The Stellated Icosahedra

by
Roman E. Maeder
ETH Zurich
Institute of Theoretical Computer Science
ETH Zentrum IFW
8092 Zurich
SWITZERLAND
maeder@inf.ethz.ch

Mathematica in Education
Volume 3 Number 1
Winter 1994
(c) TELOS/Springer-Verlag

Abstract

The enumeration of all stellations of the icosahedron was accomplished in 1938. The geometric constructions and combinatorial algorithms used can easily be programmed
in Mathematica. Its symbolic  and graphic capabilities make it well suited to render the solids in a variety of formats.

Reference

Title

The Stellated Icosahedra

Author

Roman E. Maeder

Summary

Contains the code for the examples in the article "The Stellated Icosahedra".

Package Version

1.1

Copyright

ã Copyright 1994, Roman E. Maeder.

Permission is granted to distribute this file for any purpose except for inclusion in commercial software or program collections. This copyright notice must remain intact..
  
The use of this code for the preparation of graphics images for publication or commercial purposes requires prior written permission by the author.

History

Version 1.1 for Mathematica in Education, Winter 1994.

Keywords

Stellated Icosahedra, Stellation, Polyhedra

Source

Maeder, Roman E. 1993. The Stellated Icosahedra. Mathematica in Education, 3(1).

Mathematica Version

2.2

Requirements

Icosahedra.m

Graphics`Polyhedra`

Graphics`Shapes`

Init

You may have to add the directory with Icosahedra.m to your $Path or do something like this:

SetDirectory["full path name of directory"];

Imports

Needs["Icosahedra`"]

Needs["Graphics`Polyhedra`"]

Needs["Graphics`Shapes`"]

Option Settings

SetOptions[GraphicsArray, GraphicsSpacing->0];
SetOptions[Graphics3D, Boxed->False];

Auxiliary Functions

Show 3D image and planar map next to each other.

plotit[n_, opts___] :=
GraphicsArray[
	{ Icosahedra[n, opts],
      FaceGraphics[n, opts, Frame->True, FrameTicks->None]}
]

Examples from the Article (and More)

Stella Octangula

The stella octangula is the only stellation of the octahedron.

tetra = Polyhedron[Tetrahedron];
dual  = AffineShape[ tetra, {-1, -1, -1} ];

Show[{tetra, dual}];

Regular Star Polygons

The first three of these are the only stellations of the dodecahedron. The fourth is a stellation of the icosahedron known to Poinsot. All of them are regular polyhedra (but not cconvex).

starpoly =
Map[ Polyhedron,
     {{SmallStellatedDodecahedron, GreatStellatedDodecahedron},
      {GreatDodecahedron, GreatIcosahedron}
     }, {2} ];

Show[ GraphicsArray[starpoly] ];

Maps of Lines and Facets

The first two show the intersection figures of all planes of the icosahedron. The inner graphics   leaves out the outermost points and allows a better view into the complicated interior.

Show[ lineGraphics ];

Show[ lineGraphicsInner ];

The different kinds of facets are color coded.

Show[ faceGraphics ];

Show[ faceGraphicsInner ];

Here is an example of a planar map of a stellation.

Show[ FaceGraphics[59] ];

Left and Right Forms

Th left and right form of the compound of five tetrahedra.

Show[ GraphicsArray[{Icosahedra[36], Icosahedra[36, True]}] ];

Their combination is another stellation: the compound of ten tentrahedra.

Show[ Icosahedra[18] ];

All 59 Pictures

This command produces all 59 pictures. To the left is a 3D view of the solid, to the right is its planar map.

Scan[ Show[plotit[#], PlotLabel->#]&,
      Range[Length[icosahedra]]
]

The Cover Picture

The journal cover shows these twelve stellations:

icosa = {
	{{ 2}, { 3}, {11}},
	{{36, False},  {18}, {36, True}},
	{{ 8}, {33}, {30}},
	{{23}, {45}, { 4}}
};

The data for ray tracing was derived from the following Mathematica graphics.

Show[ GraphicsArray[
  Apply[ Icosahedra[##, Boxed->False]&, icosa, {2} ]
] ] ;