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D7 polytope

[[File:7-demicube_t0_D7.svg160px]]7-demicube[[File:7-cube_t6_B6.svg160px]]7-orthoplex

In 7-dimensional geometry, there are 95 uniform polytopes with D7 symmetry; 32 are unique, and 63 are shared with the B7 symmetry. There are two regular forms, the 7-orthoplex, and 7-demicube with 14 and 64 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 32 polytopes can be made in the D7, D6, D5, D4, D3, A5, A3, Coxeter planes. Ak has [k+1] symmetry, Dk has [2(k-1)] symmetry. B7 is also included although only half of its [14] symmetry exists in these polytopes.

These 32 polytopes are each shown in these 8 symmetry planes, with vertices and edges drawn, and vertices colored by the number of overlapping vertices in each projective position.

#Coxeter plane graphsCoxeter diagramNamesB7[14/2]D7[12]D6[10]D5[8]D4[6]D3[4]A5[6]A3[4]1234567891011121314151617181920212223242526272829303132
[[File:7-demicube_t0_B7.svg80px]][[File:7-demicube_t0_D7.svg80px]][[File:7-demicube_t0_D6.svg80px]][[File:7-demicube_t0_D5.svg80px]][[File:7-demicube_t0_D4.svg80px]][[File:7-demicube_t0_D3.svg80px]][[File:7-demicube_t0_A5.svg80px]][[File:7-demicube_t0_A3.svg80px]]= 7-demicubeDemihepteract (Hesa)
[[File:7-demicube_t01_B7.svg80px]][[File:7-demicube_t01_D7.svg80px]][[File:7-demicube_t01_D6.svg80px]][[File:7-demicube_t01_D5.svg80px]][[File:7-demicube_t01_D4.svg80px]][[File:7-demicube_t01_D3.svg80px]][[File:7-demicube_t01_A5.svg80px]][[File:7-demicube_t01_A3.svg80px]]= Cantic 7-cubeTruncated demihepteract (Thesa)
[[File:7-demicube_t02_B7.svg80px]][[File:7-demicube_t02_D7.svg80px]][[File:7-demicube_t02_D6.svg80px]][[File:7-demicube_t02_D5.svg80px]][[File:7-demicube_t02_D4.svg80px]][[File:7-demicube_t02_D3.svg80px]][[File:7-demicube_t02_A5.svg80px]][[File:7-demicube_t02_A3.svg80px]]= Runcic 7-cubeSmall rhombated demihepteract (Sirhesa)
[[File:7-demicube_t03_B7.svg80px]][[File:7-demicube_t03_D7.svg80px]][[File:7-demicube_t03_D6.svg80px]][[File:7-demicube_t03_D5.svg80px]][[File:7-demicube_t03_D4.svg80px]][[File:7-demicube_t03_D3.svg80px]][[File:7-demicube_t03_A5.svg80px]][[File:7-demicube_t03_A3.svg80px]]= Steric 7-cubeSmall prismated demihepteract (Sphosa)
[[File:7-demicube_t04_B7.svg80px]][[File:7-demicube_t04_D7.svg80px]][[File:7-demicube_t04_D6.svg80px]][[File:7-demicube_t04_D5.svg80px]][[File:7-demicube_t04_D4.svg80px]][[File:7-demicube_t04_D3.svg80px]][[File:7-demicube_t04_A5.svg80px]][[File:7-demicube_t04_A3.svg80px]]= Pentic 7-cubeSmall cellated demihepteract (Sochesa)
[[File:7-demicube_t05_B7.svg80px]][[File:7-demicube_t05_D7.svg80px]][[File:7-demicube_t05_D6.svg80px]][[File:7-demicube_t05_D5.svg80px]][[File:7-demicube_t05_D4.svg80px]][[File:7-demicube_t05_D3.svg80px]][[File:7-demicube_t05_A5.svg80px]][[File:7-demicube_t05_A3.svg80px]]= Hexic 7-cubeSmall terated demihepteract (Suthesa)
[[File:7-demicube_t012_B7.svg80px]][[File:7-demicube_t012_D7.svg80px]][[File:7-demicube_t012_D6.svg80px]][[File:7-demicube_t012_D5.svg80px]][[File:7-demicube_t012_D4.svg80px]][[File:7-demicube_t012_D3.svg80px]][[File:7-demicube_t012_A5.svg80px]][[File:7-demicube_t012_A3.svg80px]]= Runcicantic 7-cubeGreat rhombated demihepteract (Girhesa)
