Schoenberg Plane-Filling Curve

The definition of the Schoenberg curve begins with a piecewise sawtooth-like function whose values lie between 0 and 1. The plane-filling curve is defined parametrically using sums of scaled copies of the original function. In the limit, the Schoenberg curve touches every point in the unit square.


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


Snapshot 1: first term of the summation, the upper-right corner of which is the point ; the line color changes from blue to red to help the eye track the progress of the curve from the lower left-corner to the upper-right corner
Snapshot 2: the Schoenberg curve differs from such plane-filling curves as the Peano curves, Hilbert and Moore curves, Lebesgue curve, and so forth, in that the approximations intersect themselves and indeed double back on themselves
Snapshot 3: greater iterations reach more points in the unit square
I. J. Schoenberg, "On the Peano Curve of Lebesgue," I. J. Schoenberg: Selected Papers, Vol. 1 (C. de Boor, ed.), Boston: Birkhäuser, 1988.
    • 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.

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 © 2017 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+