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D6 polytope

[[File:6-demicube t0 D6.svg160px]][6-demicube](6-demicube)[[File:6-cube t5 B5.svg160px]][6-orthoplex](6-orthoplex)

In 6-dimensional geometry, there are 47 uniform polytopes with D6 symmetry, of which 16 are unique and 31 are shared with the B6 symmetry. There are two regular forms, the 6-orthoplex, and 6-demicube with 12 and 32 vertices respectively.

They can be visualized as symmetric orthographic projections in Coxeter planes of the D6 Coxeter group, and other subgroups. TOC

Graphs

Symmetric orthographic projections of these 16 polytopes can be made in the D6, D5, D4, D3, A5, A3, Coxeter planes. Ak has [k+1] symmetry, Dk has [2(k-1)] symmetry. B6 is also included although only half of its [12] symmetry exists in these polytopes.

These 16 polytopes are each shown in these 7 symmetry planes, with vertices and edges drawn, and vertices colored by the number of overlapping vertices in each projective position.

#Coxeter plane graphsCoxeter diagramNamesB6[12/2]D6[10]D5[8]D4[6]D3[4]A5[6]A3[4]12345678910111213141516
[[File:6-demicube t0 B6.svg80px]][[File:6-demicube t0 D6.svg80px]][[File:6-demicube t0 D5.svg80px]][[File:6-demicube t0 D4.svg80px]][[File:6-demicube t0 D3.svg80px]][[File:6-demicube t0 A5.svg80px]][[File:6-demicube t0 A3.svg80px]]= [6-demicube](6-demicube)Hemihexeract (hax)
[[File:6-demicube t01 B6.svg80px]][[File:6-demicube t01 D6.svg80px]][[File:6-demicube t01 D5.svg80px]][[File:6-demicube t01 D4.svg80px]][[File:6-demicube t01 D3.svg80px]][[File:6-demicube t01 A5.svg80px]][[File:6-demicube t01 A3.svg80px]]= cantic 6-cubeTruncated hemihexeract (thax)
[[File:6-demicube t02 B6.svg80px]][[File:6-demicube t02 D6.svg80px]][[File:6-demicube t02 D5.svg80px]][[File:6-demicube t02 D4.svg80px]][[File:6-demicube t02 D3.svg80px]][[File:6-demicube t02 A5.svg80px]][[File:6-demicube t02 A3.svg80px]]= runcic 6-cubeSmall rhombated hemihexeract (sirhax)
[[File:6-demicube t03 B6.svg80px]][[File:6-demicube t03 D6.svg80px]][[File:6-demicube t03 D5.svg80px]][[File:6-demicube t03 D4.svg80px]][[File:6-demicube t03 D3.svg80px]][[File:6-demicube t03 A5.svg80px]][[File:6-demicube t03 A3.svg80px]]= steric 6-cubeSmall prismated hemihexeract (sophax)
[[File:6-demicube t04 B6.svg80px]][[File:6-demicube t04 D6.svg80px]][[File:6-demicube t04 D5.svg80px]][[File:6-demicube t04 D4.svg80px]][[File:6-demicube t04 D3.svg80px]][[File:6-demicube t04 A5.svg80px]][[File:6-demicube t04 A3.svg80px]]= pentic 6-cubeSmall cellated demihexeract (sochax)
[[File:6-demicube t012 B6.svg80px]][[File:6-demicube t012 D6.svg80px]][[File:6-demicube t012 D5.svg80px]][[File:6-demicube t012 D4.svg80px]][[File:6-demicube t012 D3.svg80px]][[File:6-demicube t012 A5.svg80px]][[File:6-demicube t012 A3.svg80px]]= runcicantic 6-cubeGreat rhombated hemihexeract (girhax)
[[File:6-demicube t013 B6.svg80px]][[File:6-demicube t013 D6.svg80px]][[File:6-demicube t013 D5.svg80px]][[File:6-demicube t013 D4.svg80px]][[File:6-demicube t013 D3.svg80px]][[File:6-demicube t013 A5.svg80px]][[File:6-demicube t013 A3.svg80px]]= stericantic 6-cubePrismatotruncated hemihexeract (pithax)
[[File:6-demicube t023 B6.svg80px]][[File:6-demicube t023 D6.svg80px]][[File:6-demicube t023 D5.svg80px]][[File:6-demicube t023 D4.svg80px]][[File:6-demicube t023 D3.svg80px]][[File:6-demicube t023 A5.svg80px]][[File:6-demicube t023 A3.svg80px]]= steriruncic 6-cubePrismatorhombated hemihexeract (prohax)
