Stellation is a process of constructing new polygons (in two dimensions), new polyhedra in three dimensions, or, in general, new polytopes in n dimensions. The process consists of extending elements such as edges or face planes, usually in a symmetrical way, until they meet each other again. The new polyhedron is a stellation of the original.

A partial stellation is one where not all elements of a given dimensionality are extended.

A sub-symmetric stellation is one where not all elements are extended symmetrically.

Stellation of a regular polygon forms a star polygon or polygon compound. Stellation of a pentagon, for instance, forms a pentagram.

The face planes of a polyhedron divide space into many discrete cells. For a symmetrical polyhedron, these cells will lie in layers, or shells, of congruent cells - we say that the cells in such a congruent set are of the same type. A set of cells forming a closed layer around the core is called a shell. A common method of finding stellations involves selecting one or more cell types.

This can lead to a huge number of possible forms, so further criteria are often imposed to reduce the set to those stellations that are significant and unique in some way. In "The Fifty-Nine Icosahedra" Coxeter, Du Val, Flather and Petrie record five rules suggested by Miller. Although these rules refer specifically to the icosahedron's geometry, they can easily be extended to work for arbitrary polyhedra. They ensure, among other things, that the rotational symmetry of the original polyhedron is preserved, and that each stellation is different in appearance.

Under Miller's rules we find:

• There are no stellations of the tetrahedron, because all faces are adjacent
• There are no stellations of the cube, because non-adjacent faces are parallel and thus cannot be extended to meet in new edges
• There is 1 stellation of the octahedron, the stella octangula
• There are 3 stellations of the dodecahedron: the small stellated dodecahedron, the great dodecahedron and the great stellated dodecahedron, all of which are Kepler-Poinsot solids.
• There are 58 stellations of the icosahedron, including the great icosahedron (one of the Kepler-Poinsot solids), and the 2nd and final stellations of the icosahedron. The 59th model in "The 59 Icosahedra" is the original icosahedron itself.

The Archimedean solids and their duals can also be stellated. Here we usually add the rule that all of the original faces must "contribute" to the stellation, so the cube is not considered a stellation of the cuboctahedron. There are:

• 4 stellations of the rhombic dodecahedron
• 187 stellations of the triakis tetrahedron
• 358,833,097 stellations of the rhombic triacontahedron
• 17 stellations of the cuboctahedron (4 are shown in Wenninger's "Polyhedron Models")
• Unknown stellations of the icosidodecahedron, but many more than above! (19 are shown in Wenninger's "Polyhedron Models")

Seventeen of the nonconvex uniform polyhedra are stellations of Archimedean solids.

Miller's rules by no means represent the "correct" way to enumerate stellations however. They are based on combining parts within the stellation diagram in certain ways, and don't take into account the topology of the resulting faces. As such there are some quite reasonable stellations of the icosahedron that are not part of their list - one was identified by James Bridge in 1974, while some "Miller stellations" are questionable as to whether they should be regarded as stellations at all - one of the icosahedral set comprises several quite disconnected cells floating symmetrically in space.

As yet an alternative set of rules that takes this into account has not been fully developed. Most progress has been made based on the notion that stellation is the reciprocal process to facetting, whereby parts are removed from a polyhedron without creating any new vertices. For every stellation of some polyhedron, there is a dual facetting of the dual polyhedron, and vice versa. By studying facettings of the dual, we gain insights into the stellations of the original. Bridge found his new stellation of the icosahedron by studying the facettings of its dual, the dodecahedron.

Some polyhedronists take the view that stellation is a two-way process, such that any two polyhedra sharing the same face planes are stellations of each other. This is understandable if one is devising a general algorithm suitable for use in a computer program, but is otherwise not particularly helpful.

Many examples of stellations can be found in the list of Wenninger's stellation models.

See also

• List of Wenninger polyhedron models Includes 44 stellated forms of the octahedron, dodecahedron, icosahedron, and icosidodecahedron, enumerated the 1974 book "Polyhedron Models" by Magnus Wenninger
• Polyhedral compound Includes 5 regular compounds and 4 dual regular compounds.

External links

• Stellating the Icosahedron and Facetting the Dodecahedron
• Stella: Polyhedron Navigator - Software for exploring polyhedra and printing nets for their physical construction. Includes uniform polyhedra, stellations, compounds, Johnson solids, etc.
• Enumeration of stellations

Polygons | Polyhedra | Polytopes

星型多面体 | Estrelamento de um poliedro