Snub polyhedron

In geometry, a snub polyhedron is a polyhedron obtained by performing a snub operation: alternating a corresponding omnitruncated or truncated polyhedron, depending on the definition. Some, but not all, authors include antiprisms as snub polyhedra, as they are obtained by this construction from a degenerate "polyhedron" with only two faces (a dihedron).

Class	Number and properties
Polyhedron
Platonic solids	(5, convex, regular)
Archimedean solids	(13, convex, uniform)
Kepler–Poinsot polyhedra	(4, regular, non-convex)
Uniform polyhedra	(75, uniform)
Prismatoid: prisms, antiprisms etc.	(4 infinite uniform classes)
Polyhedra tilings	(11 regular, in the plane)
Quasi-regular polyhedra	(8)
Johnson solids	(92, convex, non-uniform)
Pyramids and Bipyramids	(infinite)
Stellations	Stellations
Polyhedral compounds	(5 regular)
Deltahedra	(Deltahedra, equilateral triangle faces)
Snub polyhedra	(12 uniform, not mirror image)
Zonohedron	(Zonohedra, faces have 180°symmetry)
Dual polyhedron
Self-dual polyhedron	(infinite)
Catalan solid	(13, Archimedean dual)

Chiral snub polyhedra do not always have reflection symmetry and hence sometimes have two enantiomorphous (left- and right-handed) forms which are reflections of each other. Their symmetry groups are all point groups.

For example, the snub cube:

Snub polyhedra have Wythoff symbol $| p q r$ and by extension, vertex configuration $3. p .3. q .3. r$ . Retrosnub polyhedra (a subset of the snub polyhedron, containing the great icosahedron, small retrosnub icosicosidodecahedron, and great retrosnub icosidodecahedron) still have this form of Wythoff symbol, but their vertex configurations are instead ${\frac {(3.-p.3.-q.3.-r)}{2}}.$

List of snub polyhedra

Uniform

There are 12 uniform snub polyhedra, not including the antiprisms, the icosahedron as a snub tetrahedron, the great icosahedron as a retrosnub tetrahedron and the great disnub dirhombidodecahedron, also known as Skilling's figure.

When the Schwarz triangle of the snub polyhedron is isosceles, the snub polyhedron is not chiral. This is the case for the antiprisms, the icosahedron, the great icosahedron, the small snub icosicosidodecahedron, and the small retrosnub icosicosidodecahedron.

In the pictures of the snub derivation (showing a distorted snub polyhedron, topologically identical to the uniform version, arrived at from geometrically alternating the parent uniform omnitruncated polyhedron) where green is not present, the faces derived from alternation are coloured red and yellow, while the snub triangles are blue. Where green is present (only for the snub icosidodecadodecahedron and great snub dodecicosidodecahedron), the faces derived from alternation are red, yellow, and blue, while the snub triangles are green.

Snub polyhedron	Original omnitruncated polyhedron	Image	Snub derivation	Symmetry group	Wythoff symbol Vertex description
Icosahedron (snub tetrahedron)	Truncated octahedron			I_h (T_h)	\| 3 3 2 3.3.3.3.3
Great icosahedron (retrosnub tetrahedron)	Truncated octahedron			I_h (T_h)	\| 2 ³/₂ ³/₂ ^(3.3.3.3.3)/₂
Snub cube or snub cuboctahedron	Truncated cuboctahedron			O	\| 4 3 2 3.3.3.3.4
Snub dodecahedron or snub icosidodecahedron	Truncated icosidodecahedron			I	\| 5 3 2 3.3.3.3.5
Small snub icosicosidodecahedron	Doubly covered truncated icosahedron			I_h	\| 3 3 ⁵/₂ 3.3.3.3.3.⁵/₂
Snub dodecadodecahedron	Small rhombidodecahedron with extra 12{¹⁰/₂} faces			I	\| 5 ⁵/₂ 2 3.3.⁵/₂.3.5
Snub icosidodecadodecahedron	Icositruncated dodecadodecahedron			I	\| 5 3 ⁵/₃ 3.⁵/₃.3.3.3.5
Great snub icosidodecahedron	Rhombicosahedron with extra 12{¹⁰/₂} faces			I	\| 3 ⁵/₂ 2 3.3.⁵/₂.3.3
Inverted snub dodecadodecahedron	Truncated dodecadodecahedron			I	\| 5 2 ⁵/₃ 3.⁵/₃.3.3.3.5
Great snub dodecicosidodecahedron	Great dodecicosahedron with extra 12{¹⁰/₂} faces		no image yet	I	\| 3 ⁵/₂ ⁵/₃ 3.⁵/₃.3.⁵/₂.3.3
Great inverted snub icosidodecahedron	Great truncated icosidodecahedron			I	\| 3 2 ⁵/₃ 3.⁵/₃.3.3.3
Small retrosnub icosicosidodecahedron	Doubly covered truncated icosahedron		no image yet	I_h	\| ⁵/₂ ³/₂ ³/₂ ^{(3.3.3.3.3.⁵/₂)}/₂
Great retrosnub icosidodecahedron	Great rhombidodecahedron with extra 20{⁶/₂} faces		no image yet	I	\| 2 ⁵/₃ ³/₂ ^{(3.3.3.⁵/₂.3)}/₂
Great dirhombicosidodecahedron	—	—	—	I_h	\| ³/₂ ⁵/₃ 3 ⁵/₂ ^{(4.³/₂.4.⁵/₃.4.3.4.⁵/₂)}/₂
Great disnub dirhombidodecahedron	—	—	—	I_h	\| (³/₂) ⁵/₃ (3) ⁵/₂ ^{(³/₂.³/₂.³/₂.4.⁵/₃.4.3.3.3.4.⁵/₂.4)}/₂

