Snub polyhedron

Polyhedron resulting from the snub operation From Wikipedia, the free encyclopedia

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).

More information Class, Number and properties ...
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)
Bipyramids (infinite)
Pyramids (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)
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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:

Thumb Thumb

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

List of snub polyhedra

Summarize
Perspective

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.

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

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

Non-uniform

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

More information Image, Original polyhedron ...
Snub polyhedron Image Original polyhedron Image Symmetry group
Snub disphenoid Thumb Disphenoid Thumb D2d
Snub square antiprism Thumb Square antiprism Thumb D4d
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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, Bibcode:1954RSPTA.246..401C, 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, Bibcode:1975RSPTA.278..111S, 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.
More information Seed, Truncation ...
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}
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