List of polyhedral stellations

In three-dimensional space, applying the operation of stellation to a polyhedron extends its faces (or edges and planes) until they generate new vertices that bound a newly formed figure. Stellation represents the dual action to faceting a polyhedron.^[1]

Originating from studies of star polyhedra in 14th century Europe, a proper mathematical account of polyhedral stellations was given by Johannes Kepler in his 1619 classic work, Harmonices Mundi.^[2] Progress on detailing and enumerating stellations of prominent stars, such as the regular Kepler-Poinsot polyhedra, later ensued,^[3] with developments on different stellation methods occurring in the 1900s – principally from Coxeter et al. (1938) and soon afterward, Pawley (1975).

A short generalized table of the most notable polyhedral stellations belonging to convex uniform polyhedra is provided, with complete sets of stellations for the Platonic solids (including the fifty-nine icosahedral stellations), as well as for select Catalan solids (e.g., the rhombic dodecahedron and the rhombic triacontahedron). Stellations featuring unbounded vertices, per Wenninger (1983), are also included; these are stellations to infinity which conform to extensions on traditional definitions of polyhedra.

Experimentation with star polygons and star polyhedra since the fourteenth century AD led the way to formal theories for stellating polyhedra:

It was in 1619 that the first geometric description of a stellation was given, by Johannes Kepler in his landmark book, Harmonices Mundi: the process of extending the edges (or faces) of a figure until new vertices are generated, which collectively form a new figure.^[14][15][b] Using this method, Kepler was able to discover the small stellated dodecahedron and the great stellated dodecahedron.^[16][17][18] In 1809, Louis Poinsot rediscovered Kepler's star figures and discovered a further two, the great icosahedron and great dodecahedron;^[19] he achieved this by experimenting assembling regular star polygons and convex regular polygons on vertices of the regular icosahedron and dodecahedron (i.e., pentagons, pentagrams and equilateral triangles).^[20] Three years later, Augustin-Louis Cauchy proved, using concepts of symmetry, that these four stellations are the only regular star-polyhedra,^[21][22] eventually termed the Kepler–Poinsot polyhedra. As with most non-convex polyhedra including stellations and other star polyhedra, the Kepler-Poinsot polyhedra, with regular self-intersecting faces, are now known to be inequivalent to the topological sphere as a simple connected surface^[23] (this is in contrast with the traditional convex uniform polyhedra and their corresponding homotopy invariance).^[c]

Coxeter et al. (1938) details, for the first time, all stellations of the regular icosahedron with specific rules proposed by J. C. P. Miller.^[24] Generalizing these (Miller's rules) for stellating any uniform polyhedron yields the following:^[25]

These rules are ideal for stellating smaller uniform solids, such as the regular polyhedra; however, when assessing stellations of other larger uniform polyhedra, this method can quickly become overwhelming. (For example, there are a total of 358,833,072 stellations to the rhombic triacontahedron using this set of rules.)^[26] To address this, Pawley (1973) proposed a set of rules that restrict the number of stellations to a more manageable set of fully supported stellations that are radially convex,^[27][28] such that an outward ray from the center of the original polyhedro (in any direction) crosses the stellation surface only once^[29] (that is to say, all visible parts of a face are seen from the same side).^[d]

In the 1948 first edition of Regular Polytopes, H. S. M. Coxeter describes the stellation process as the reciprocal action to faceting,^[1] identifying the four Kepler-Poinsot polyhedra as stellations and facetings of the regular dodecahedron and icosahedron.^[32][33] He specifies the construction of a star polyhedron as a stellation of its core (with congruent face-planes), or by faceting its case — the former requires the addition of solid pieces that generate new vertices, while the latter involves the removal of solid pieces, without forming any new vertices (the core of a star polyhedron or compound is the largest convex solid that can be drawn inside them, while their case is the smallest convex solid that contains them).^[34]

Lists for polyhedral stellations contain non-convex polyhedra; some of the most notable examples include:

Stellations that topologically do not fit into standard definitions of uniform polyhedra are listed further down (i.e. stellations of hemipolyhedra).^[35]

KEY

* Kepler-Poinsot polyhedron (star polyhedron with regular facets)
† Regular compound polyhedron (vertex, edge, and face-transitive compound)
‡ Compound of dual regular polyhedra (Platonic or Kepler-Poinsot duals)
¶ First/outermost stellation of stellation core

"Stellation core" describes a stellated regular (Platonic), semi-regular (Archimedean), or dual to a semi-regular (Catalan) figure.
"Face diagram" represents the lines of intersection from extended polyhedral edges that are used in the stellation process.
"Refs." (references) such as indexes found in Coxeter et al. (1999) using the Crennells' illustration notation (C), and Wenninger (1989) (W).

