Alternation (geometry)

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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.

In geometry, an alternation or partial truncation, is an operation on a polyhedron or tiling that removes alternate vertices.[1] 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[edit]

Further information: Snub (geometry)

A snub 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}.

Alternated polytopes[edit]

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:

Altered polyhedra[edit]

Coxeter also used the operator a, which contains both halfs, 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 CDel node h3.pngCDel p.pngCDel node.pngCDel q.pngCDel node.png, CDel node.pngCDel p.pngCDel node h3.pngCDel q.pngCDel node.png, and CDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node h3.png respectively.

The compound polyhedron, stellated octahedron can be represented by a{4,3}, and CDel node h3.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png, Compound of two tetrahedra.png.

The star-polyhedron, small ditrigonal icosidodecahedron, can be represented by a{5,3}, and CDel node h3.pngCDel 5.pngCDel node.pngCDel 3.pngCDel node.png, Small ditrigonal icosidodecahedron.png.

Alternate truncations[edit]

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
Hexahedron.jpg Alternate truncated cube.png Uniform polyhedron-43-t01.png Alternate truncated cube
Rhombic dodecahedron
Dual of cuboctahedron
Rhombicdodecahedron.jpg Truncated rhombic dodecahedron2.png StellaTruncRhombicDodeca.png Truncated rhombic dodecahedron
Rhombic triacontahedron
Dual of icosidodecahedron
Rhombictriacontahedron.svg Truncated rhombic triacontahedron.png StellaTruncRhombicTriaconta.png Truncated rhombic triacontahedron
Triakis tetrahedron
Dual of truncated tetrahedron
Triakistetrahedron.jpg Truncated triakis tetrahedron.png StellaTruncTriakisTetra.png Truncated triakis tetrahedron
Triakis octahedron
Dual of truncated cube
Triakisoctahedron.jpg Truncated triakis octahedron.png StellaTruncTriakisOcta.png Truncated triakis octahedron
Triakis icosahedron
Dual of truncated dodecahedron
Triakisicosahedron.jpg Truncated triakis icosahedron.png Truncated triakis icosahedron

See also[edit]

References[edit]

  1. ^ Coxeter, Regular polytopes, pp. 154–156 8.6 Partial truncation, or alternation

External links[edit]

Polyhedron operators
Seed Truncation Rectification Bitruncation Dual Expansion Omnitruncation Alternations
CDel node 1.pngCDel p.pngCDel node n1.pngCDel q.pngCDel node n2.png CDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.png CDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node.png CDel node.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node 1.png CDel node.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.png CDel node 1.pngCDel p.pngCDel node.pngCDel q.pngCDel node 1.png CDel node 1.pngCDel p.pngCDel node 1.pngCDel q.pngCDel node 1.png CDel node h.pngCDel p.pngCDel node.pngCDel q.pngCDel node.png CDel node.pngCDel p.pngCDel node h.pngCDel q.pngCDel node h.png CDel node h.pngCDel p.pngCDel node h.pngCDel q.pngCDel node h.png
Uniform polyhedron-43-t0.png Uniform polyhedron-43-t01.png Uniform polyhedron-43-t1.png Uniform polyhedron-43-t12.png Uniform polyhedron-43-t2.png Uniform polyhedron-43-t02.png Uniform polyhedron-43-t012.png Uniform polyhedron-33-t0.png Uniform polyhedron-43-h01.png Uniform polyhedron-43-s012.png
t0{p,q}
{p,q}
t01{p,q}
t{p,q}
t1{p,q}
r{p,q}
t12{p,q}
2t{p,q}
t2{p,q}
2r{p,q}
t02{p,q}
rr{p,q}
t012{p,q}
tr{p,q}
ht0{p,q}
h{q,p}
ht12{p,q}
s{q,p}
ht012{p,q}
sr{p,q}