Snub polyhedron

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Polyhedron
Class Number and properties
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,
equalatial 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)


A snub polyhedron is a polyhedron obtained by alternating a corresponding omnitruncated polyhedron.

Chiral snub polyhedra do not have reflection symmetry and hence have two enantiomorphous forms which are reflections of each other. Their symmetry groups are all point groups and are one of:

For example, the snub cube:

Snubhexahedronccw.gif Snubhexahedroncw.gif

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 (3.−p.3.−q.3.−r)/2.

Among the snub polyhedra that cannot be otherwise generated, only the pentagonal antiprism, pentagrammic antiprism, pentagrammic crossed-antiprism, small snub icosicosidodecahedron and small retrosnub icosicosidodecahedron are known to occur in any non-prismatic uniform polychoron. The tetrahedron, octahedron, icosahedron, and great icosahedron appear commonly in non-prismatic uniform polychora, but not in their snub constructions. Every snub polyhedron however can appear in the polyhedral prism based on them.

List of snub polyhedra[edit]

Uniform[edit]

There are 12 uniform snub polyhedra, not including 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 icosahedron, great icosahedron, small snub icosicosidodecahedron, and 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) Snub tetrahedron.png Truncated octahedron Omnitruncated tetrahedron.png Snub-polyhedron-icosahedron.png Ih (Th) | 3 3 2
3.3.3.3.3
Great icosahedron (retrosnub tetrahedron) Retrosnub tetrahedron.png Truncated octahedron Omnitruncated tetrahedron.png Snub-polyhedron-great-icosahedron.png Ih (Th) | 2 3/2 3/2
(3.3.3.3.3)/2
Snub cube Snub hexahedron.png Truncated cuboctahedron Great rhombicuboctahedron.png Snub-polyhedron-snub-cube.png O | 4 3 2
3.3.3.3.4
Snub dodecahedron Snub dodecahedron ccw.png Truncated icosidodecahedron Great rhombicosidodecahedron.png Snub-polyhedron-snub-dodecahedron.png I | 5 3 2
3.3.3.3.5
Small snub icosicosidodecahedron Small snub icosicosidodecahedron.png Doubly covered truncated icosahedron Truncated icosahedron.png Snub-polyhedron-small-snub-icosicosidodecahedron.png Ih | 3 3 5/2
3.3.3.3.3.5/2
Snub dodecadodecahedron Snub dodecadodecahedron.png Small rhombidodecahedron with extra 12{10/2} faces Small rhombidodecahedron.png Snub-polyhedron-snub-dodecadodecahedron.png I | 5 5/2 2
3.3.5/2.3.5
Snub icosidodecadodecahedron Snub icosidodecadodecahedron.png Icositruncated dodecadodecahedron Icositruncated dodecadodecahedron.png Snub-polyhedron-snub-icosidodecadodecahedron.png I | 5 3 5/3
3.5/3.3.3.3.5
Great snub icosidodecahedron Great snub icosidodecahedron.png Rhombicosahedron with extra 12{10/2} faces Rhombicosahedron.png Snub-polyhedron-great-snub-icosidodecahedron.png I | 3 5/2 2
3.3.5/2.3.3
Inverted snub dodecadodecahedron Inverted snub dodecadodecahedron.png Truncated dodecadodecahedron Truncated dodecadodecahedron.png Snub-polyhedron-inverted-snub-dodecadodecahedron.png I | 5 2 5/3
3.5/3.3.3.3.5
Great snub dodecicosidodecahedron Great snub dodecicosidodecahedron.png Great dodecicosahedron with extra 12{10/2} faces Great dodecicosahedron.png no image yet I | 3 5/2 5/3
3.5/3.3.5/2.3.3
Great inverted snub icosidodecahedron Great inverted snub icosidodecahedron.png Great truncated icosidodecahedron Great truncated icosidodecahedron.png Snub-polyhedron-great-inverted-snub-icosidodecahedron.png I | 3 2 5/3
3.5/3.3.3.3
Small retrosnub icosicosidodecahedron Small retrosnub icosicosidodecahedron.png Doubly covered truncated icosahedron Truncated icosahedron.png no image yet Ih | 5/2 3/2 3/2
(3.3.3.3.3.5/2)/2
Great retrosnub icosidodecahedron Great retrosnub icosidodecahedron.png Great rhombidodecahedron with extra 20{6/2} faces Great rhombidodecahedron.png no image yet I | 2 5/3 3/2
(3.3.3.5/2.3)/2
Great dirhombicosidodecahedron Great dirhombicosidodecahedron.png Ih | 3/2 5/3 3 5/2
(4.3/2.4.5/3.4.3.4.5/2)/2
Great disnub dirhombidodecahedron Great disnub dirhombidodecahedron.png 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.

Snub polyhedron Image Original omnitruncated polyhedron Image Snub derivation Symmetry group Wythoff symbol
Vertex description
Tetrahedron Linear antiprism.png Cube Uniform polyhedron 222-t012.png Snub-polyhedron-tetrahedron.png Td (D2d) | 2 2 2
3.3.3
Octahedron Trigonal antiprism.png Hexagonal prism Uniform polyhedron-23-t012.png Snub-polyhedron-octahedron.png Oh (D3d) | 3 2 2
3.3.3.3
Square antiprism Square antiprism.png Octagonal prism Octagonal prism.png Snub-polyhedron-square-antiprism.png D4d | 4 2 2
3.4.3.3
Pentagonal antiprism Pentagonal antiprism.png Decagonal prism Decagonal prism.png Snub-polyhedron-pentagonal-antiprism.png D5d | 5 2 2
3.5.3.3
Pentagrammic antiprism Pentagrammic antiprism.png Doubly covered pentagonal prism Pentagonal prism.png Snub-polyhedron-pentagrammic-antiprism.png D5h | 5/2 2 2
3.5/2.3.3
Pentagrammic crossed-antiprism Pentagrammic crossed antiprism.png Decagrammic prism Prism 10-3.png Snub-polyhedron-pentagrammic-crossed-antiprism.png D5d | 2 2 5/3
3.5/3.3.3
Hexagonal antiprism Hexagonal antiprism.png Dodecagonal prism Dodecagonal prism.png Snub-polyhedron-hexagonal-antiprism.png D6d | 6 2 2
3.6.3.3

Notes:

Non-uniform[edit]

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 Snub disphenoid.png Disphenoid Disphenoid tetrahedron.png D2d
Snub square antiprism Snub square antiprism.png Square antiprism Square antiprism.png D4d

References[edit]

Seed Truncation Rectification Bitruncation Dual Expansion Omnitruncation Alternations
CDel node 1.pngCDel p.pngCDel node n1.pngCDel q.pngCDel node n2.png CDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.png CDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.png CDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node 1.png CDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.png CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.png CDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node 1.png CDel node h.pngCDel p.pngCDel node.pngCDel q.pngCDel node.png CDel node.pngCDel p.pngCDel node h.pngCDel q.pngCDel node h.png CDel node h.pngCDel p.pngCDel node h.pngCDel q.pngCDel node h.png
Uniform polyhedron-43-t0.png Uniform polyhedron-43-t01.png Uniform polyhedron-43-t1.png Uniform polyhedron-43-t12.png Uniform polyhedron-43-t2.png Uniform polyhedron-43-t02.png Uniform polyhedron-43-t012.png Uniform polyhedron-33-t0.png Uniform polyhedron-43-h01.png Uniform polyhedron-43-s012.png
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}