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Snub polyhedron

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Title: Snub polyhedron  
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Subject: Omnitruncation, List of uniform polyhedra by Wythoff symbol, List of uniform polyhedra by spherical triangle, Polyhedron types, Snub (geometry)
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Snub polyhedron


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:

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 4-polytope. The tetrahedron, octahedron, icosahedron, and great icosahedron appear commonly in non-prismatic uniform 4-polytopes, but not in their snub constructions. Every snub polyhedron however can appear in the polyhedral prism based on them.

Contents

  • List of snub polyhedra 1
    • Uniform 1.1
    • Non-uniform 1.2
  • References 2

List of snub polyhedra

Uniform

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) 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 Truncated cuboctahedron O | 4 3 2
3.3.3.3.4
Snub dodecahedron 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
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 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.

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

  • Mäder, R. E. Uniform Polyhedra. Mathematica J. 3, 48-57, 1993.
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