This article mainly lists various stellations belonging to uniform polyhedra, such as those of the regular Platonic solids and the semiregular Archimidean solids. It also lists stellations featuring unbounded vertices.
The faces must lie in face-planes, i.e., the bounding planes of the regular solid.
All parts composing the faces must be the same in each plane, although they may be quite disconnected.
The parts included in any one plane must be symmetric about corresponding point groups, without or with reflection. This secures polyhedral symmetry for the whole solid.
All parts included in planes must be "accessible" in the completed solid (i.e. they must be on the "outside").
Cases where the parts can be divided into two sets, each giving a solid with as much symmetry as the whole figure, are excluded from consideration; combination of enantiomorphous pairs having no common part (which actually occurs in just one case) are included.
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.)[21] 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,[22][23] such that an outward ray from the center of the original polyhedron (in any direction) crosses the stellation surface only once[24] (that is to say, all visible parts of a face are seen from the same side).[c]
H. S. M. Coxeter was the first to describe the stellation process as the reciprocal action to faceting, in his 1948 first edition of Regular Polytopes.[26] He further 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.[27][d]
Lists
Lists for polyhedral stellations contain non-convex polyhedra; some of the most notable examples include:[e]
Examples of stellations that topologically do not fit into standard definitions of uniform polyhedra are listed further down (i.e. stellations of hemipolyhedra).[29]
Different from the larger, regular self-dual polyhedral enantiomorphisms (such as in the compound of five cubes), the tetrahedron is the only Platonic solid to generate a stellation (and regular polyhedron compound) from a single intersecting copy of itself.[f] Like the cube, the regular tetrahedron does not generate stellations when extending its faces, since all are adjacent (this yields only one possible convex hull).[34]
Coxeter et al. (1938) details stellations of the regular icosahedron with (aformentioned) rules proposed by J. C. P. Miller. The following table lists all such stellations per the Crennells' indexing, as found in Coxeter et al. (1999). In this list,[g] the regular icosahedron (or snub octahedron) stellation core is indexed as "1" (press "show" to open the table):
Wenninger (1989) includes a subset of these as formal stellations, primarily based on illustrative methods of construction of stellated polyhedral models (and extending to stellations of the icosidodecahedron).[36] While only one stellation of the icosahedron is a Kepler-Poinsot polyhedron, all stellations of the dodecahedron are Kepler-Poinsot polyhedra (the remaining).
Enumerations
The table below is adapted from research by Robert Webb, using his program Stella.[37] 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.[h] 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.[i]
"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).
In Wenninger (1983), a unique family of stellations with unbounded vertices are identified.[29] 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).[45] Of these, only the tetrahemihexahedron would produce a stellation without another coincidental figure, the tetrahemihexacron.
Notes
↑ More specifically, Pacioli's "elevation" of polyhedra involved truncating (or rectifying) the Platonic solids, afterwhich pyramids of different bases are systematically attached to faces of the polyhedra in order to augment them into a "star-like" polyhedron.[5] In this same work, da Vinci illustrates a concaved triakis icosahedron, which shares its outer shell with the great stellated dodecahedron.[6]
↑ Kepler (1997, Book I: II. Definitions; p. 17) defines a star polygon via stellation of a convex polygon: "Some of these [figures] are primary and basic, not extending beyond their boundaries, and it is to these that the previous definition properly applies: others are augmented, as if it were extending beyond their sides, and if two non-neighboring sides of one of the basic figures are produced they meet [to form a vertex of the augmented figure]: these are called Stars."
↑ Alternatively, McKeown & Badler (1980) presents a computer algorithm for manually generating stellations of convex polyhedra.[25]
↑ 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.[28]
↑ "Cell" corresponds to the internal spaces formed by extending face-planes of the regular icosahedron (du Val notation). In a symmetric figure such as the regular icosahedron, these regions form groups or sets of congruent cells, unique to each stellation — a set of cells forming a closed layer around its core forms a shell, which can be made of multiple types (e.g., e comprises e1 and e2).
↑ A "chiral" stellation is enantiomorphous, while a "reflexible" stellation maintains the same group symmetry as its stellation core, yet is achiral. For a count of these separately, visit the referenced source directly (Webb).[37]
↑ 358833072 from earlier sources,[21] and extending to 358833106 per a deeper analysis by Webb of Miller's fifth rule.[44]
Third edition: Coxeter, H. S. M.; etal. (1999). The Fifty-Nine Icosahedra. Illustrations by Kate and David Crennell. Tarquin. pp.1–72. ISBN978-1899618323.
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