List of polyhedral stellations

Short description: Stellations of convex and non-convex uniform polyhedra

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.

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. Progress later ensued on detailing and enumerating stellations of prominent stars, such as the regular Kepler-Poinsot polyhedra, with developments on different stellation methods occurring in the 1900s, principally from (Coxeter du Val) 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 of non-convex uniform polyhedra of structure with facial planes passing through their centers (i.e., hemipolyhedral), which render unbounded vertices, are also included; these are stellations to infinity — per (Wenninger 1983) — conforming to extensions on traditional definitions of polyhedra.

Background

Star polytopes

The small stellated dodecahedron (left), and great stellated dodecahedron (right), as found in Harmonices Mundi by Johannes Kepler.

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

• 14th c. AD: Thomas Bradwardine was the first to detail studies of star polygons by extending the sides of polygons.
• 15th – 16th c.: Charles de Bovelles became the first to study star polyhedra.
• 1509: Luca Pacioli publishes Divina proportione, featuring woodcut illustrations by Leonardo da Vinci of various polyhedra, including "elevated" polyhedra that have been augmented by attaching pyramids to faces, as showcased with the stellated octahedron.

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. Using this method, Kepler was able to discover the small stellated dodecahedron and the great stellated dodecahedron. In 1809, Louis Poinsot rediscovered Kepler's star figures and discovered a further two, the great icosahedron and great dodecahedron; 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). Three years later, Augustin-Louis Cauchy proved, using concepts of symmetry, that these four stellations are the only regular star-polyhedra, 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 (this is in contrast with the traditional convex uniform polyhedra and their corresponding homotopy invariance).

Stellation process

(Coxeter du Val) details, for the first time, all stellations of the regular icosahedron with specific rules proposed by J. C. P. Miller. Generalizing these (Miller's rules) for stellating any uniform polyhedron yields the following:

• 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.) 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, such that an outward ray from the center of the original polyhedron (in any direction) crosses the stellation surface only once (that is to say, all visible parts of a face are seen from the same side).

In the 1948 first edition of Regular Polytopes, H. S. M. Coxeter describes the stellation process as the reciprocal action to faceting, identifying the four Kepler-Poinsot polyhedra as stellations and facetings of the regular dodecahedron and icosahedron. 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).

Lists

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

• Kepler-Poinsot polyhedra (W20, W21, W22, and W41/C7)
• Regular compound polyhedra (W19, W2/C3, W24/C47, W25/C22, UC₉) and
• Platonic or Kepler-Poinsot compound dual polyhedra (W19, W47, W61, W43, and the great icosahedron and great stellated dodecahedron compound)

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

Notable stellations of various polyhedra

Image	Name	Stellation core	Face diagram	Refs.	Notes
75px	Great dodecahedron	Regular dodecahedron	60px	W21	*, ¶
75px	Great stellated dodecahedron		60px	W22
75px	Small stellated dodecahedron		60px	W20	*
75px	Great icosahedron	Regular icosahedron	60px	W41,C7
75px	Stella octangula (compound of two tetrahedra)	Regular octahedron	60px	W19	† (‡), ¶
75px	Compound of five tetrahedra	Regular icosahedron	60px	W24,C47	†
75px	Compound of ten tetrahedra		60px	W25,C22
75px	Compound of five octahedra		60px	W2,C3
75px	Compound of five cubes	Rhombic triacontahedron	60px
75px	Compound of cube and octahedron	Cuboctahedron	60px	W43	‡, ¶
75px	Compound of dodecahedron and icosahedron	Icosidodecahedron	60px	W47
75px	Compound of great icosahedron and great stellated dodecahedron		60px	W61	‡
75px	Compound of great dodecahedron and small stellated dodecahedron		60px
75px	Small triambic icosahedron	Regular icosahedron	60px	W1,C2	¶
75px	Final stellation of the icosahedron		60px	W13,C8
75px	First stellation of the rhombic dodecahedron	Rhombic dodecahedron	60px

Final stellation of the icosahedron, as represented by Max Brückner in his 1900 book, Vielecke und Vielflache: Theorie und Geschichte

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 du Val) using the Crennells' illustration notation (C), and (Wenninger 1989) (W).

Enumerations

The table below is adapted from research by Robert Webb, using his program Stella. 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).

