- Alternation (geometry)
In geometry, an

**alternation**(also called "partial truncation") is an operation on apolyhedron 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 atetrahedron ), and h{4,4} is an alternatedsquare 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 anicosahedron or adodecahedron depending on which vertices are removed.**Examples****Platonic solid generators**Three forms: regular --> omnitruncated --> snub.

The

Coxeter-Dynkin diagram s 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 alternatedcubic honeycomb is thetetrahedral-octahedral honeycomb .

*# An alternatedhexagonal prismatic honeycomb is thegyrated alternated cubic honeycomb .

* Polychora

*# An alternatedtruncated 24-cell is thesnub 24-cell .

* Ahypercube can always be alternated into a uniformdemihypercube .

*#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

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