses and Utilities Crossing Problem
The Hydraulic Press
The Hydrogen Atom in Parabolic Coordinates
The Hypergeometric Distribution
The Incircle and the Altitudes of a Triangle
The Integral Mean Value Theorem: An Illustration
The Integral Test and Error Estimation Using p-Series
The Interplanetary Magnetic Field (Parker Spiral)
The Intersection of an Angle Bisector and a Perpendicular Bisector
The Intersection of Circumcircles of Medial Triangles
The Intersections of Extended Cevians with Three Circumcircles of Subtriangles
The Itô Integral and Itô's Lemma
The Johnson Circles
The Josephus Problem in Both Directions
The Kappa Distribution
The Keynesian IS-LM Model
The Knight's Tour
The Kronig-Penney Model
The Law of Cosines
The Law of Mass Action
The Law of Sines
The Law of the Iterated Logarithm in Probability Theory
The Linear Josephus Problem
The Linear Josephus Problem in Two Directions
The Line through the Incenter and Circumcenter
The Locker Problem
The Log Normal Distribution
The Long Tail
The Lotka-Volterra Equations for Competition between Two Species
The Lunes of Hippocrates
The Malfatti Problem
The Mass Spectrometer
The Mating Dance of the Alexander Horned Spheres
The Maximum Rectangular Area under Different Curves
The Maxwell Speed Distribution
The Mayan Calendar
The Medial Triangle and Concurrency at the Nagel Point
The Menger Sponge
The Method of Common Random Numbers: An Example
The Midpoint Quadrilateral Theorem
The Midpoints of Two Cevians
The Minimal Model of the Complexity of Financial Security Prices
The Minkowski Sum of a Disk and a Polygon
The Minkowski Sum of Two Triangles
The Modified Dirichlet Function
The Monodromy Group of an Algebraic Function
The Moran Process
The Mug That Holds the Most Liquid
The Multiplication Parabola
The Murder Mystery Method for Identifying and Solving Exact Differential Equations
The Mystery of the Disappearing Square
The Mystic Rose
The Nagel Point
The Natural Logarithm is the Limit of the Integrals of Powers
The Negative Binomial Distribution
The Normal Distribution
The Normal Inverse Gaussian Lévy Process
The Number of Binomial Coefficients Divisible by a Fixed Power of a Prime
The Number of Distinct Real Roots of a Real Polynomial
The Number of Fixed Points in a Random Permutation
The Number of Partitions with a Given Largest Part
Théorčme de Pappus (French)
Théorčme de Pascal (French)
Theorem of the Equal Incircles
Theorem of the Owl's Eyes
Theorems of Pappus on Surfaces of Revolution
The Orthocenter of the Medial Triangle
The Orthocentroidal Circle
The OZI Rule in Meson Decay
The Paradox of Thrift in a Simple Stock-Flow Consistent Model
The Parametrized Szilassi Polyhedron
The Path of a Ray on a Cube
The Penrose Unilluminable Room
The Perfect Venn Diagram
The Perpendicular Bisector of a Chord
The Perpendicular Bisectors of a Triangle
The Persuasion Effect: A Traditional Two-Stage Jury Model
The Perturbed Rat
The Photoelectric Effect
The Pigeonhole Principle - Disk Coverings
The Pigeonhole Principle - Repunits
The Pigford Problem
The Poisson Process
The Polar Equations of Hearts and Flowers
The Polar Planimeter
The Power of a Test Concerning the Mean of a Normal Population
The Present Value of Future Gas Use
The Price Elasticity of Demand
The Price of a Call Option on Electrical Power
The Price-Terms Tradeoff
The Prime Number Theorem
The Primes Are Infinite
The Principle of Archimedes
The Prisoner's Dilemma
The Product of an Altitude and the Circumradius of a Triangle