Stellation totals of convex polyhedra by group symmetry and order (Td, Oh, Ih)Template:Hair-space
	▼	Polyhedron	Cell types	Fully supported stellations	Miller stellations
	P L A T O N I C	Tetrahedron	1	1	1
		Cube	1	1	1
		Octahedron	2	2	2
		Dodecahedron	4	4	4
		Icosahedron	11	18	59
	A R C H I M E D E A N	Truncated tetrahedron	4	6	10
		Cuboctahedron	8	13	21
		Truncated octahedron	9	18	45
		Truncated cube	9	18	45
		Rhombicuboctahedron	48	18827	? (128723453647 reflexible)
		Truncated cuboctahedron	49	22632	? (317650001638 reflexible)
		Snub cube	274	299050957776	?
		Icosidodecahedron	41	847	70841855109
		Truncated icosahedron	45	1117	3082649548558
		Truncated dodecahedron	45	1141	2645087084526
		Rhombicosidodecahedron	273	298832037395	?
		Truncated icosidodecahedron	294	1016992138164	?
		Snub dodecahedron	1940	? (579 reflexible)	?
	C A T A L A N	Triakis tetrahedron	9	188
		Rhombic dodecahedron	4	4	5
		Tetrakis hexahedron	10	1762	143383367876
		Triakis octahedron	32	3083	218044256331
		Deltoidal icositetrahedron	32	1201	253811894971
		Disdyakis dodecahedron	292	? (14728897413 reflexible)	?
		Pentagonal icositetrahedron	69	72621	?
		Rhombic triacontahedron	29	227	358833098
		Pentakis dodecahedron	253	71112946668	?
		Triakis icosahedron	241	13902332663	?
		Deltoidal hexecontahedron	226	7146284014	?
		Disdyakis triacontahedron	2033	? (~ 1012 reflexible)	?
		Pentagonal hexecontahedron	536	30049378413796	?

"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).

Stellations of Platonic solids

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.

Stellations of the octahedron

Main article: Stellated octahedron

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

Figure	Stellation
Regular octahedron	Compound of two tetrahedra stella octangula
60px	60px
60px	60px
deltahedra, antiprisms

Stellations of the dodecahedron

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

Stellations of the regular dodecahedron

Platonic solid	Kepler–Poinsot solids
Regular dodecahedron	Small stellated dodecahedron	Great dodecahedron	Great stellated dodecahedron
60px	60px	60px	60px
60px	60px	60px	60px

Stellations of the icosahedron

Main article: The Fifty-Nine Icosahedra

This is the stellation diagram of the regular icosahedron, with face sets labelled, 0-13.

(Coxeter du Val) detailed the stellations of the regular icosahedron with rules proposed by J. C. P. Miller. As found in (Coxeter du Val), 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"):