[[File:7-demicube_t013_B7.svg80px]][[File:7-demicube_t013_D7.svg80px]][[File:7-demicube_t013_D6.svg80px]][[File:7-demicube_t013_D5.svg80px]][[File:7-demicube_t013_D4.svg80px]][[File:7-demicube_t013_D3.svg80px]][[File:7-demicube_t013_A5.svg80px]][[File:7-demicube_t013_A3.svg80px]]= Stericantic 7-cubePrismatotruncated demihepteract (Pothesa)
[[File:7-demicube_t023_B7.svg80px]][[File:7-demicube_t023_D7.svg80px]][[File:7-demicube_t023_D6.svg80px]][[File:7-demicube_t023_D5.svg80px]][[File:7-demicube_t023_D4.svg80px]][[File:7-demicube_t023_D3.svg80px]][[File:7-demicube_t023_A5.svg80px]][[File:7-demicube_t023_A3.svg80px]]= Steriruncic 7-cubePrismatorhomated demihepteract (Prohesa)
[[File:7-demicube_t014_B7.svg80px]][[File:7-demicube_t014_D7.svg80px]][[File:7-demicube_t014_D6.svg80px]][[File:7-demicube_t014_D5.svg80px]][[File:7-demicube_t014_D4.svg80px]][[File:7-demicube_t014_D3.svg80px]][[File:7-demicube_t014_A5.svg80px]][[File:7-demicube_t014_A3.svg80px]]= Penticantic 7-cubeCellitruncated demihepteract (Cothesa)
[[File:7-demicube_t024_B7.svg80px]][[File:7-demicube_t024_D7.svg80px]][[File:7-demicube_t024_D6.svg80px]][[File:7-demicube_t024_D5.svg80px]][[File:7-demicube_t024_D4.svg80px]][[File:7-demicube_t024_D3.svg80px]][[File:7-demicube_t024_A5.svg80px]][[File:7-demicube_t024_A3.svg80px]]= Pentiruncic 7-cubeCellirhombated demihepteract (Crohesa)
[[File:7-demicube_t034_B7.svg80px]][[File:7-demicube_t034_D7.svg80px]][[File:7-demicube_t034_D6.svg80px]][[File:7-demicube_t034_D5.svg80px]][[File:7-demicube_t034_D4.svg80px]][[File:7-demicube_t034_D3.svg80px]][[File:7-demicube_t034_A5.svg80px]][[File:7-demicube_t034_A3.svg80px]]= Pentisteric 7-cubeCelliprismated demihepteract (Caphesa)
[[File:7-demicube_t015_B7.svg80px]][[File:7-demicube_t015_D7.svg80px]][[File:7-demicube_t015_D6.svg80px]][[File:7-demicube_t015_D5.svg80px]][[File:7-demicube_t015_D4.svg80px]][[File:7-demicube_t015_D3.svg80px]][[File:7-demicube_t015_A5.svg80px]][[File:7-demicube_t015_A3.svg80px]]= Hexicantic 7-cubeTeritruncated demihepteract (Tuthesa)
[[File:7-demicube_t025_B7.svg80px]][[File:7-demicube_t025_D7.svg80px]][[File:7-demicube_t025_D6.svg80px]][[File:7-demicube_t025_D5.svg80px]][[File:7-demicube_t025_D4.svg80px]][[File:7-demicube_t025_D3.svg80px]][[File:7-demicube_t025_A5.svg80px]][[File:7-demicube_t025_A3.svg80px]]= Hexiruncic 7-cubeTerirhombated demihepteract (Turhesa)
[[File:7-demicube_t035_B7.svg80px]][[File:7-demicube_t035_D7.svg80px]][[File:7-demicube_t035_D6.svg80px]][[File:7-demicube_t035_D5.svg80px]][[File:7-demicube_t035_D4.svg80px]][[File:7-demicube_t035_D3.svg80px]][[File:7-demicube_t035_A5.svg80px]][[File:7-demicube_t035_A3.svg80px]]= Hexisteric 7-cubeTeriprismated demihepteract (Tuphesa)
[[File:7-demicube_t045_B7.svg80px]][[File:7-demicube_t045_D7.svg80px]][[File:7-demicube_t045_D6.svg80px]][[File:7-demicube_t045_D5.svg80px]][[File:7-demicube_t045_D4.svg80px]][[File:7-demicube_t045_D3.svg80px]][[File:7-demicube_t045_A5.svg80px]][[File:7-demicube_t045_A3.svg80px]]= Hexipentic 7-cubeTericellated demihepteract (Tuchesa)