Notes:

The icosahedron, snub cube and snub dodecahedron are the only three convex ones. They are obtained by snubification of the truncated octahedron, truncated cuboctahedron and the truncated icosidodecahedron - the three convex truncated quasiregular polyhedra.
The only snub polyhedron with the chiral octahedral group of symmetries is the snub cube.
Only the icosahedron and the great icosahedron are also regular polyhedra. They are also deltahedra.
Only the icosahedron, great icosahedron, small snub icosicosidodecahedron, small retrosnub icosicosidodecahedron, great dirhombicosidodecahedron, and great disnub dirhombidodecahedron also have reflective symmetries.

There is also the infinite set of antiprisms. They are formed from prisms, which are truncated hosohedra, degenerate regular polyhedra. Those up to hexagonal are listed below. In the pictures showing the snub derivation, the faces derived from alternation (of the prism bases) are coloured red, and the snub triangles are coloured yellow. The exception is the tetrahedron, for which all the faces are derived as red snub triangles, as alternating the square bases of the cube results in degenerate digons as faces.

Snub polyhedron	Original omnitruncated polyhedron	Symmetry group	Wythoff symbol Vertex description
Tetrahedron	Cube	T_d (D_2d)	\| 2 2 2 3.3.3
Octahedron	Hexagonal prism	O_h (D_3d)	\| 3 2 2 3.3.3.3
Square antiprism	Octagonal prism	D_4d	\| 4 2 2 3.4.3.3
Pentagonal antiprism	Decagonal prism	D_5d	\| 5 2 2 3.5.3.3
Pentagrammic antiprism	Doubly covered pentagonal prism	D_5h	\| ⁵/₂ 2 2 3.⁵/₂.3.3
Pentagrammic crossed-antiprism	Decagrammic prism	D_5d	\| 2 2 ⁵/₃ 3.⁵/₃.3.3
Hexagonal antiprism	Dodecagonal prism	D_6d	\| 6 2 2 3.6.3.3

Notes:

Two of these polyhedra may be constructed from the first two snub polyhedra in the list starting with the icosahedron: the pentagonal antiprism is a parabidiminished icosahedron and a pentagrammic crossed-antiprism is a parabidiminished great icosahedron, also known as a parabireplenished great icosahedron.

Non-uniform

Two Johnson solids are snub polyhedra: the snub disphenoid and the snub square antiprism. Neither is chiral.

Snub polyhedron	Image	Original polyhedron	Image	Symmetry group
Snub disphenoid		Disphenoid		D_2d
Snub square antiprism		Square antiprism		D_4d

References

Coxeter, Harold Scott MacDonald; Longuet-Higgins, M. S.; Miller, J. C. P. (1954), "Uniform polyhedra", Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences, 246 (916): 401–450, doi:10.1098/rsta.1954.0003, ISSN 0080-4614, JSTOR 91532, MR 0062446, S2CID 202575183
Wenninger, Magnus (1974). Polyhedron Models. Cambridge University Press. ISBN 0-521-09859-9.
Skilling, J. (1975), "The complete set of uniform polyhedra", Philosophical Transactions of the Royal Society of London. Series A. Mathematical and Physical Sciences, 278 (1278): 111–135, doi:10.1098/rsta.1975.0022, ISSN 0080-4614, JSTOR 74475, MR 0365333, S2CID 122634260
Mäder, R. E. Uniform Polyhedra. Mathematica J. 3, 48-57, 1993.

Polyhedron operators
Seed	Truncation	Rectification	Bitruncation	Dual	Expansion	Omnitruncation	Alternations


$t 0 {p, q} {p, q}$	$t 01 {p, q} t{p, q}$	$t 1 {p, q} r{p, q}$	$t 12 {p, q} 2t{p, q}$	$t 2 {p, q} 2r{p, q}$	$t 02 {p, q} rr{p, q}$	$t 012 {p, q} tr{p, q}$	$ht 0 {p, q} h{q, p}$	$ht 12 {p, q} s{q, p}$	$ht 012 {p, q} sr{p, q}$

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