Snub (geometry)

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 Two snub cubes from great rhombicuboctahedronSee 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.
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Two snub cubes from great rhombicuboctahedron

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.

A snub is an operation on a polyhedron which fully truncates alternate vertices. Only zonohedra can have this operation performed so every 2n-sided face becomes n-sided.

A snubbed regular polyhedron is generated in two steps. First a {p,q} regular polyhedron is omnitruncated. This creates a vertex configuration 4.2p.2q. Then this form is snubbed. The squares become degenerated into edges, and new triangle faces form at each original vertex, creating vertex configuration 3.3.p.3.q.

A snub operations has two choices of vertices and so for some polyhedra, it can create two chiral forms.

Non-uniform zonohedra can also snubbed. For instance, the Rhombic triacontahedron can be snubbed into either an icosahedron or a dodecahedron depending on which vertices are rectified.

Contents

[edit] Examples

[edit] Platonic solid generators

Three steps: regular --> omnitruncated --> snubbed

Symmetry Regular Omnitruncated Snub
Tetrahedral
(3 3 2)

Tetrahedron

truncated octahedron

icosahedron
Octahedral
(4 3 2)

Cube

Great rhombicuboctahedron

snub cube
Icosahedral
(5 3 2)

Dodecahedron

Great rhombicosidodecahedron

snub dodecahedron

[edit] Regular tiling generators

Symmetry Regular Omnitruncated Snub
Square
(4 4 2)

(4.4.4.4)

(4.8.8)

(3.3.4.3.4)
Hexagonal
(6 3 2)

(6.6.6)

(3.4.6.4)

3.3.3.3.6

[edit] Uniform prism generators (dihedral symmetry)

Alternate truncations can be applied to prisms, although the term snub is usually reserved for omnitruncated regular polyhedra. (A square antiprism may be called a snubbed octagonal prism, but naming a snubbed cube would be ambiguous for the use above.)

Two steps: 2n-gonal prisms --> n-gonal antiprism.

[edit] Higher dimensions

This alternate truncation (snub) operation applies to higher dimensional polytopes as well, however in general most forms won't have uniform solution.

Examples: