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Alternation (geometry)
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Alternation (geometry)
please note:
 the content below is remote from Wikipedia
 it has been imported raw for GetWiki
{{multiple image
 the content below is remote from Wikipedia
 it has been imported raw for GetWiki
 align = right  total_width = 400
 image1 = Polyhedron 4a.png
 image2 = Polyhedron 44 dual blue.png
 image3 = Polyhedron 4b.png
 footer = Alternation of a cube creates a tetrahedron.
}}{{multiple image
 image1 = Polyhedron 4a.png
 image2 = Polyhedron 44 dual blue.png
 image3 = Polyhedron 4b.png
 footer = Alternation of a cube creates a tetrahedron.
 align = right  total_width = 400
 image1 = Polyhedron great rhombi 68 subsolid snub left maxmatch.png
 image2 = Polyhedron great rhombi 68 max.png
 image3 = Polyhedron great rhombi 68 subsolid snub right maxmatch.png
 footer = Alternation of a truncated cuboctahedron creates a nonuniform snub cube.
}}In geometry, an alternation or partial truncation, is an operation on a polygon, polyhedron, tiling, or higher dimensional polytope that removes alternate vertices.Coxeter, Regular polytopes, pp. 154â€“156 8.6 Partial truncation, or alternationCoxeter 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 evensided faces. An alternated square face becomes a digon, and being degenerate, is usually reduced to a single edge.More generally any vertexuniform polyhedron or tiling with a vertex configuration consisting of all evennumbered 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. image1 = Polyhedron great rhombi 68 subsolid snub left maxmatch.png
 image2 = Polyhedron great rhombi 68 max.png
 image3 = Polyhedron great rhombi 68 subsolid snub right maxmatch.png
 footer = Alternation of a truncated cuboctahedron creates a nonuniform snub cube.
Snub
{{furtherSnub (geometry)}}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 evensided 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
This alternation operation applies to higherdimensional 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 24cell from the truncated 24cell.Examples: Honeycombs
 An alternated cubic honeycomb is the tetrahedraloctahedral honeycomb.
 An alternated hexagonal prismatic honeycomb is the gyrated alternated cubic honeycomb.
 4polytope
 An alternated truncated 24cell is the snub 24cell.
 4honeycombs:
 An alternated truncated 24cell honeycomb is the snub 24cell honeycomb.
 A hypercube can always be alternated into a uniform demihypercube.
 Cube â†’ Tetrahedron (regular)
 missing image!â†’
 hexahedron.png 
50pxmissing image!
 tetrahedron.png 
50px

 Tesseract (8cell) â†’ 16cell (regular)
 missing image!â†’
 Schlegel wireframe 8cell.png 
50pxmissing image!
 Schlegel wireframe 16cell.png 
50px

 Penteract â†’ demipenteract (semiregular)
 Hexeract â†’ demihexeract (uniform)
 ...
 Cube â†’ Tetrahedron (regular)
Altered polyhedra
Coxeter also used the operator a, which contains both halves, so retains the original symmetry. For evensided regular polyhedra, a{2p,q} represents a compound polyhedron with two opposite copies of h{2p,q}. For oddsided, 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 {{CDDnode_h3pnodeqnode}}, {{CDDnodepnode_h3qnode}}, and {{CDDnodepnodeqnode_h3}} respectively.The compound polyhedron, stellated octahedron can be represented by a{4,3}, and {{CDDnode_h34node3node}}, (File:Compound_of_two_tetrahedra.png40px).The starpolyhedron, small ditrigonal icosidodecahedron, can be represented by a{5,3}, and {{CDDnode_h35node3node}}, (File:Small ditrigonal icosidodecahedron.png40px). Here all the pentagons have been alternated into pentagrams, and triangles have been inserted to take up the resulting free edges.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:{ class="wikitable"!Name!Original!Alternatedtruncation!Truncation!Truncated nameCubeDual of rectified tetrahedronImage:hexahedron.jpg>50px  Image:Alternate truncated cube.png>50px  50px)Alternate truncated cube 
Image:rhombicdodecahedron.jpg>50px  Image:Truncated rhombic dodecahedron2.png>50px  50px)Truncated rhombic dodecahedron 
Image:Rhombictriacontahedron.svg>50px  Image:Truncated rhombic triacontahedron.png>50px  50px)Truncated rhombic triacontahedron 
Image:Triakistetrahedron.jpg>50px  Image:Truncated triakis tetrahedron.png>50px  50px)Truncated triakis tetrahedron 
Image:Triakisoctahedron.jpg>50px  Image:Truncated triakis octahedron.png>50px  50px)Truncated triakis octahedron 
Image:Triakisicosahedron.jpg>50px  Image:Truncated triakis icosahedron.png>50pxTruncated triakis icosahedron 
See also
References
{{reflist}} Coxeter, H.S.M. Regular Polytopes, (3rd edition, 1973), Dover edition, {{ISBN0486614808}}
 Norman Johnson Uniform Polytopes, Manuscript (1991)
 N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966
 {{mathworld  urlname = Snubification  title = Snubification}}
 Richard Klitzing, Snubs, alternated facetings, and StottCoxeterDynkin diagrams, Symmetry: Culture and Science, Vol. 21, No.4, 329344, (2010) weblink
External links
 {{GlossaryForHyperspace  anchor=Alternation  title=Alternation }}
 Polyhedra Names, snub
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