The Product of the Circumradius and an Exradius
The Product of the Distances of the Incenter to the Vertices
The Product of the Inradius and Semiperimeter of a Triangle
The Product of the Inradius and Three Exradii
The Product of the Perpendiculars to the Sides of a Triangle
The Product of the Side Lengths of a Triangle
The Product Rule
The P-Series Theorem
The Purpose of the Law
The Puzzle of the Bridge
The Puzzle of the Two Bridges
The Quadrilateral of Centroids
The Quantum-Tobogganic Paths
The Quotient Rule
The Rabbits Game
The Radiation Pressure of Light
The Radii of Four Incircles
The Ratio of a Side of a Triangle to the Corresponding Side of the Orthic Triangle
The Ratio of Surface Area to Volume for a Cube and a Sphere
The Ratio Theorem
The r-Distribution
The Repeated Power Function
The Residue Curve Map (RCM) of a Ternary Mixture at Various Pressures
The Return Distribution of the Variance Gamma Process
Thermal Energy
Thermal Equilibrium
Thermal Expansion of Solids
Thermal Motion in a Solid
The Rogers-Ramanujan Continued Fraction and Generalized Elliptic Integrals
The Roots of Unity in the Complex Plane
The Rossler Attractor
The Rule 30 Fault Line
The Russian Option: Reduced Regret
The Schiffler Point
The Schwarzian Derivative of Iterated Functions
The Second Lemoine Circle
The See-Saw Lemma
The Semiperimeter and Tangents to Excircles
The Sensitivity of Page Rank to Connection Errors
The Seven Circles Theorem
The Seven Crystal Classes
The Simplest Non-Periodic Tiling
The Simultaneous Contrast Effect
The Smith Chart
The Solar Wind
The Solenoid
The Sound of Sine Waves
The Spatial Midpoint Quadrilateral Theorem
The Square of a Median in Terms of the Triangle's Sides
The Stains Cellular Automaton
The Statue of Regiomontanus
The Straight Line as a Roulette
The Structure of the Real Roots of a Quintic Polynomial
The Sum of the Circumradii of Three Subtriangles
The Sum of the Distances from the Orthocenter to the Vertices
The Sum of the Exradii Minus the Inradius
The Sum of the Harmonic Progression as an Integral
The Sum of the Interior Angles of a Triangle Equals 180 Degrees by Paper Folding
The Sum of the Perimeters of Three Subtriangles
The Sum of the Reciprocals of the Exradii
The Sum of the Square Roots of the Areas of Three Subtriangles
The Sum of the Squares of the Distances from the Vertices to the Orthocenter
The Sum of Two Cantor Sets
The Tangent Line Problem
The Taylor Circle
The Third Pedal Triangle of a Triangle
The Three-Chord Lemma
The Time-Dependent Electromagnetic Fields of a Relativistic Circular Current
The Topology of Costa's Minimal Surface
The Torricelli Point
The Trapezoidal Rule for Increasing Functions
The Tree of All Fractions
The Tree of Life
The Triangle Formed by the Centers of the Miquel Circles
The Triangle Inequality
The Triangles Cut Off by the Sides of the Orthic Triangle
The Triangles Formed by the Endpoints and Midpoints of Cevians
The Trinomial Distribution
The Troublesome Twelve Circle Problem
The "True" Self-Avoiding Random Walk in Z
The Turbine Pattern in the Game of Life
The Twin Circles of Archimedes and Their Families
The Two Triangles Theorem
The Ulam Prime Spiral
The Variance Gamma Process
The Venturi Effect
The Vibrating String
The Wheatstone Bridge
The Wire Problem
The Wolfram 2,3 Turing Machine
The Wolfram 2,3 Turing Machine Left-Compressed Evolution
The Wolfram 2,3 Turing Machine with Random Conditions
The x^x Spindle
Think Big, Really Big!
