List of polyhedral stellations

In three-dimensional space, a stellation extends facets of a polyhedron to form a new figure. Usually, this is achieved by extending faces or edges and planes of a polyhedron, until they generate new vertices that bound a newly formed figure.

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.

Background

Star polytopes

Further information: Kepler–Poinsot polyhedron § History
Model of the final stellation of the icosahedron by Max Brückner, as part of his 1900 book, Vielecke und Vielflache: Theorie und Geschichte[1]

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 mathematical 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.[11][12][b] Using this method, Kepler was able to discover the small stellated dodecahedron and the great stellated dodecahedron[13][14][15] (the latter, a solid Jamnitzer previously studied).[8] In 1809, Louis Poinsot rediscovered Kepler's star figures and discovered a further two, the great icosahedron and great dodecahedron;[16] he achieved this by experimenting assembling regular star polygons and convex regular polygons (i.e. pentagons, pentagrams and equilateral triangles) on vertices of the regular icosahedron and dodecahedron.[17] Three years later, Augustin-Louis Cauchy proved, using concepts of symmetry, that these four stellations are the only regular star-polyhedra,[18][19] eventually termed the Kepler–Poinsot polyhedra.

Stellation process

Further information: Stellation § Stellating polyhedra

Coxeter et al. (1938) details, for the first time, all stellations of the regular icosahedron with specific rules proposed by J. C. P. Miller.[20] Generalizing these (Miller's rules) for stellating any uniform polyhedron yields the following:[21]

  • 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.)[22] 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,[23][24] such that an outward ray from the center of the original polyhedro (in any direction) crosses the stellation surface only once[25] (that is to say, all visible parts of a face are seen from the same side).[c]

In the 1948 first edition of Regular Polytopes, H. S. M. Coxeter describes the stellation process as the reciprocal action to faceting,[28] identifying the four Kepler-Poinsot polyhedra as stellations and facetings of the regular dodecahedron and icosahedron.[29][30] 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.[31][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).[33]

Stellations of various polyhedra
Image Name Stellation core Refs. Notes
Great dodecahedron Regular dodecahedron
W21
*, ¶
Great stellated dodecahedron
W22
Small stellated dodecahedron
W20
*
Great icosahedron Regular icosahedron
W41,C7
Compound of two tetrahedra Regular octahedron
W19
† (‡), ¶
Compound of five tetrahedra Regular icosahedron
W24,C47
†
Compound of ten tetrahedra
W25,C22
Compound of five octahedra
W2,C3
Compound of five cubes Rhombic triacontahedron
[34]
Compound of cube and octahedron Cuboctahedron
W43
‡, ¶
Compound of dodecahedron and icosahedron Icosidodecahedron
W47
Compound of great icosahedron and great stellated dodecahedron
W61
‡
Compound of great dodecahedron and small stellated dodecahedron Rhombic triacontahedron
[35]
Small triambic icosahedron Regular icosahedron
W1,C2
Final stellation of the icosahedron
W13,C8
First stellation of the rhombic dodecahedron Rhombic dodecahedron
[36]
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)
First/outermost stellation of polyhedron

"Refs." (references) such as indexes found in Coxeter et al. (1999) using the Crennells' illustration notation (C), and Wenninger (1989) (W).
"Stellation core" describes a stellated regular (Platonic), semi-regular (Archimedean), or dual to a semi-regular (Catalan) figure.

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. 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.[f]

Stellation totals of convex polyhedra by group symmetry (Td, Oh, Ih) [37]
▼ Polyhedron Cell types Fully supported stellations Miller stellations
 P L A T O N I C  Tetrahedron 1 1[22] 1[38]
Cube 1 1[22] 1[38]
Octahedron 2 2[39] 2[38]
Dodecahedron 4 4[39] 4[40]
Icosahedron 11 18[39][21] 59[41]
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 21[39][42] 188
Rhombic dodecahedron 4 4[39][43] 5
Tetrakis hexahedron 10 1762[39] 143383367876
Triakis octahedron 32 3083[39] 218044256331
Deltoidal icositetrahedron 32 1201 253811894971
Disdyakis dodecahedron 292 ? (14728897413 reflexible) ?
Pentagonal icositetrahedron 69 72621[39] ?
Rhombic triacontahedron 29 227[44][39] 358833098[g]
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 the octahedron

Main article: Stellated octahedron

The stella octangula (or stellated octahedron), is the only stellation of the regular octahedron.[38] This stellation is made of self-dual tetrahedra, as the simplest regular polyhedral compound:[46]

Figure Stellation
Regular octahedron
Stellated octahedron
stella octangula
Regular polytope or regular compound,
deltahedra, antiprisms

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.[h] 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).[38]

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 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 (press "show" to open the table), the regular icosahedron (or snub octahedron) stellation core is indexed as "1', where cells (du Val notation) correspond to the internal congruent spaces formed by extending face-planes of the regular icosahedron, and face diagrams the lines of intersection from extended polyhedral edges that are used in the stellation process:

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

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).[47] While only one stellation of the icosahedron is a Kepler-Poinsot polyhedron, all stellations of the dodecahedron are Kepler-Poinsot polyhedra (the remaining).

