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User:Tomruen/Snub (geometry)

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Two snub cubes from truncated cuboctahedron

See that red and green dots are placed at alternate vertices. A snub cube is generated from deleting either set of vertices, one resulting in clockwise gyrated squares, and other counterclockwise.

In geometry, a snub is an operation on a regular or quasi-regular polyhedron or tiling that defines a related polyhedron or tiling with the same faces with new triangle faces. The term originates from two semiregular polyhedron, named by Kepler named as snub cube (cubus simum) and snub dodecahedron (dodekaeder simum). Snubs contain chiral symmetry that arises from an alternation operation.

History

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In Wythoff constructions there are two competing definitions that are consistent for polyhedra, but diverge on higher polytopes. The first definition follows Kepler's naming from the regular forms. In this construction the snub cube and snub octahedron are two names for the same polyhedron, containing the 6 faces of the cube, the 8 faces of the octahedron, and two triangles in place of each of the 12 edges of either regular polyhedron.

Coxeter recognized both regular polyhedra as generators and wrote the snub cube as a vertical Schläfli symbol and Coxeter diagram . He gave the cuboctahedron the symbol , Coxeter diagram , and truncated cuboctahedron as , Coxeter diagram . The rhombitruncated cuboctahedron he represented by , Coxeter diagram . From these constructions is is apparent that a snub is an alternation of a truncated quasiregular polyhedron. So he considered the snub cube more constructionally explicitly named from the quasiregular polyhedron, the cuboctahedron as a snub cuboctahedron.

This naming aided Coxeter in naming the only uniform snub in 4D, the snub 24-cell, , as an alternated truncated 24-cell, with the 24-cell and truncated 24-cell .

From this interpretation the snub octahedron is not identical to the snub cube (snub cuboctahedron), but construcated as an alternated truncated octahedron, , which is a lower symmetry form of the icosahedron. In addition a snub tetratetrahedron is another name with the tetratetrahedron as a bicolored octahedron, which can be truncated, and then alternated also into an icosahedron.

Constructions

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The snubs can be seen as created in two topological constructive steps.

First truncated creating the truncated cuboctahedron, and lastly alternated into the snub cube. You can see from the picture on the right that there are two ways to alternate the vertices, and they are mirror images of each other, creating two chiral forms. Finally all edges would be rescaled to unity. - Note: this latter part depends on the degree of freedom.

The Coxeter-Dynkin diagrams are given showing the active mirrors in the Wythoff construction. The truncated quasiregular form is also called an omnitruncation with all of the mirrors active (ringed). The alternation is shown as rings with holes.

The original regular form faces are show in red. The regular dual faces are in yellow. The quasiregular polyhedron has all the red and yellow faces combined. The truncated quasiregular form has new square faces in blue. In the snub, the blue squares are reduced to edges, and new blue triangles are shown in the alternated vertex gaps.

Snub polyhedron

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Symmetry
(p q 2)
Regular
Dual
regular
Quasiregular
Truncated quasiregular
Snub
Tetrahedral
(3 3 2)

Tetrahedron

Tetrahedron

Tetratetrahedron

Truncated tetratetrahedron

Icosahedron
(Snub tetratetrahedron)
Octahedral
(4 3 2)

Cube

Octahedron

Cuboctahedron

Truncated cuboctahedron

Snub cube
(Snub cuboctahedron)
Icosahedral
(5 3 2)

Dodecahedron

Icosahedron

Icosidodecahedron

Truncated icosidodecahedron

Snub dodecahedron
(Snub icosidodecahedron)

Snub Euclidean tilings

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Symmetry
(p q 2)
Regular
Dual
regular
Quasiregular
Truncated quasiregular
Snub
Square
(4 4 2)

Square tiling

Square tiling

Square tiling

Truncated square tiling

Snub square tiling
Hexagonal
(6 3 2)

Hexagonal tiling

Triangular tiling

Trihexagonal tiling

Truncated trihexagonal tiling

Snub trihexagonal tiling

Snub hyperbolic tilings

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Symmetry
(p q 2)
Regular
Dual
regular
Quasiregular
Truncated quasiregular
Snub
Tetrapentagonal
(5 4 2)

Order-4 pentagonal tiling

Order-5 square tiling

tetrapentagonal tiling

Truncated tetrapentagonal tiling

Snub tetrapentagonal tiling
Pentapentagonal
(5 5 2)

Order-5 pentagonal tiling

Order-5 pentagonal tiling

Pentapentagonal tiling

Truncated pentapentagonal tiling

Snub pentapentagonal tiling
Tetrahexagonal
(6 4 2)

Order-4 hexagonal tiling

Order-6 square tiling

Tetrahexagonal tiling

Truncated tetrahexagonal tiling

Snub tetrahexagonal tiling
Triheptagonal
(7 3 2)

Heptagonal tiling

Order-7 triangular tiling

Triheptagonal tiling

Truncated triheptagonal tiling

Snub triheptagonal tiling
Tetraheptagonal
(7 4 2)

Order-4 heptagonal tiling

Order-7 square tiling

Tetraheptagonal tiling

Truncated tetraheptagonal tiling

Snub tetraheptagonal tiling
Trioctagonal
(8 3 2)

Octagonal tiling

Order-8 triangular tiling

Trioctagonal tiling

Truncated trioctagonal tiling

Snub trioctagonal tiling
Tetraoctagonal
(8 4 2)

Order-4 octagonal tiling

Order-8 square tiling

Tetraoctagonal tiling

Truncated tetraoctagonal tiling

Snub tetraoctagonal tiling

References

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  • Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 (pp. 154–156 8.6 Partial truncation, or alternation)
  • Norman Johnson Uniform Polytopes, Manuscript (1991)
    • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
  • Weisstein, Eric W. "Snubification". MathWorld.
  • Richard Klitzing, Snubs, alternated facetings, and Stott-Coxeter-Dynkin diagrams, Symmetry: Culture and Science, Vol. 21, No.4, 329-344, (2010) [1]
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Category:Polyhedra Category:Polychora Category:Polytopes