[[File:7-demicube_t0123_B7.svg80px]][[File:7-demicube_t0123_D7.svg80px]][[File:7-demicube_t0123_D6.svg80px]][[File:7-demicube_t0123_D5.svg80px]][[File:7-demicube_t0123_D4.svg80px]][[File:7-demicube_t0123_D3.svg80px]][[File:7-demicube_t0123_A5.svg80px]][[File:7-demicube_t0123_A3.svg80px]]= Steriruncicantic 7-cubeGreat prismated demihepteract (Gephosa)
[[File:7-demicube_t0124_B7.svg80px]][[File:7-demicube_t0124_D7.svg80px]][[File:7-demicube_t0124_D6.svg80px]][[File:7-demicube_t0124_D5.svg80px]][[File:7-demicube_t0124_D4.svg80px]][[File:7-demicube_t0124_D3.svg80px]][[File:7-demicube_t0124_A5.svg80px]][[File:7-demicube_t0124_A3.svg80px]]= Pentiruncicantic 7-cubeCelligreatorhombated demihepteract (Cagrohesa)
[[File:7-demicube_t0134_B7.svg80px]][[File:7-demicube_t0134_D7.svg80px]][[File:7-demicube_t0134_D6.svg80px]][[File:7-demicube_t0134_D5.svg80px]][[File:7-demicube_t0134_D4.svg80px]][[File:7-demicube_t0134_D3.svg80px]][[File:7-demicube_t0134_A5.svg80px]][[File:7-demicube_t0134_A3.svg80px]]= Pentistericantic 7-cubeCelliprismatotruncated demihepteract (Capthesa)
[[File:7-demicube_t0234_B7.svg80px]][[File:7-demicube_t0234_D7.svg80px]][[File:7-demicube_t0234_D6.svg80px]][[File:7-demicube_t0234_D5.svg80px]][[File:7-demicube_t0234_D4.svg80px]][[File:7-demicube_t0234_D3.svg80px]][[File:7-demicube_t0234_A5.svg80px]][[File:7-demicube_t0234_A3.svg80px]]= Pentisteriruncic 7-cubeCelliprismatorhombated demihepteract (Coprahesa)
[[File:7-demicube_t0125_B7.svg80px]][[File:7-demicube_t0125_D7.svg80px]][[File:7-demicube_t0125_D6.svg80px]][[File:7-demicube_t0125_D5.svg80px]][[File:7-demicube_t0125_D4.svg80px]][[File:7-demicube_t0125_D3.svg80px]][[File:7-demicube_t0125_A5.svg80px]][[File:7-demicube_t0125_A3.svg80px]]= Hexiruncicantic 7-cubeTerigreatorhombated demihepteract (Tugrohesa)
[[File:7-demicube_t0135_B7.svg80px]][[File:7-demicube_t0135_D7.svg80px]][[File:7-demicube_t0135_D6.svg80px]][[File:7-demicube_t0135_D5.svg80px]][[File:7-demicube_t0135_D4.svg80px]][[File:7-demicube_t0135_D3.svg80px]][[File:7-demicube_t0135_A5.svg80px]][[File:7-demicube_t0135_A3.svg80px]]= Hexistericantic 7-cubeTeriprismatotruncated demihepteract (Tupthesa)
[[File:7-demicube_t0235_B7.svg80px]][[File:7-demicube_t0235_D7.svg80px]][[File:7-demicube_t0235_D6.svg80px]][[File:7-demicube_t0235_D5.svg80px]][[File:7-demicube_t0235_D4.svg80px]][[File:7-demicube_t0235_D3.svg80px]][[File:7-demicube_t0235_A5.svg80px]][[File:7-demicube_t0235_A3.svg80px]]= Hexisteriruncic 7-cubeTeriprismatorhombated demihepteract (Tuprohesa)
[[File:7-demicube_t0145_B7.svg80px]][[File:7-demicube_t0145_D7.svg80px]][[File:7-demicube_t0145_D6.svg80px]][[File:7-demicube_t0145_D5.svg80px]][[File:7-demicube_t0145_D4.svg80px]][[File:7-demicube_t0145_D3.svg80px]][[File:7-demicube_t0145_A5.svg80px]][[File:7-demicube_t0145_A3.svg80px]]= Hexipenticantic 7-cubeTericellitruncated demihepteract (Tucothesa)