The table below is adapted from research by Robert Webb, using his program Stella.^[40] It enumerates fully supported stellations and stellations per Miller's process, of the regular Platonic solids as well as the semi-regular Archimedean solids and their Catalan duals. In this list, the elongated square gyrobicupola and its dual polyhedron are not included (these are sometimes considered a fourteenth Archimedean and Catalan solid, respectively). The base polyhedron stellation core is included as a zeroth convex stellation following the Crennells' indexing, with stellation totals the sum of chiral and reflexible stellations (a "chiral" stellation is enantiomorphous, while a "reflexible" stellation maintains the same group symmetry as its stellation core, yet remains achiral – for a count of these separately, visit the parent source).

"Cell types" are sets of symmetrically equivalent stellation cells, where "stellation cells" are the minimal 3D spaces enclosed on all sides by the original polyhedron's extended facial planes.
"?" denotes an unknown total number of stellations; however, the number of reflexible stellations are sometimes known for these (where chiral stellations are excluded).

Only three of the five Platonic solids produce stellations: the regular octahedron, regular dodecahedron, and regular icosahedron. The regular tetrahedron and cube are unable to generate stellations when extending their faces, since extending their vertices only form one possible convex hull.^[41]

The stella octangula (or stellated octahedron) is the only stellation of the regular octahedron.^[41] This stellation is made of self-dual tetrahedra, as the simplest regular polyhedral compound:^[49]

All stellations of the regular dodecahedron are Kepler-Poinsot polyhedra:

Coxeter et al. (1938) detailed the stellations of the regular icosahedron with rules proposed by J. C. P. Miller. As found in Coxeter et al. (1999), the following table lists all stellations of the icosahedron per the Crennells' indexing (in it, the regular icosahedron (or snub octahedron) stellation core is indexed as "1"):

"Cells" (du Val notation) correspond to the internal congruent spaces formed by extending face-planes of the regular icosahedron.

A subset of these are illustrated in Wenninger (1989), alongside constructions for physical models (W19–W66).^[51]

The rhombic dodecahedron produces three fully supported stellations, described in Luke (1957):^[53][54]

An additional fourth stellation is possible under Miller's rules.^[55] The first stellation of the rhombic dodecahedron is notable for being able to form a honeycomb in three-dimensional space, using copies of itself.^[52]

Pawley (1975) shows the rhombic triacontahedron produces 227 fully supported stellations, including the rhombic triacontahedron itself.^[42] Some of these are shown in the table below:

Of these, the compound of five cubes is notable for being a regular compound polyhedron. The medial rhombic triacontahedron and the great rhombic triacontahedron are also notable for being star (non-convex) isotoxal polyhedra.

In Wenninger (1983), a unique family of stellations with unbounded vertices are identified.^[56] These originate from orthogonal edges of faces that pass through centers of their corresponding dual hemipolyhedra. The following is a list of these stellations; specifically, of non-convex uniform hemipolyhedra (with coincidental figures in parentheses):

This family of stellations does not strictly fulfill the definition of a polyhedron that is bound by vertices, and Wenninger notes that at the limit their facets can be interpreted as forming unbounded elongated pyramids, or equivalently, prisms (indistinguishably).^[61] As with their dual polyhedra, these hemipolyhedral stellations are isotoxal polyhedra (in their case, at infinity). The final polyhedron on this list, the great dirhombicosidodecacron,^[62] is the only stellation whose dual figure — the last-indexed and most complex uniform polyhedron, the great dirhombicosidodecahedron (U₇₅) — is constructed using a spherical quadrilateral Wythoff construction (rather than with spherical triangles).^[63][f]

The tetrahemihexahedron is the only hemipolyhedron to produce a dual hemipolychron without a coincidental figure, the tetrahemihexacron.

• Stellation and Facetting - a Brief History from Guy's Polyhedral Pages (Guy Inchbald) for a brief chronological listing regarding stellation
• Stellations of the Rhombic Triacontahedron from Virtual Polyhedra (The Encyclopedia of Polyhedra) (George W. Hart)