In geometry, an alternation (also called partial truncation) is an operation on a polyhedron or tiling that fully truncates alternate vertices. Only even-sided polyhedra can be alternated, for example the zonohedra. Every 2n-sided face becomes n-sided. Square faces disappear into new edges.
An alternation of a regular polyhedron or tiling is sometimes labeled by the regular form, prefixed by an h, standing for half. For example h{4,3} is an alternated cube (creating a tetrahedron), and h{4,4} is an alternated square tiling (still a square tiling).
Snub
A snub is a related operation. It is an alternation applied to an omnitruncated regular polyhedron. An omnitruncated regular polyhedron or tiling always has even-sided faces and so can always be alternated.For instance the snub cube is created in two steps. First it is omnitruncated, creating the great rhombicuboctahedron. Secondly that polyhedron is alternated into a 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.
Another example is the uniform antiprisms. A uniform n-gonal antiprism can be constructed as an alternation of a 2n-gonal prism, and the snub of an n-edge hosohedron. In the case of prisms both alternated forms are identical.
Non-uniform zonohedra can also be alternated. For instance, the Rhombic triacontahedron can be snubbed into either an icosahedron or a dodecahedron depending on which vertices are removed.
Examples
Platonic solid generators
Three forms: regular --> omnitruncated --> snub.
The Coxeter-Dynkin diagrams are given as well. The omnitruncation actives all of the mirrors (ringed). The alternation is shown as rings with holes.
| Symmetry (p q 2) | ! Omnitruncated ! Snub >- | Tetrahedral (3 3 2) | Tetrahedron > truncated octahedron | icosahedron (snub tetrahedron) |- | Octahedral (4 3 2) | Cube > Great rhombicuboctahedron | snub cube |- | Icosahedral (5 3 2) | Dodecahedron > Great rhombicosidodecahedron | snub dodecahedron |
|---|
Regular tiling generators
| Symmetry (p q 2) | ! 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 |
|---|
Uniform prism generators (dihedral symmetry)
Alternate truncations can be applied to prisms. (A square antiprism may be called a snubbed 4-edge hosohedron, as well as an alternated octagonal prism.)
Two steps: 2n-gonal prisms --> n-gonal antiprism.
- square prism --> dihedral antiprism
- -->
- hexagonal prism --> triangular antiprism
- -->
- octagonal prism --> square antiprism
- -->
- decagonal prism --> pentagonal antiprism
- -->
- ....
Alternate truncations
A similar operation can truncate alternate vertices, rather than just removing them. Below is a set of polyhedra that can be generated from the duals of Catalan solids. These have two types of vertices which can be alternately truncated. Truncating the "higher order" vertices produces these forms:
| Name | Original | Truncation | Truncated name |
|---|---|---|---|
| Cube Dual of rectified tetrahedron | Alternate truncated cube | ||
| Rhombic dodecahedron Dual of cuboctahedron | Truncated rhombic dodecahedron | ||
| Rhombic triacontahedron Dual of icosidodecahedron | Truncated rhombic triacontahedron | ||
| Triakis tetrahedron Dual of truncated tetrahedron | Truncated triakis tetrahedron | ||
| Triakis octahedron Dual of truncated cube | Truncated triakis octahedron | ||
| Triakis icosahedron Dual of truncated dodecahedron | Truncated triakis icosahedron |
Higher dimensions
This alternation operation applies to higher dimensional polytopes and honeycombs as well, however in general most forms won't have uniform solution. The voids created by the deleted vertices will not in general create uniform facets.
Examples:
- Honeycombs
- An alternated cubic honeycomb is the tetrahedral-octahedral honeycomb.
- An alternated hexagonal prismatic honeycomb is the gyrated alternated cubic honeycomb.
- Polychora
- An alternated truncated 24-cell is the snub 24-cell.
- A hypercube can always be alternated into a uniform demihypercube.
- Cube --> Tetrahedron (regular)
- -->
- Tesseract (8-cell) --> 16-cell (regular)
- -->
- Penteract --> demipenteract (semiregular)
- Hexeract --> demihexeract (uniform)
- ...
See also
- Other operators on uniform polytopes:
- Truncation (geometry)
- Rectification (geometry)
- Omnitruncation (geometry)
- Cantellation (geometry)
- Runcination (geometry)
- Conway polyhedral notation
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)
External links
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Last updated on Friday July 25, 2008 at 10:54:21 PDT (GMT -0700)
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