Threading Cylinders On a Torus Coil
Three-Asset Efficient Frontier
Three Circles Defined by Chords
Three Circumcenters Concyclic with a Fourth Point
Three Classes of Levers
Three Concyclic Sets of Points Associated with the Orthic Triangle
Three Cylinders in a Cube
Three-Dimensional Isotropic Harmonic Oscillator
Three-Dimensional Linear System of DEs
Three-Distance Theorem
Three Forks Puzzle
Three Hinged Dissections of Squares
Three Holes in a Cube
Three-Piece Dissection of a Hill Tetrahedron into a Triangular Prism
Three Points Determine a Circle
Three Polyhedra with Rectangular Nets
Three Rings of Half Rhombohedra
Three Sequences with Limit e
Three-Stage Batch Rectification with Constant Reflux
Three Subtriangles of a Triangle
Three-Uniform Hypergraphs
Three Views of the Logistic Map
Three Ways to Fold a Net into a Snub Disphenoid
Three Ways to Fold a Net of Eight Triangles
Three-Way Switches
Through Any Three Points
Throw a Hammer
Throwing a Baseball from the Outfield to Home Plate
Throw off a Cliff
Thrust Reverser
Tic-Tac-Toe
Tightly Packed Squares
Tile Murals
Tiles around a Swimming Pool
Tiles with Finite Convex Spectra
Tiling a Room Irrationally
Tilings of 1 by n Chessboards Using Squares, Dominos, and Triominos
Tiling Space with a Schmitt-Conway Biprism
Time and Weight in the Solar System's Planets
Time-Averaged Moiré Fringes
Time-Dependent Scattering in the Causal Interpretation of Quantum Theory
Time-Dependent Superposition of 2D Particle-in-a-Box Eigenstates
Time-Dependent Superposition of Harmonic Oscillator Eigenstates
Time-Dependent Superposition of Particle-in-a-Box Eigenstates
Time-Dependent Superposition of Rigid Rotor Eigenstates
Time-Displacement Plots
Time Evolution of a Four-Spring Three-Mass System
Time Evolution of a Quantum Free Particle in 1D
Time Evolution of a Quantum Free Particle in 2D
Time-Evolution of a Wavepacket in a Square Well
Time Evolution of Quantum-Mechanical Harmonic Oscillator with Time-Dependent Frequency
Time Series from Cellular Automata
Tinbergen-Solow Production Function
Tin Box with Maximum Volume
Tip Calculator
Tippe Top
Tipping Point of a Cylinder
Titchener Illusion
Titius-Bode Law
Titration of Weak Acids with Strong Bases
Toolbox
Tornado Activity in the U.S. 1950-2007
Toroidal Wrapping
Torque Exerted Opening a Door
Torricelli's Law for Tank Draining
Torricelli's Theorem
Torsional Pendulums
Torsion Pendulum
Torus Coil
Torus in Nil-Space
Torus Knot
Total Areas of Alternating Subtriangles in a 2n-gon
Total Differential of the First Order
Total Internal Reflection
Totalistic Automata with Hexagonal Neighborhoods
Totalistic Diffusion Elementary Reaction Automata
Totalistic K3 R 1/2 Cellular Automata
Total Probability and Bayes's Theorem
Touching Planar Polycubes
Touching Polygon Concurrence
Touch Tones
Tournaments
Tower of Dodecahedra
Towers of Hanoi
Toy Model of an Inflating Wormhole
Toy Model of a Reissner-Nordstrom Wormhole
tq-System Explorer
Tracing Doodle
Tracking and Separation (Visual Depth Perception 11)
Trader Dynamics in Minimal Models of Financial Complexity
Traffic Lights
Trajectories of Deformable Self-Propelled Particles
Trajectories of the Logistic Map
Trajectory of a Point in an Iterated Torus Map
Trajectory of a Soccer Ball
Transfer Function from Poles and Zeroes
Transformable Letterforms
Transformación de funciones (Spanish)
