Snub (geometry)
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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.
-
- cube --> tetrahedron
- hexagonal prism --> octahedron
- octagonal prism --> square antiprism
- decagonal prism --> pentagonal 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:
- Honeycombs
- An alternately truncated cubic honeycomb is the tetrahedral-octahedral honeycomb.
- An alternately truncated hexagonal prismatic honeycomb is the gyrated alternated cubic honeycomb.
- Polychora
- An alternately truncated truncated 24-cell is the snub 24-cell.
- A Measure polytope can always be alternatingly truncated into a uniform half measure polytope.
- Cube --> Tetrahedron (regular)
- Tesseract (8-cell) --> 16-cell (regular)
- Penteract --> demipenteract (semiregular)
- Hexeract --> demihexeract (semiregular)
- ...