Alternation (geometry)

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".

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

*GlossaryForHyperspace | anchor=Alternation | title=Alternation