[[File:7-demicube_t0245_B7.svg80px]][[File:7-demicube_t0245_D7.svg80px]][[File:7-demicube_t0245_D6.svg80px]][[File:7-demicube_t0245_D5.svg80px]][[File:7-demicube_t0245_D4.svg80px]][[File:7-demicube_t0245_D3.svg80px]][[File:7-demicube_t0245_A5.svg80px]][[File:7-demicube_t0245_A3.svg80px]]= Hexipentiruncic 7-cubeTericellirhombated demihepteract (Tucrohesa)
[[File:7-demicube_t0345_B7.svg80px]][[File:7-demicube_t0345_D7.svg80px]][[File:7-demicube_t0345_D6.svg80px]][[File:7-demicube_t0345_D5.svg80px]][[File:7-demicube_t0345_D4.svg80px]][[File:7-demicube_t0345_D3.svg80px]][[File:7-demicube_t0345_A5.svg80px]][[File:7-demicube_t0345_A3.svg80px]]= Hexipentisteric 7-cubeTericelliprismated demihepteract (Tucophesa)
[[File:7-demicube_t01234_B7.svg80px]][[File:7-demicube_t01234_D7.svg80px]][[File:7-demicube_t01234_D6.svg80px]][[File:7-demicube_t01234_D5.svg80px]][[File:7-demicube_t01234_D4.svg80px]][[File:7-demicube_t01234_D3.svg80px]][[File:7-demicube_t01234_A5.svg80px]][[File:7-demicube_t01234_A3.svg80px]]= Pentisteriruncicantic 7-cubeGreat cellated demihepteract (Gochesa)
[[File:7-demicube_t01235_B7.svg80px]][[File:7-demicube_t01235_D7.svg80px]][[File:7-demicube_t01235_D6.svg80px]][[File:7-demicube_t01235_D5.svg80px]][[File:7-demicube_t01235_D4.svg80px]][[File:7-demicube_t01235_D3.svg80px]][[File:7-demicube_t01235_A5.svg80px]][[File:7-demicube_t01235_A3.svg80px]]= Hexisteriruncicantic 7-cubeTerigreatoprimated demihepteract (Tugphesa)
[[File:7-demicube_t01245_B7.svg80px]][[File:7-demicube_t01245_D7.svg80px]][[File:7-demicube_t01245_D6.svg80px]][[File:7-demicube_t01245_D5.svg80px]][[File:7-demicube_t01245_D4.svg80px]][[File:7-demicube_t01245_D3.svg80px]][[File:7-demicube_t01245_A5.svg80px]][[File:7-demicube_t01245_A3.svg80px]]= Hexipentiruncicantic 7-cubeTericelligreatorhombated demihepteract (Tucagrohesa)
[[File:7-demicube_t01345_B7.svg80px]][[File:7-demicube_t01345_D7.svg80px]][[File:7-demicube_t01345_D6.svg80px]][[File:7-demicube_t01345_D5.svg80px]][[File:7-demicube_t01345_D4.svg80px]][[File:7-demicube_t01345_D3.svg80px]][[File:7-demicube_t01345_A5.svg80px]][[File:7-demicube_t01345_A3.svg80px]]= Hexipentistericantic 7-cubeTericelliprismatotruncated demihepteract (Tucpathesa)
[[File:7-demicube_t02345_B7.svg80px]][[File:7-demicube_t02345_D7.svg80px]][[File:7-demicube_t02345_D6.svg80px]][[File:7-demicube_t02345_D5.svg80px]][[File:7-demicube_t02345_D4.svg80px]][[File:7-demicube_t02345_D3.svg80px]][[File:7-demicube_t02345_A5.svg80px]][[File:7-demicube_t02345_A3.svg80px]]= Hexipentisteriruncic 7-cubeTericellprismatorhombated demihepteract (Tucprohesa)
[[File:7-demicube_t012345_B7.svg80px]][[File:7-demicube_t012345_D7.svg80px]][[File:7-demicube_t012345_D6.svg80px]][[File:7-demicube_t012345_D5.svg80px]][[File:7-demicube_t012345_D4.svg80px]][[File:7-demicube_t012345_D3.svg80px]][[File:7-demicube_t012345_A5.svg80px]][[File:7-demicube_t012345_A3.svg80px]]= Hexipentisteriruncicantic 7-cubeGreat terated demihepteract (Guthesa)

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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