Transformation of s-Reflection Coefficient between Normal and Oblique Incidence
Transformations of Complex, Dual, and Hyperbolic Numbers
Transformations of Relativistic Temperature: Planck-Einstein, Ott, Landsberg, and Doppler Formulas
Transforming One Rhombic 210-hedron into Another
Transient Behavior of a Couette Flow
Transient Diffusion in a Bilayer System
Transient Heat Conduction with No Steady State
Transient Response of a Semiconductor Laser
Transition Matrices of Markov Chains
Transition Strengths of Alkali-Metal Atoms
Translucent Caliper
Transmission and Reflection Coefficients of Quantum Particles
Transversal Oscillations and Resonance in a Four-Spring, Three-Mass System
Transverse Standing Waves
Traveling Salesman Game
Traveling Salesman Problem
Travelling Pulses (Wave Packets)
Travel with Waiting Time and Distance Distributions
Traversing the Inside of a B-Spline Curve
Trawler Problem
Tree Bender
Tree Branching in 4D
Tree Form of Recursive Function Evaluation Steps
Triacontahedron Stellations
Triangle Altitudes and Circumradius
Triangle Altitudes and Inradius
Triangle Centroid
Triangle Circles and Spirals
Triangle-Closed Planar Sets
Triangle Fractal
Triangle Interior
Triangles: Acute, Right, and Obtuse
Triangles in the Coordinate Plane
Triangles in the Poincaré Disk
Triangles on a Sphere
Triangles: Scalene, Isosceles, and Equilateral
Triangle to Parallelogram
Triangular Koch Fractal Surface
Tricks for Finding Small Divisors of Large Integers
Tridimensional Trivalent Graph
Tries
Trigonometric Functions Illustrated
Trigonometric Functions with Four Parameters
Trigonometric Integral Functions
Trigonometric Sums as Parametric Curves
Triple Touch 3D Pad
Triple Touch Trace Pad
Truchet Tiles
Truck Packing
Truel World
Truncated Dodecahedron to Great Rhombicosidodecahedron
Truncated Icosahedron to Great Rhombicosidodecahedron
Truncated Octahedra Outer Totalistic Cellular Automaton
Truncated Rhombic Triacontahedron
Truth Tables
Tube in a Sphere
Tube Made of Segments
Tumor Growth Model
Tuned Mass Damper
Tuning Fork
Turing Machine Causal Networks
Turing Machine Enumeration
Turing Machine Runs
Turing Machines with Cyclic Tapes
Turing Snakes
Turmites
Turning a Punctured Torus Inside Out
Turning a Punctured Torus Inside Out (Variation)
Turn Off the Lights
Turtle-Graphics
Twelve Decagons
Twelve-Tone Equal-Tempered Scales
Twenty Dodecahedra and a Golden Rhombohedron
Twenty Dodecahedron Towers
Twenty Prolate Golden Rhombohedra
Twilight Calendar
Twin Pythagorean Triples
Twisted Möbius Klein Bottle
Twisted Polygonal Torus
Twisting Rose Surfaces
Two-Asset Markowitz Feasible Set
Two-Body Orbits with Lagrange Point L4
Two-by-Two Linear Systems Problem Generator
Two Circular Windows
Two-Color Left/Right Mobile Automata
Two-Color Mobile Automata with Neighbors
Two Convex Polyhedra with the Same Net
Two Dice with Histogram
Two-Dimensional Cellular Automata from One-Dimensional Rules
Two-Dimensional Fractional Brownian Motion
Two-Dimensional Linear Systems
Two-Dimensional Optimal Control Problem
Two Geometric Views of Newton's Method
Two Hinged Dissections
Two Hinged Dissections of a Gnomon to a Square
Two Hinged Dissections of a Greek Cross to a Square
Two Integral Mean Value Theorems
Two Integral Mean Value Theorems of Flett Type
Two Intersecting Cylinders
Two Isothermal Continuous Stirred-Tank Reactors with a Recycle Stream