Stellations of the regular icosahedron
Crennell	Cells	Faces		Figure	Face diagram
1	A	0		60px	60px
2	B	1		60px	60px
3	C	2		60px	60px
4	D	3 4		60px	60px
5	E	5 6 7		60px	60px
6	F	8 9 10		60px	60px
7	G	11 12		60px	60px
8	H	13		60px	60px
9	e1	3' 5		60px	60px
10	f1	5' 6' 9 10		60px	60px
11	g1	10' 12		60px	60px
12	e1f1	3' 6' 9 10		60px	60px
13	e1f1g1	3' 6' 9 12		60px	60px
14	f1g1	5' 6' 9 12		60px	60px
15	e2	4' 6 7		60px	60px
16	f2	7' 8		60px	60px
17	g2	8' 9'11		60px	60px
18	e2f2	4' 6 8		60px	60px
19	e2f2g2	4' 6 9' 11		60px	60px
20	f2g2	7' 9' 11		60px	60px
21	De1	4 5		60px	60px
22	Ef1	7 9 10		60px	60px
23	Fg1	8 9 12		60px	60px
24	De1f1	4 6' 9 10		60px	60px
25	De1f1g1	4 6' 9 12		60px	60px
26	Ef1g1	7 9 12		60px	60px
27	De2	3 6 7		60px	60px
28	Ef2	5 6 8		60px	60px
29	Fg2	10 11		60px	60px
30	De2f2	3 6 8		60px	60px
31	De2f2g2	3 6 9' 11		60px	60px
32	Ef2g2	5 6 9' 11		60px	60px
33	f1	5' 6' 9 10		60px	60px
34	e1f1	3' 5 6' 9 10		60px	60px
35	De1f1	4 5 6' 9 10		60px	60px
36	f1g1	5' 6' 9 10' 12		60px	60px
37	e1f1g1	3' 5 6' 9 10' 12		60px	60px
38	De1f1g1	4 5 6' 9 10' 12		60px	60px
39	f1g2	5' 6' 8' 9' 10 11		60px	60px
40	e1f1g2	3' 5 6' 8' 9' 10 11		60px	60px
41	De1f1g2	4 5 6' 8' 9' 10 11		60px	60px
42	f1f2g2	5' 6' 7' 9' 10 11		60px	60px
43	e1f1f2g2	3' 5 6' 7' 9' 10 11		60px	60px
44	De1f1f2g2	4 5 6' 7' 9' 10 11		60px	60px
45	e2f1	4' 5' 6 7 9 10		60px	60px
46	De2f1	3 5' 6 7 9 10		60px	60px
47	Ef1	5 6 7 9 10		60px	60px
48	e2f1g1	4' 5' 6 7 9 10' 12		60px	60px
49	De2f1g1	3 5' 6 7 9 10' 12		60px	60px
50	Ef1g1	5 6 7 9 10' 12		60px	60px
51	e2f1f2	4' 5' 6 8 9 10		60px	60px
52	De2f1f2	3 5' 6 8 9 10		60px	60px
53	Ef1f2	5 6 8 9 10		60px	60px
54	e2f1f2g1	4' 5' 6 8 9 10' 12		60px	60px
55	De2f1f2g1	3 5' 6 8 9 10' 12		60px	60px
56	Ef1f2g1	5 6 8 9 10' 12		60px	60px
57	e2f1f2g2	4' 5' 6 9' 10 11		60px	60px
58	De2f1f2g2	3 5' 6 9' 10 11		60px	60px
59	Ef1f2g2	5 6 9' 10 11		60px	60px

"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).

Stellations of Catalan solids

Stellations of the rhombic dodecahedron

Escher's solid, or the first stellation of the rhombic dodecahedron, tessellates three-dimensional space with copies of itself.

The rhombic dodecahedron produces three fully supported stellations, described in (Luke 1957):

Rhombic dodecahedron stellations

Stellation	Figure	Face diagram
Rhombic dodecahedron
First stellation of the rhombic dodecahedron
Second stellation of the rhombic dodecahedron
Final stellation of the rhombic dodecahedron

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

Stellations of the rhombic triacontahedron

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

Select stellations to the rhombic triacontahedron

Stellation	Figure	Face diagram
Rhombic triacontahedron	65px	60px
First stellation of the rhombic triacontahedron	65px	60px
Medial rhombic triacontahedron	65px	60px
Rhombic hexecontahedron	65px	60px
Compound of five cubes	65px	60px
Great rhombic triacontahedron	65px	60px
Final stellation of the rhombic triacontahedron	65px	60px

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.

Hemipolychrons

In (Wenninger 1983), a unique family of stellations with unbounded vertices are identified. 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):

Stellations to infinityTemplate:Hair-space

Image	Name	Dual figure	Stellation coreTemplate:Hair-space
60px	Tetrahemihexacron	Tetrahemihexahedron	Cube
60px	Octahemioctacron (hexahemioctacron)	Octahemioctahedron (cubohemioctahedron)	Rhombic dodecahedron
60px	Small icosihemidodecacron    (small dodecahemidodecacron)	Small icosihemidodecahedron (small dodecahemidodecahedron)	Rhombic triacontahedron
60px	Great dodecahemidodecacron (great icosihemidodecacron)	Great dodecahemidodecahedron (great icosihemidodecahedron)
60px	Great dodecahemicosacron (small dodecahemicosacron)	Great dodecahemicosahedron (small icosihemidodecahedron)
60px	Great dirhombicosidodecacron (great disnub dirhombidodecacron)	Great dirhombicosidodecahedron (great disnub dirhombidodecahedron)	Deltoidal hexecontahedron

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). 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, 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).

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

References

Works cited

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External links

• 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)