User:Tomruen/Snub (geometry)

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

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 $s{\begin{Bmatrix}4\\3\end{Bmatrix}}$ and Coxeter diagram . He gave the cuboctahedron the symbol ${\begin{Bmatrix}4\\3\end{Bmatrix}}$ , Coxeter diagram , and truncated cuboctahedron as $t{\begin{Bmatrix}4\\3\end{Bmatrix}}$ , Coxeter diagram . The rhombitruncated cuboctahedron he represented by $r{\begin{Bmatrix}4\\3\end{Bmatrix}}$ , 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

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

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

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

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

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]

External links

Polyhedra Names, snub

Category:Polyhedra Category:Polychora Category:Polytopes