[[File:6-demicube t014 B6.svg80px]][[File:6-demicube t014 D6.svg80px]][[File:6-demicube t014 D5.svg80px]][[File:6-demicube t014 D4.svg80px]][[File:6-demicube t014 D3.svg80px]][[File:6-demicube t014 A5.svg80px]][[File:6-demicube t014 A3.svg80px]]= Stericantic 6-cubeCellitruncated hemihexeract (cathix)
[[File:6-demicube t024 B6.svg80px]][[File:6-demicube t024 D6.svg80px]][[File:6-demicube t024 D5.svg80px]][[File:6-demicube t024 D4.svg80px]][[File:6-demicube t024 D3.svg80px]][[File:6-demicube t024 A5.svg80px]][[File:6-demicube t024 A3.svg80px]]= Pentiruncic 6-cubeCellirhombated hemihexeract (crohax)
[[File:6-demicube t034 B6.svg80px]][[File:6-demicube t034 D6.svg80px]][[File:6-demicube t034 D5.svg80px]][[File:6-demicube t034 D4.svg80px]][[File:6-demicube t034 D3.svg80px]][[File:6-demicube t034 A5.svg80px]][[File:6-demicube t034 A3.svg80px]]= Pentisteric 6-cubeCelliprismated hemihexeract (cophix)
[[File:6-demicube t0123 B6.svg80px]][[File:6-demicube t0123 D6.svg80px]][[File:6-demicube t0123 D5.svg80px]][[File:6-demicube t0123 D4.svg80px]][[File:6-demicube t0123 D3.svg80px]][[File:6-demicube t0123 A5.svg80px]][[File:6-demicube t0123 A3.svg80px]]= Steriruncicantic 6-cubeGreat prismated hemihexeract (gophax)
[[File:6-demicube t0124 B6.svg80px]][[File:6-demicube t0124 D6.svg80px]][[File:6-demicube t0124 D5.svg80px]][[File:6-demicube t0124 D4.svg80px]][[File:6-demicube t0124 D3.svg80px]][[File:6-demicube t0124 A5.svg80px]][[File:6-demicube t0124 A3.svg80px]]= Pentiruncicantic 6-cubeCelligreatorhombated hemihexeract (cagrohax)
[[File:6-demicube t0134 B6.svg80px]][[File:6-demicube t0134 D6.svg80px]][[File:6-demicube t0134 D5.svg80px]][[File:6-demicube t0134 D4.svg80px]][[File:6-demicube t0134 D3.svg80px]][[File:6-demicube t0134 A5.svg80px]][[File:6-demicube t0134 A3.svg80px]]= Pentistericantic 6-cubeCelliprismatotruncated hemihexeract (capthix)
[[File:6-demicube t0234 B6.svg80px]][[File:6-demicube t0234 D6.svg80px]][[File:6-demicube t0234 D5.svg80px]][[File:6-demicube t0234 D4.svg80px]][[File:6-demicube t0234 D3.svg80px]][[File:6-demicube t0234 A5.svg80px]][[File:6-demicube t0234 A3.svg80px]]= Pentisteriruncic 6-cubeCelliprismatorhombated hemihexeract (caprohax)
[[File:6-demicube t01234 B6.svg80px]][[File:6-demicube t01234 D6.svg80px]][[File:6-demicube t01234 D5.svg80px]][[File:6-demicube t01234 D4.svg80px]][[File:6-demicube t01234 D3.svg80px]][[File:6-demicube t01234 A5.svg80px]][[File:6-demicube t01234 A3.svg80px]]= Pentisteriruncicantic 6-cubeGreat cellated hemihexeract (gochax)

References

  • H.S.M. Coxeter:
    • H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973
  • Kaleidoscopes: Selected Writings of H.S.M. Coxeter, edited by F. Arthur Sherk, Peter McMullen, Anthony C. Thompson, Asia Ivic Weiss, Wiley-Interscience Publication, 1995,
    • (Paper 22) H.S.M. Coxeter, Regular and Semi Regular Polytopes I, [Math. Zeit. 46 (1940) 380-407, MR 2,10]
    • (Paper 23) H.S.M. Coxeter, Regular and Semi-Regular Polytopes II, [Math. Zeit. 188 (1985) 559-591]
    • (Paper 24) H.S.M. Coxeter, Regular and Semi-Regular Polytopes III, [Math. Zeit. 200 (1988) 3-45]
  • N.W. Johnson: The Theory of Uniform Polytopes and Honeycombs, Ph.D. Dissertation, University of Toronto, 1966

Notes

References

  1. "Wiley::Kaleidoscopes: Selected Writings of H.S.M. Coxeter".
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