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D7 polytope

{{DISPLAYTITLE:D7 polytope}}

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|+ Orthographic projections in the D7 Coxeter plane

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|160px
7-demicube
{{CDD|nodes_10ru|split2|node|3|node|3|node|3|node|3|node}}

|160px
7-orthoplex
{{CDD|nodes|split2|node|3|node|3|node|3|node|3|node_1}}

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.

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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.

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!rowspan=2|#

!colspan=8|Coxeter plane graphs

!rowspan=2|Coxeter diagram
Names

B7
[14/2]||D7
[12]|| D6
[10]|| D5
[8]|| D4
[6]|| D3
[4]|| A5
[6]|| A3
[4]
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!1

80px80px80px80px80px80px80px80px{{CDD|nodes_10ru|split2|node|3|node|3|node|3|node|3|node}} = {{CDD|node_h|4|node|3|node|3|node|3|node|3|node|3|node}}
7-demicube
Demihepteract (Hesa)
align=center

!2

80px80px80px80px80px80px80px80px{{CDD|nodes_10ru|split2|node_1|3|node|3|node|3|node|3|node}} = {{CDD|node_h|4|node|3|node_1|3|node|3|node|3|node|3|node}}
Cantic 7-cube
Truncated demihepteract (Thesa)
align=center

!3

80px80px80px80px80px80px80px80px{{CDD|nodes_10ru|split2|node|3|node_1|3|node|3|node|3|node}} = {{CDD|node_h|4|node|3|node|3|node_1|3|node|3|node|3|node}}
Runcic 7-cube
Small rhombated demihepteract (Sirhesa)
align=center

!4

80px80px80px80px80px80px80px80px{{CDD|nodes_10ru|split2|node|3|node|3|node_1|3|node|3|node}} = {{CDD|node_h|4|node|3|node|3|node|3|node_1|3|node|3|node}}
Steric 7-cube
Small prismated demihepteract (Sphosa)
align=center

!5

80px80px80px80px80px80px80px80px{{CDD|nodes_10ru|split2|node|3|node|3|node|3|node_1|3|node}} = {{CDD|node_h|4|node|3|node|3|node|3|node|3|node_1|3|node}}
Pentic 7-cube
Small cellated demihepteract (Sochesa)
align=center

!6

80px80px80px80px80px80px80px80px{{CDD|nodes_10ru|split2|node|3|node|3|node|3|node|3|node_1}} = {{CDD|node_h|4|node|3|node|3|node|3|node|3|node|3|node_1}}
Hexic 7-cube
Small terated demihepteract (Suthesa)
align=center

!7

80px80px80px80px80px80px80px80px{{CDD|nodes_10ru|split2|node_1|3|node_1|3|node|3|node|3|node}} = {{CDD|node_h|4|node|3|node_1|3|node_1|3|node|3|node|3|node}}
Runcicantic 7-cube
Great rhombated demihepteract (Girhesa)
align=center

!8

80px80px80px80px80px80px80px80px{{CDD|nodes_10ru|split2|node_1|3|node|3|node_1|3|node|3|node}} = {{CDD|node_h|4|node|3|node_1|3|node|3|node_1|3|node|3|node}}
Stericantic 7-cube
Prismatotruncated demihepteract (Pothesa)
align=center

!9

80px80px80px80px80px80px80px80px{{CDD|nodes_10ru|split2|node|3|node_1|3|node_1|3|node|3|node}} = {{CDD|node_h|4|node|3|node_1|3|node_1|3|node|3|node|3|node}}
Steriruncic 7-cube
Prismatorhomated demihepteract (Prohesa)
align=center

!10

80px80px80px80px80px80px80px80px{{CDD|nodes_10ru|split2|node_1|3|node|3|node|3|node_1|3|node}} = {{CDD|node_h|4|node|3|node_1|3|node|3|node|3|node_1|3|node}}
Penticantic 7-cube
Cellitruncated demihepteract (Cothesa)
align=center