Stellations of the rhombic dodecahedron

Further information: Rhombic dodecahedron § Stellations
Escher's solid, or the first stellation of the rhombic dodecahedron, tessellates three-dimensional space with copies of itself.[48]

The rhombic dodecahedron produces three fully supported stellations; these were described in Luke (1957):[43][49]

Rhombic dodecahedron stellations
Stellation Figure Face diagram
1
2
3
4

An additional fourth stellation is possible under Miller's rules.[50] The first stellation of the rhombic dodecahedron is notable for being able to form a honeycomb in three-dimensional space, using copies of itself.[48]

Hemipolychrons

Further information: Hemipolyhedron § Duals of the hemipolyhedra

In Wenninger (1983), a unique family of stellations with unbounded vertices are identified.[33] 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).

Table of hemipolyhedral stellations
Image Name Stellation core
  Tetrahemihexacron Tetrahemihexahedron
  Hexahemioctacron
(octahemioctacron)		 Octahemioctahedron
  Small dodecahemidodecacron   
(small icosihemidodecacron)		 Small icosihemidodecahedron
  Great icosihemidodecacron
(great dodecahemidodecacron)		 Great dodecahemidodecahedron
  Small dodecahemicosacron
(great dodecahemicosacron)		 Great dodecahemicosahedron

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).[51] Of these, only the tetrahemihexahedron would produce a stellation without another coincidental figure, the tetrahemihexacron.

Notes

  1. ^ 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]
  2. ^ 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."
  3. ^ McKeown & Badler (1980) presented an early computer algorithm to generate and visualize stellations of convex polyhedra,[26] as for the 227 stellations of the rhombic triacontahedron that Pawley (1975) formally described.[27]
  4. ^ 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.[32]
  5. ^ Using index notation from Coxeter et al. (1999) (C), the Crennells' third edition of The Fifty-Nine Icosahedra, and Magnus Wenninger's notation as found in Wenninger (1989) (W), where applicable.
  6. ^ 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 parent source.
  7. ^ 358833072 from earlier sources,[22] and extending to 358833106 per a deeper analysis by Webb of Miller's fifth rule.[45]
  8. ^ Regular compound polyhedra larger than the stellated octahedron are made of larger sets of regular polyhedra with chiral symmetry.

References

Works cited

  1. ^ Brückner (1900), p. 260.
  2. ^ Coxeter (1969), p. 37.
  3. ^ Chasles (1875), pp. 480, 481.
  4. ^ Pacioli (1509), pls. XIX, XX.
  5. ^ Innocenzi (2018), p. 248.
  6. ^ Pacioli (1509), pls. XXV, XXVI.
  7. ^ Jamnitzer (1568), eng. F.IIII.
  8. ^ a b Innocenzi (2018), pp. 256, 257.
  9. ^ Jamnitzer (1568), eng. C.V.
  10. ^ Hart (1996). "Wentzel Jamnitzer's Polyhedra".
  11. ^ Kepler (1619), Liber I: II. Definitio (pp. 6, 7).
  12. ^ Kepler (1997), Book I: II. Definitions (p. 17).
  13. ^ Wenninger (1965), pp. 244.
  14. ^ Kepler (1619), Liber II: XXVI Propositio (p. 60).
  15. ^ Kepler (1997), Book II: XXVI Proposition (pp. 116, 117).
  16. ^ Wenninger (1965), pp. 244, 245.
  17. ^ Poinsot (1810), pp. 39–42.
  18. ^ Wenninger (1965), p. 245.
  19. ^ Cauchy (1813), pp. 68–75.
  20. ^ Coxeter et al. (1938), pp. 7, 8.
  21. ^ a b Webb (2000).
  22. ^ a b c d Messer (1995), p. 26.
  23. ^ Wenninger (1983), pp. 36, 153.
  24. ^ Messer (1995), p. 27.
  25. ^ Webb (2001). "Stella Polyhedral Glossary".
  26. ^ McKeown & Badler (1980), pp. 19–24.
  27. ^ Lansdown (1982), p. 55.
  28. ^ Coxeter (1948), pp. 95.
  29. ^ Cundy (1949), p. 48.
  30. ^ Coxeter (1948), pp. 96.
  31. ^ Coxeter (1948), p. 99.
  32. ^ Coxeter (1948), p. 98.
  33. ^ a b Wenninger (1983), pp. 101–119.
  34. ^ Pawley (1975), p. 225.
  35. ^ Weisstein (1999). "Great Dodecahedron-Small Stellated Dodecahedron Compound".
  36. ^ Holden (1971), p. 134.
  37. ^ a b Webb (2001). "Enumeration of Stellations (Research)".
  38. ^ a b c d e Coxeter (1973), p. 96.
  39. ^ a b c d e f g h i Messer (1995), p. 32.
  40. ^ Wenninger (1989), pp. 35, 38–40.
  41. ^ Coxeter et al. (1938).
  42. ^ Hart (1996). "Stellations of the Triakis Tetrahedron".
  43. ^ a b Cundy & Rollett (1961), pp. 149–151.
  44. ^ Pawley (1975).
  45. ^ Webb (2001). "Miller's Fifth Rule".
  46. ^ Coxeter (1973), pp. 48, 49.
  47. ^ Wenninger (1989), pp. 34–36, 41–65.
  48. ^ a b Holden (1971), p. 165.
  49. ^ Hart (1996). "Stellations".
  50. ^ Weisstein (1999). "Rhombic Dodecahedron Stellations".
  51. ^ Wenninger (1983), pp. 101, 103, 104.

Secondary sources

Primary sources

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