9717

Dynkin Diagrams

This Demonstration lets you create and modify multiple Dynkin and Coxeter–Dynkin diagrams. Some topological patterns can be recognized for a known simple Lie group (up to rank 8) and its designated type, including finite, affine, hyperbolic, and very extended. You can use the dropdown "Dynkin diagrams" to get various diagrams and geometric permutations.
You can also modify, recognize, and name the (binary) Coxter–Dynkin geometric permutations on uniform polyhedra and name their Wythoff construction operator. These permutations are indicated by filled nodes. When more than one diagram is created, they can be interpreted as a geometric Cartesian product, such as a duoprism.
The Cartan matrix that defines the Lie algebra is calculated directly from the last diagram entered.

SNAPSHOTS

  • [Snapshot]
  • [Snapshot]
  • [Snapshot]

DETAILS

The main window is a clickable pane.
Clicking more than one and a half cells away from all nodes creates a new diagram.
Clicking less than one and a half cells away from all nodes creates a node and a single line to its nearest neighboring node.
Clicking a node in the last diagram created toggles the node fill on/off and changes the geometric permutation (unless it is an already filled node that is adjacent to an unlinked node, which implies the creation of an affine loop).
Clicking a line between nodes in the last diagram created toggles the symmetry angle or the directionality between its nodes. When more than one diagram is in the clickable pane, only the last diagram is active. Clicking previous diagrams creates a new overlapping diagram.
For better Dynkin diagram topology recognition, select the affine level early and create the linear (A-type diagram) nodes before any off-linear nodes (for D and E diagrams).
For more information on Dynkin diagrams, see the Wikipedia entry for "Dynkin Diagram".
For more information on simple Lie groups, see the Wikipedia entry for "Simple Lie Group".
For more information on Coxeter–Dynkin diagrams, see the Wikipedia entry for "Coxeter–Dynkin Diagram".
For more information on Coxter–Dynkin geometric permutations on uniform polyhedron and their Wythoff construction operator naming, see the Wikipedia page for "Uniform Polyhedron".
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.









 
RELATED RESOURCES
Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2014 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+