!11

80px80px80px80px80px80px80px80px{{CDD|nodes_10ru|split2|node|3|node_1|3|node|3|node_1|3|node}} = {{CDD|node_h|4|node|3|node|3|node_1|3|node|3|node_1|3|node}}
Pentiruncic 7-cube
Cellirhombated demihepteract (Crohesa)
align=center

!12

80px80px80px80px80px80px80px80px{{CDD|nodes_10ru|split2|node|3|node|3|node_1|3|node_1|3|node}} = {{CDD|node_h|4|node|3|node|3|node|3|node_1|3|node_1|3|node}}
Pentisteric 7-cube
Celliprismated demihepteract (Caphesa)
align=center

!13

80px80px80px80px80px80px80px80px{{CDD|nodes_10ru|split2|node_1|3|node|3|node|3|node|3|node_1}} = {{CDD|node_h|4|node|3|node_1|3|node|3|node|3|node|3|node_1}}
Hexicantic 7-cube
Teritruncated demihepteract (Tuthesa)
align=center

!14

80px80px80px80px80px80px80px80px{{CDD|nodes_10ru|split2|node|3|node_1|3|node|3|node|3|node_1}} = {{CDD|node_h|4|node|3|node|3|node_1|3|node|3|node|3|node_1}}
Hexiruncic 7-cube
Terirhombated demihepteract (Turhesa)
align=center

!15

80px80px80px80px80px80px80px80px{{CDD|nodes_10ru|split2|node|3|node|3|node_1|3|node|3|node_1}} = {{CDD|node_h|4|node|3|node|3|node|3|node_1|3|node|3|node_1}}
Hexisteric 7-cube
Teriprismated demihepteract (Tuphesa)
align=center

!16

80px80px80px80px80px80px80px80px{{CDD|nodes_10ru|split2|node|3|node|3|node|3|node_1|3|node_1}} = {{CDD|node_h|4|node|3|node|3|node|3|node|3|node_1|3|node_1}}
Hexipentic 7-cube
Tericellated demihepteract (Tuchesa)
align=center

!17

80px80px80px80px80px80px80px80px{{CDD|nodes_10ru|split2|node_1|3|node_1|3|node_1|3|node|3|node}} = {{CDD|node_h|4|node|3|node_1|3|node_1|3|node_1|3|node|3|node}}
Steriruncicantic 7-cube
Great prismated demihepteract (Gephosa)
align=center

!18

80px80px80px80px80px80px80px80px{{CDD|nodes_10ru|split2|node_1|3|node_1|3|node|3|node_1|3|node}} = {{CDD|node_h|4|node|3|node_1|3|node_1|3|node|3|node_1|3|node}}
Pentiruncicantic 7-cube
Celligreatorhombated demihepteract (Cagrohesa)
align=center

!19

80px80px80px80px80px80px80px80px{{CDD|nodes_10ru|split2|node_1|3|node|3|node_1|3|node_1|3|node}} = {{CDD|node_h|4|node|3|node_1|3|node|3|node_1|3|node_1|3|node}}
Pentistericantic 7-cube
Celliprismatotruncated demihepteract (Capthesa)
align=center

!20

80px80px80px80px80px80px80px80px{{CDD|nodes_10ru|split2|node|3|node_1|3|node_1|3|node_1|3|node}} = {{CDD|node_h|4|node|3|node|3|node_1|3|node_1|3|node_1|3|node}}
Pentisteriruncic 7-cube
Celliprismatorhombated demihepteract (Coprahesa)
align=center

!21

80px80px80px80px80px80px80px80px{{CDD|nodes_10ru|split2|node_1|3|node_1|3|node|3|node|3|node_1}} = {{CDD|node_h|4|node|3|node_1|3|node_1|3|node|3|node|3|node_1}}
Hexiruncicantic 7-cube
Terigreatorhombated demihepteract (Tugrohesa)
align=center