Two Lines Determine a Circle
Two Midpoints of a Trapezoid
Two Models of Projective Geometry
Two-Octagons-into-One Dissection in Two Steps
Two Operations on Rhombic Solids
Two Points Determine a Line
Two Polyhedra with D2 Symmetry
Two Proofs of the Equality of the Volumes of a Box and a Parallelepiped
Two-Regime Threshold Autoregressive Model Simulation
Two Solids Made from Golden Rhombic Solids
Two-Soliton Collision for the Gross-Pitaevskii Equation in the Causal Interpretation
Two-State Protein Melting Curve (N, P, T Ensemble)
Two Stellations of the Rhombic Triacontahedron
Two-Step Equations
Two Tangent Lines to a Parabola
Two Trains Puzzle
Two Triangles of Equal Area on Either Side of an Angle Bisector
Two Ways of Folding a Sphinx into a Triangular Dipyramid
Two Ways to Fold a Net into a Biaugmented Triangular Prism
Two Wheel Belt
Typeset Sierpinski Sieve
Types of Angles
Typical Predicate Calculus Statements
Ullman's Theorem in Two Dimensions
Unbiased and Biased Estimators
Uncertain Calculus: Mean versus Likelihood Mean
Uncertainty of Measurement and Areas Over and Under the ROC Curves
Uncertainty of Measurement and Diagnostic Accuracy Measures
Understanding 2D Reflection
Understanding 2D Rescaling
Understanding 2D Rotation
Understanding 2D Scaling
Understanding 2D Shearing
Understanding 2D Translation
Understanding 3D Reflection
Understanding 3D Rescaling
Understanding 3D Rotation
Understanding 3D Scaling
Understanding 3D Shearing
Understanding 3D Translation
Understanding Polarization with an Analogy
Understanding Runge-Kutta
Un Divisor Silábico (Spanish)
Unfold Bit Mask along Axes
Unfolding a Cone
Unfolding a Dodecahedron to Produce a Stage
Unfolding Fractal Structures
Unfolding Polyhedron Nets
Unforced, Damped, Simple Harmonic Motion
Uniform Acceleration within Special Relativity
Uniform Circular Motion
Uniform Circular Motion Using a Tetherball
Uniform Continuity
Unilateral Accident Model
Unit Converter
Universal Joint
Universal Rhombic Dodecahedron
Unsöld's Theorem
Unsteady Heat Transfer over a Porous Flat Plate
Unsteady-State Conduction of Heat in Two Dimensions
Unsteady-State Diffusion, Convection, and Reaction in a Fluid Film
Unsteady-State Diffusion in a Slab
Unsteady-State Evaporation in an Infinite Tube
Unsteady-State Flow in a Tube by Orthogonal Collocation Method
Unsteady-State Interphase Diffusion
Unweaving the Rainbow
Upper-Tailed Critical Values of the Fisher-Snedecor Distribution
Upside Down Words
Urn Problem
U.S. City Tours
Using Common Denominators to Add Fractions
Using Different Types of Filters
Using Rule 30 to Generate Pseudorandom Real Numbers
Using Sampled Data to Estimate Derivatives, Integrals, and Interpolated Values
Using the Area under a Curve to Compute a Limit
Using the Distance Transform
Using ViewMatrix
Using Zeta Zeros to Compute a Summatory Liouville Function
Using Zeta Zeros to Compute the Chebyshev Psi Function
Using Zeta Zeros to Compute the Mertens Function
Using Zeta Zeros to Count the Squarefree Integers
Using Zeta Zeros to Tally Squarefree Divisors
Using Zeta Zeros to Tally the Euler Phi Function
U.S. Net and Total Patent Citation Flows, 1963-2002
U.S. Net Patent Citation Flows, 1963-2002
Valence Shell Electron Pair Repulsion (VSEPR) Theory
Valuation and Management of Bonds
Value Added Growth Model
Value of the Zeta Function along the Critical Line
Values of Combinatorial Functions
Van Aubel's Theorem for Quadrilaterals
Van Aubel's Theorem for Triangles
Van de Graaff Generator
Van der Pol Oscillator
Van der Waals' Equation of State for a Non-Ideal Gas
Van der Waals Isotherms
Van der Waals Isotherms for Real and Ideal Gases
Van der Waals Surface
Vapor-Liquid Data for Acetic Acid-Water Mixture: Effect of Gas Phase Dimerization
Vapor Liquid Equilibrium Data for an Ideal Binary Mixture
Vapor Liquid Equilibrium Data in Tert-Amyl Methyl Ether (TAME) Production
Vapor Pressure and Density of Alkali Metals
Variable States of Polarization Incident on a Wave Plate
Variance-Bias Tradeoff
Variance-Gamma Distribution
Variational Calculations on the Helium Isoelectronic Series
Variational Principle for Quantum Particle in a Box
Variation of Conversion with Reaction Order in a Plug Flow Reactor
Variation of the Equation of Time with the Precession of the Equinoxes
Variations on a Tetrix
Variations on State Transition Diagrams of Elementary Cellular Automata
Varying Moving Averages
Vector Addition in a Plane
Vector Addition is Commutative
Vector Field Flow through and around a Circle
Vector Fields: Plot Examples
Vector Fields: Streamline through a Point
Vector Field with Sources and Sinks
Vectorgrams of Conway's Look and Say Sequence
Vector Median Filter
Vector Projection
Vector Projections
Vector Rotations in 3D
Vectors in 3D
Velocity of a Falling Object
Velocity Stream Lines from Superposition of Elementary Fluid Flows
Velocity versus Displacement Plots in Uniformly Accelerated Linear Motion
Venn Diagrams
Venn Diagrams for Two Sets
Venn Diagrams for Two-Variable Boolean Logic Circuits
Vernam Cipher (One Time Pad)
Vertical Blinds
Vertical Pendulum Seismometer
Vibrating Circular Membrane
Vibration of Masses on a String
Vibrations of a Rectangular Membrane
Vickers Hardness of Materials
Vičte's Nested Square Root Representation of Pi
Vieth-Müller Circles (Visual Depth Perception 8)
View of Our Solar System
Views of the Costa Minimal Surface
Villarceau Circles
Virtual Abacus
Virtual Flowers with Crispate Petals
Virtual Image Scanning
Virtual Oscilloscope
Visible Divisibility Tests
Visual Computation of an Integral
Visual Computation of an Integral (II)
Visual Computation of Three Geometric Sums
Visual Encryption Pad
Visualizations of a Recursive Sequence
Visualization Tools for the Design of Digits
Visualizing Atomic Orbitals
Visualizing Conway's Game of Life
Visualizing Correlations
Visualizing Euler's Constant
Visualizing Jerk: Change of Acceleration
Visualizing Legal Rules: A Homicide Case
Visualizing Legal Rules: Battle of the Forms
Visualizing Simple Addition and Multiplication
Visualizing Square Root and Absolute Value
Visualizing Superellipses
Visualizing the Coarse Graining of a Cellular Automaton
Visualizing the Exact Median Rank
Visualizing the Gradient Vector
Visualizing the Poisson Kernel
Visualizing the Riemann Zeta Function
Visualizing the Solution of Two Linear Differential Equations
Visualizing Vector Fields
Visual Proof of Two Integrals
Viviani's Theorem
Volatility Surface in the Heston Model
Volpert Graph
Volpert Graph of Chemical Reactions
Volume of a Parallelopiped
Volume of Cones, Pyramids, and Spheres
Volume of Cylinders
Volume of Triangular Prisms
Volumes of Revolution Using Cylindrical Shells
