In geometry, an **alternation** or *partial truncation*, is an operation on a polygon, polyhedron, tiling, or higher dimensional polytope that removes alternate vertices.^{ [1] }

- Snub
- Alternated polytopes
- Altered polyhedra
- Alternate truncations
- See also
- References
- External links

Coxeter labels an *alternation* by a prefixed *h*, standing for *hemi* or *half*. Because alternation reduces all polygon faces to half as many sides, it can only be applied to polytopes with all even-sided faces. An alternated square face becomes a digon, and being degenerate, is usually reduced to a single edge.

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 of 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 divides 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 as the degenerate faces of the original cube.

A **snub** (in Coxeter's terminology) can be seen as an **alternation** of a truncated regular or truncated quasiregular polyhedron. In general a polyhedron can be snubbed if its truncation has only even-sided faces. All truncated rectified polyhedra can be snubbed, not just from regular polyhedra.

The snub square antiprism is an example of a general snub, and can be represented by ss{2,4}, with the square antiprism, s{2,4}.

This *alternation* operation applies to higher-dimensional polytopes and honeycombs as well, but in general most of the results of this operation will not be uniform. The voids created by the deleted vertices will not in general create uniform facets, and there are typically not enough degrees of freedom to allow an appropriate rescaling of the new edges. Exceptions do exist, however, such as the derivation of the snub 24-cell from the truncated 24-cell.

Examples:

- Honeycombs
- An alternated cubic honeycomb is the tetrahedral-octahedral honeycomb.
- An alternated hexagonal prismatic honeycomb is the gyrated alternated cubic honeycomb.

- 4-polytope
- An alternated truncated 24-cell is the snub 24-cell.

- 4-honeycombs:
- An alternated truncated 24-cell honeycomb is the snub 24-cell honeycomb.

- 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)
- ...

Coxeter also used the operator *a*, which contains both halves, so retains the original symmetry. For even-sided regular polyhedra, a{2p,q} represents a compound polyhedron with two opposite copies of h{2p,q}. For odd-sided, greater than 3, regular polyhedra a{p,q}, becomes a star polyhedron.

Norman Johnson extended the use of the **altered** operator *a*{p,q}, *b*{p,q} for **blended**, and *c*{p,q} for **converted**, as , , and respectively.

The compound polyhedron known as the stellated octahedron can be represented by a{4,3} (an altered cube), and , .

The star polyhedron known as the small ditrigonal icosidodecahedron can be represented by a{5,3} (an altered dodecahedron), and , . Here all the pentagons have been alternated into pentagrams, and triangles have been inserted to take up the resulting free edges.

The star polyhedron known as the great ditrigonal icosidodecahedron can be represented by a{5/2,3} (an altered great stellated dodecahedron), and , . Here all the pentagrams have been alternated back into pentagons, and triangles have been inserted to take up the resulting free edges.

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 | Alternated truncation | 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 |

In geometry, a **polyhedral compound** is a figure that is composed of several polyhedra sharing a common centre. They are the three-dimensional analogs of polygonal compounds such as the hexagram.

In geometry, the **Schläfli symbol** is a notation of the form that defines regular polytopes and tessellations.

In geometry, a **vertex figure**, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off.

In geometry, a **uniform 4-polytope** is a 4-dimensional polytope which is vertex-transitive and whose cells are uniform polyhedra, and faces are regular polygons.

In geometry of 4 dimensions or higher, a **duoprism** is a polytope resulting from the Cartesian product of two polytopes, each of two dimensions or higher. The Cartesian product of an n-polytope and an m-polytope is an (*n*+*m*)-polytope, where n and m are dimensions of 2 (polygon) or higher.

In Euclidean geometry, **rectification**, also known as **critical truncation** or **complete-truncation**, is the process of truncating a polytope by marking the midpoints of all its edges, and cutting off its vertices at those points. The resulting polytope will be bounded by vertex figure facets and the rectified facets of the original polytope.

In geometry, a **uniform polyhedron** has regular polygons as faces and is vertex-transitive. It follows that all vertices are congruent.

In geometry, the **great icosahedron** is one of four Kepler-Poinsot polyhedra, with Schläfli symbol {3,5⁄2} and Coxeter-Dynkin diagram of . It is composed of 20 intersecting triangular faces, having five triangles meeting at each vertex in a pentagrammic sequence.

The **cubic honeycomb** or **cubic cellulation** is the only proper regular space-filling tessellation in Euclidean 3-space made up of cubic cells. It has 4 cubes around every edge, and 8 cubes around each vertex. Its vertex figure is a regular octahedron. It is a self-dual tessellation with Schläfli symbol {4,3,4}. John Horton Conway called this honeycomb a **cubille**.

In geometry, a **snub polyhedron** is a polyhedron obtained by performing a snub operation: alternating a corresponding omnitruncated or truncated polyhedron, depending on the definition. Some, but not all, authors include antiprisms as snub polyhedra, as they are obtained by this construction from a degenerate "polyhedron" with only two faces.

In geometry, a **truncation** is an operation in any dimension that cuts polytope vertices, creating a new facet in place of each vertex. The term originates from Kepler's names for the Archimedean solids.

In four-dimensional geometry, a **runcinated 24-cell** is a convex uniform 4-polytope, being a runcination of the regular 24-cell.

In geometry, a **uniform polytope** of dimension three or higher is a vertex-transitive polytope bounded by uniform facets. The uniform polytopes in two dimensions are the regular polygons.

In geometry, a **quasiregular polyhedron** is a uniform polyhedron that has exactly two kinds of regular faces, which alternate around each vertex. They are vertex-transitive and edge-transitive, hence a step closer to regular polyhedra than the semiregular, which are merely vertex-transitive.

In geometry, a **snub** is an operation applied to a polyhedron. The term originates from Kepler's names of two Archimedean solids, for the snub cube and snub dodecahedron. In general, snubs have chiral symmetry with two forms: with clockwise or counterclockwise orientation. By Kepler's names, a snub can be seen as an expansion of a regular polyhedron: moving the faces apart, twisting them about their centers, adding new polygons centered on the original vertices, and adding pairs of triangles fitting between the original edges.

- ↑ Coxeter, Regular polytopes, pp. 154–156 8.6 Partial truncation, or alternation

- Coxeter, H.S.M.
*Regular Polytopes*, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8 - Norman Johnson
*Uniform Polytopes*, Manuscript (1991)- N.W. Johnson:
*The Theory of Uniform Polytopes and Honeycombs*, Ph.D. Dissertation, University of Toronto, 1966

- N.W. Johnson:
- 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)

- Olshevsky, George. "Alternation".
*Glossary for Hyperspace*. Archived from the original on 4 February 2007. - Polyhedra Names, snub

Seed | Truncation | Rectification | Bitruncation | Dual | Expansion | Omnitruncation | Alternations | ||
---|---|---|---|---|---|---|---|---|---|

t_{0}{p,q} { p,q} | t_{01}{p,q} t{ p,q} | t_{1}{p,q} r{ p,q} | t_{12}{p,q} 2t{ p,q} | t_{2}{p,q} 2r{ p,q} | t_{02}{p,q} rr{ p,q} | t_{012}{p,q} tr{ p,q} | ht_{0}{p,q} h{ q,p} | ht_{12}{p,q} s{ q,p} | ht_{012}{p,q} sr{ p,q} |

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