Playing with Stellations of the Icosahedron

Play with the facets and cells of stellations of the icosahedron. Cut off a segment (along an axis through a vertex, edge, or face) and see inside. Different color schemes help identify the symmetry, facets, and cells of icosahedral stellations.


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


What is a stellation? Roughly, it's a bit of space contained by the face planes of the icosahedron that has the same rotational symmetry as the icosahedron. But that's not all. The planes containing the faces divide space into regions and they divide other face planes into polygons. The bounded regions are called "cells" and the bounded polygons "facets". But whether any symmetric selection of cells or any symmetric configuration of facets deserves to be called a stellation is, to some extent, a matter of taste. See [1] and [2] and use this Demonstration to get started on your own definition. Snapshots 1 and 2 show two stellations colored according to the cells and facets. Snapshots 3 and 4 show cut-away segments, the 4th revealing the faceting of the stellation.
This Demonstration was adapted from Mathematica code originally developed by Michael Rogers for a talk at Colby College in 1993.
[1] H. S. M. Coxeter, P. Du Val, H. T. Flather, and J. F. Petrie, The Fifty-Nine Icosahedra, Toronto: University of Toronto Press, 1938. Reprint, New York: Springer-Verlag, 1982.
[2] G. Inchbald, "In Search of the Lost Icosahedra," The Mathematical Gazette 86(506), July 2002 pp. 208-215.
    • 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 © 2018 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+