Volumes of Solids of Revolution: Shell Method
Volumes Using the Disc Method
Volume Units of Measurement
Von Mises Yield Criterion
Von Neumann Exponent Calculator
Voronoi Diagrams
Voronoi Image
Voronoi Kaleidoscope
Voter Model
Voxel Distortion
Wagon Wheel Approximation of Pi
Walking with an Umbrella
Wallis Formula
Wallis Sieve Pi Approximation
Wallpaper Patterns
Walrasian Equilibrium or Disequilibrium
Walsh Basis Functions
Wannier Representation for Tight-Binding Hamiltonian of a Periodic Chain with N Sites
Waterman Polyhedra
Water-Pouring Problem
Watt's Curve
Waveforms and Spectrograms
Wave Functions of Identical Particles
Wave Generator
Wave Interference
Wavelet Shrinkage Denoising
Wave Packet Dispersion
Wave Packet Dynamics
Wavepacket for a Free Particle
Wave-Particle Duality in the Double-Slit Experiment
Waves at Different Angles
Waves from a Moving Source
Waves from a Moving Source in 3D
Wave Spring
Wave Spring Washer
Ways of Stepping One, Two, or Three Stairs Up a Stairway
Weak Values in an Electronic Mach-Zehnder Interferometer
Weather Browser
Wedding Ring
Wedge Operated Jig
Weibullian Inactivation Rate as a Function of Temperature
Weibullian Inactivation Rate as a Function of Time
Weibull Statistics for Fracture Data
Weighted Random Graph
Weight of a Person Riding in an Elevator
Weitzenböck's Inequality
Wertheimer-Koffka Ring
What Color Is the Bear?
What Color is the Moving Circle?
Wheel Illusion
Wheels on Wheels on Wheels
When Do the Three Hands of an Analog Clock Overlap?
Where Are My Roots?
Where in Pi Is My Prime?
Which Country Is Bigger?
Whirling Polygons
Whitney Umbrella
Why a Number Is Prime
Why Location Matters: The Bid Rent Curve
Wigner Function of a Canonical Ensemble of Harmonic Oscillators at a Given Temperature
Wigner Function of Harmonic Oscillator
Wigner Function of Two-Dimensional Isotropic Harmonic Oscillator
Wilson's Theorem in Disguise
Wimshurst Machine
Windmill Model
Window Opacity Controlled by Cellular Automata
Wire-and-String Puzzles
WireWorld Pixel Editor
Wisdom of Crowds
Witch of Agnesi
Wittenbauer's Parallelogram
Wolfram's 2,3 Turing Machine Step-By-Step
Wooden Bucket
Wooden Stand for a Christmas Tree
Word Memory Game
Work-Hardening Models
Work in an Attractive Inverse-Square Field
World Map Projections
Woven Panel
Wrapping a Torus
Wrapping the Number Line around the Unit Circle
Wristwatch as a Compass
Writing Systems
Wunderlich Curves
XFT2D: A 2D Fast Fourier Transform
XFT: An Improved Fast Fourier Transform
X-Ray Reflection by Planes of Atoms in a Cubic Lattice
Yates's Trisector
Yield and Selectivity for Series-Reaction Network in a CSTR and a PFR
Yin Yang
Yin Yang by Fan Fun
Young Tableaux
Your Favorite Necktie
Your Town's Namesake
Zebra-Painted Egg
Zeckendorf Representation of Integers
Zeno's Paradox: Achilles and the Tortoise
Zeolite A
Zeolite Unit Cell Based on Cuboctahedron
Zernike Coefficients for Concentric, Circular, Scaled Pupils
Zernike Polynomials and Optical Aberration
Zeros of a Perturbed Polynomial
Zeros of Truncated Series of Elementary Functions
Zigzagging Over Tables of Numbers
Zipf's Law Applied to Word and Letter Frequencies
Zipf's Law for Cities
Zipf's Law for U.S. Cities
Zonogons: Minkowski Sums of Line Segments in 2D
Zonohedral Torus
Zonohedron Turned Inside Out
Zoom Characters
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