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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,
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)

A snub polyhedron is a polyhedron obtained by 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 forms which are reflections of each other. Their symmetry groups are all point groups.

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.

Contents

List of snub polyhedraEdit

UniformEdit

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
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. 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-uniformEdit

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

ReferencesEdit

  • 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: 401–450, doi:10.1098/rsta.1954.0003, ISSN 0080-4614, JSTOR 91532, MR 0062446
  • 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: 111–135, doi:10.1098/rsta.1975.0022, ISSN 0080-4614, JSTOR 74475, MR 0365333
  • 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}