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.
More generally any vertex-uniform polyhedron or tiling with a vertex configuration consisting of all even-numbered elements can be alternated. For example the alternation a vertex figure with 2a.2b.2c is a.3.b.3.c.3 where the three is the number of elements in this vertex figure. A special case is square faces whose order divide in half into degenerate digons. So for example, the cube 4.4.4 is alternated as 126.96.36.199.2.3 which is reduced to 3.3.3, being the tetrahedron, and all the 6 edges of the tetrahedra can also be seen degenerate faces of the original cube.
For instance, the snub cube is created in three topological constructive steps. First it is rectified into a cuboctahedron, second truncated creating the truncated cuboctahedron, and lastly alternated into the 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. Finally all edges would be rescaled to unity. - Note: this latter part depends on the degree of freedom.
The Coxeter-Dynkin diagrams are given showing the active mirrors in the Wythoff construction. The truncated quasiregular form is also called an omnitruncation with all of the mirrors active (ringed). The alternation is shown as rings with holes.
The original regular form faces are show in red. The regular dual faces are in yellow. The quasiregular polyhedron has all the red and yellow faces combined. The truncated quasiregular form has new square faces in blue. In the snub, the blue squares are reduced to edges, and new blue triangles are shown in the alternated vertex gaps.
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.)
- square prism → digonal antiprism (or demicube)
- hexagonal prism → triangular antiprism
- octagonal prism → square antiprism
- decagonal prism → pentagonal antiprism
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|
Dual of rectified tetrahedron
|Alternate truncated cube|
Dual of cuboctahedron
|Truncated rhombic dodecahedron|
Dual of icosidodecahedron
|Truncated rhombic triacontahedron|
Dual of truncated tetrahedron
|Truncated triakis tetrahedron|
Dual of truncated cube
|Truncated triakis octahedron|
Dual of truncated dodecahedron
|Truncated triakis icosahedron|
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 in general neither will create uniform facets nor the degree of freedom allows for an appropriate rescaling of the new edges.
- A hypercube can always be alternated into a uniform demihypercube.
- Other operators on uniform polytopes:
- Conway polyhedral notation
- Wythoff construction
- 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", MathWorld.
- Richard Klitzing, Snubs, alternated facetings, and Stott-Coxeter-Dynkin diagrams, Symmetry: Culture and Science, Vol. 21, No.4, 329-344, (2010)