Alternation (geometry)

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.

In geometry, an alternation (also called partial truncation, snub or snubification) is an operation on a polyhedron or tiling that removes 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.

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)	Regular	Omnitruncated	Snub
Tetrahedral (3 3 2)	Tetrahedron	truncated octahedron	icosahedron (snub tetrahedron)
Octahedral (4 3 2)	Cube	Truncated cuboctahedron	snub cube
Icosahedral (5 3 2)	Dodecahedron	Truncated icosidodecahedron	snub dodecahedron

Regular tiling generators

Symmetry (p q 2)	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

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 → digonal antiprism (or demicube)
  • →
• 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 Catalan solids. These have two types of vertices which can be alternately truncated. Truncating the "higher order" vertices and both vertex types produce these forms:

Name	Original	Truncation	Full 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			50px	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
  1. An alternated cubic honeycomb is the tetrahedral-octahedral honeycomb.
  2. An alternated hexagonal prismatic honeycomb is the gyrated alternated cubic honeycomb.
• Polychora
  1. An alternated truncated 24-cell is the snub 24-cell.
• A hypercube can always be alternated into a uniform demihypercube.
  1. Cube → Tetrahedron (regular)
     • →
  2. Tesseract (8-cell) → 16-cell (regular)
     • →
  3. Penteract → demipenteract (semiregular)
  4. Hexeract → demihexeract (uniform)
  5. ...

See also

• Other operators on uniform polytopes:
  • Truncation (geometry)
  • Rectification (geometry)
  • Omnitruncation (geometry)
  • Cantellation (geometry)
  • Runcination (geometry)
• Conway polyhedral notation
• Wythoff construction
• Chiral polytope

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)
• Norman Johnson Uniform Polytopes, Manuscript (1991)
  • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
• Weisstein, Eric W., "Snubification" from MathWorld.

External links

• Olshevsky, George, Alternation at Glossary for Hyperspace.
• Polyhedra Names, snub