# Alternation (geometry)

Two snub cubes from truncated cuboctahedron

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.

A rhombicuboctahedron can be transformed into a snub cube by rotating the 6 red square faces until the 12 white square become pairs of equilateral triangles.

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 2.3.2.3.2.3 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.

## Snub

The snubs can be seen as an alternation of a truncated quasiregular polyhedron.

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.

Symmetry
(p q 2)
Regular
Dual
regular
Quasiregular
Truncated quasiregular
Snub
Tetrahedral
(3 3 2)

Tetrahedron

Tetrahedron

Tetratetrahedron

Truncated tetratetrahedron

Icosahedron
(Snub tetratetrahedron)
Octahedral
(4 3 2)

Cube

Octahedron

Cuboctahedron

Truncated cuboctahedron

Snub cube
(Snub cuboctahedron)
Icosahedral
(5 3 2)

Dodecahedron

Icosahedron

Icosidodecahedron

Truncated icosidodecahedron

Snub dodecahedron
(Snub icosidodecahedron)
Euclidean tilings
Square
(4 4 2)

Square tiling

Square tiling

Square tiling

Truncated square tiling

Snub square tiling
Hexagonal
(6 3 2)

Hexagonal tiling

Triangular tiling

Trihexagonal tiling

Truncated trihexagonal tiling

Snub trihexagonal tiling
Hyperbolic tilings
Tetrapentagonal
(5 4 2)

Order-4 pentagonal tiling

Order-5 square tiling

tetrapentagonal tiling

Truncated tetrapentagonal tiling

Snub tetrapentagonal tiling
Pentapentagonal
(5 5 2)

Order-5 pentagonal tiling

Order-5 pentagonal tiling

Pentapentagonal tiling

Truncated pentapentagonal tiling

Snub pentapentagonal tiling
Tetrahexagonal
(6 4 2)

Order-4 hexagonal tiling

Order-6 square tiling

Tetrahexagonal tiling

Truncated tetrahexagonal tiling

Snub tetrahexagonal tiling
Triheptagonal
(7 3 2)

Heptagonal tiling

Order-7 triangular tiling

Triheptagonal tiling

Truncated triheptagonal tiling

Snub triheptagonal tiling
Trioctagonal
(8 3 2)

Octagonal tiling

Order-8 triangular tiling

Trioctagonal tiling

Truncated trioctagonal tiling

Snub trioctagonal tiling
Tetraoctagonal
(8 4 2)

Order-4 octagonal tiling

Order-8 square tiling

Tetraoctagonal tiling

Truncated tetraoctagonal tiling

Snub tetraoctagonal tiling

## Examples

### 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 prismsn-gonal 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
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 in general neither will create uniform facets nor the degree of freedom allows for an appropriate rescaling of the new edges.

Examples: