Stellations of the Icosahedron

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Stellation is the process of extending the face planes of a polyhedron to enclose new volumes having the same symmetry as the original polyhedron. The process is so called because it usually produces star-shaped polyhedra.

The icosahedron has many beautiful stellations and some of them turn out to be regular themselves. The compounds of five octahedra, five tetrahedra and ten tetrahedra, and the great icosahedron can be obtained as stellations of the basic icosahedron.

An attempt to catalogue all the stellations of the icosahedron was made by Coxeter, DuVal, Flather and Petrie in "The Fifty-Nine Icosahedra". They identify 59 stellated forms, but to do so they set out criteria for recognising distinct forms. More recently alternative criteria have been proposed which disallow some of their stellations and add others.

Here is just a selection of views of stellated icosahedra. At the end there is a comment on stellations of other regular polyhedra.

Click on the images to see larger, animated views.

Stellated Icosahedra

	The first stellation is the triakis icosahedron. The edges of the triangular faces of the icosahedron are still visible. At each original face, extending the planes of the surrounding faces has raised a small pyramid. The resulting new faces are irregular hexagons (one is brightly lit in the picture).
	Further extension of the original icosahedron's face planes encloses a series of different shaped volumes. Here, as the icosahedron rotates, the facets of these stellated volumes appear and disappear so that we can see the structure. Glimpses of all the icosahedral stellations can be seen.
	Choosing to add all the possible enclosed volumes leads to the final stellation of the icosahedron; a star with 60 sharp points arranged in clusters of 5.
	Intermediate stellations include some regular forms. For example, the compound of five octahedra: the 20 faces of the icosahedron extend into 8 faces of each of 5 octahedra. Each icosahedron face generates two faces of distinct octahedra. Here, the stellation is semi-transparent so that we can see the icosahedron forming the five octahedra.
	Similarly, adding some further-out carefully selected volumes forms the compound of five tetrahedra. Now each icosahedron face generates only one tetrahedron face.
	The five tetrahedra above can be selected in two enantiomorphous (mirror reflection) forms. Combining both forms by selecting more stellation volumes leads to the compound of ten tetrahedra. Now each icosahedron face generates two faces of distinct tetrahedra.
	The remaining regular stellation is the great icosahedron. Each of the original icosahedron faces has been extended into a much larger equilateral triangle. The resulting solid still has 20 faces with the same icosahedral symmetry, but they intersect and five meet at each vertex rather than three.
	The criteria of Coxeter et al for selecting volumes to give distinct stellations lead to some unexpected results. For example, can the mauve volumes here be considered a polyhedron? They are made up of 60 small volumes which are joined only at edges and vertices. The original icosahedron has been included to show how its face planes form the volumes.
	Even more surprisingly, here is a stellation in which 60 small volumes meet only at their vertices. Again, the original icosahedron has been included.
	Finally, this stellation consists of 12 unconnected volumes! Again, the original icosahedron has been included.
	Many other configurations of the elementary volumes are possible, retaining the icosahedral symmetry. In this animation, successive volumes are added, passing through a number of stellations until the final stellation is reached. The sequence starts with the triakis icosahedron and the compound of five octahedra, and passes through the compounds of of five tetrahedra and ten tetrahedra and the great icosahedron. Other sequences are possible.

Other Stellated Regular Polyhedra

Stellations of the other convex regular polyhedra are all regular polyhedra or compounds themselves:

The tetrahedron and cube have no stellations as their face planes never meet.

The octahedron stellates in only one way, forming the stella octangula, or compound of two tetrahedra.

The dodecahedron stellates to successively form the small stellated dodecahedron, the great dodecahedron and the great stellated dodecahedron. Click on the image to the left to see this in animation.

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