# Snub polyhedron

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)

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).

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 ${\displaystyle {\frac {(3.-p.3.-q.3.-r)}{2}}.}$

## 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 Image Original omnitruncated polyhedron Image Snub derivation Symmetry group Wythoff symbol
Vertex description
Icosahedron (snub tetrahedron) Truncated octahedron Ih (Th) | 3 3 2
3.3.3.3.3
Great icosahedron (retrosnub tetrahedron) Truncated octahedron Ih (Th) | 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 Ih | 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
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
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 Ih | 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 Ih | 3/2 5/3 3 5/2
(4.3/2.4.5/3.4.3.4.5/2)/2
Great disnub dirhombidodecahedron Ih | (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:

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 Image Original omnitruncated polyhedron Image Snub derivation Symmetry group Wythoff symbol
Vertex description
Tetrahedron Cube Td (D2d) | 2 2 2
3.3.3
Octahedron Hexagonal prism Oh (D3d) | 3 2 2
3.3.3.3
Square antiprism Octagonal prism D4d | 4 2 2
3.4.3.3
Pentagonal antiprism Decagonal prism D5d | 5 2 2
3.5.3.3
Pentagrammic antiprism Doubly covered pentagonal prism D5h | 5/2 2 2
3.5/2.3.3
Pentagrammic crossed-antiprism Decagrammic prism D5d | 2 2 5/3
3.5/3.3.3
Hexagonal antiprism Dodecagonal prism D6d | 6 2 2
3.6.3.3

Notes:

### 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 D2d
Snub square antiprism Square antiprism D4d

## 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
t0{p,q}
{p,q}
t01{p,q}
t{p,q}
t1{p,q}
r{p,q}
t12{p,q}
2t{p,q}
t2{p,q}
2r{p,q}
t02{p,q}
rr{p,q}
t012{p,q}
tr{p,q}
ht0{p,q}
h{q,p}
ht12{p,q}
s{q,p}
ht012{p,q}
sr{p,q}