!22

80px80px80px80px80px80px80px80px{{CDD|nodes_10ru|split2|node_1|3|node|3|node_1|3|node|3|node_1}} = {{CDD|node_h|4|node|3|node_1|3|node|3|node_1|3|node|3|node_1}}
Hexistericantic 7-cube
Teriprismatotruncated demihepteract (Tupthesa)
align=center

!23

80px80px80px80px80px80px80px80px{{CDD|nodes_10ru|split2|node|3|node_1|3|node_1|3|node|3|node_1}} = {{CDD|node_h|4|node|3|node|3|node|3|node_1|3|node|3|node_1}}
Hexisteriruncic 7-cube
Teriprismatorhombated demihepteract (Tuprohesa)
align=center

!24

80px80px80px80px80px80px80px80px{{CDD|nodes_10ru|split2|node_1|3|node|3|node|3|node_1|3|node_1}} = {{CDD|node_h|4|node|3|node_1|3|node|3|node|3|node_1|3|node_1}}
Hexipenticantic 7-cube
Tericellitruncated demihepteract (Tucothesa)
align=center

!25

80px80px80px80px80px80px80px80px{{CDD|nodes_10ru|split2|node|3|node_1|3|node|3|node_1|3|node_1}} = {{CDD|node_h|4|node|3|node|3|node_1|3|node|3|node_1|3|node_1}}
Hexipentiruncic 7-cube
Tericellirhombated demihepteract (Tucrohesa)
align=center

!26

80px80px80px80px80px80px80px80px{{CDD|nodes_10ru|split2|node|3|node|3|node_1|3|node_1|3|node_1}} = {{CDD|node_h|4|node|3|node|3|node|3|node_1|3|node_1|3|node_1}}
Hexipentisteric 7-cube
Tericelliprismated demihepteract (Tucophesa)
align=center

!27

80px80px80px80px80px80px80px80px{{CDD|nodes_10ru|split2|node_1|3|node_1|3|node_1|3|node_1|3|node}} = {{CDD|node_h|4|node|3|node|3|node_1|3|node_1|3|node_1|3|node}}
Pentisteriruncicantic 7-cube
Great cellated demihepteract (Gochesa)
align=center

!28

80px80px80px80px80px80px80px80px{{CDD|nodes_10ru|split2|node_1|3|node_1|3|node_1|3|node|3|node_1}} = {{CDD|node_h|4|node|3|node|3|node_1|3|node_1|3|node|3|node_1}}
Hexisteriruncicantic 7-cube
Terigreatoprimated demihepteract (Tugphesa)
align=center

!29

80px80px80px80px80px80px80px80px{{CDD|nodes_10ru|split2|node_1|3|node_1|3|node|3|node_1|3|node_1}} = {{CDD|node_h|4|node|3|node|3|node_1|3|node|3|node_1|3|node_1}}
Hexipentiruncicantic 7-cube
Tericelligreatorhombated demihepteract (Tucagrohesa)
align=center

!30

80px80px80px80px80px80px80px80px{{CDD|nodes_10ru|split2|node_1|3|node|3|node_1|3|node_1|3|node_1}} = {{CDD|node_h|4|node|3|node_1|3|node|3|node_1|3|node_1|3|node_1}}
Hexipentistericantic 7-cube
Tericelliprismatotruncated demihepteract (Tucpathesa)
align=center

!31

80px80px80px80px80px80px80px80px{{CDD|nodes_10ru|split2|node|3|node_1|3|node_1|3|node_1|3|node_1}} = {{CDD|node_h|4|node|3|node|3|node|3|node_1|3|node_1|3|node_1}}
Hexipentisteriruncic 7-cube
Tericellprismatorhombated demihepteract (Tucprohesa)
align=center

!32

80px80px80px80px80px80px80px80px{{CDD|nodes_10ru|split2|node_1|3|node_1|3|node_1|3|node_1|3|node_1}} = {{CDD|node_h|4|node|3|node_1|3|node|3|node_1|3|node_1|3|node_1}}
Hexipentisteriruncicantic 7-cube
Great 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, {{ISBN|978-0-471-01003-6}} [http://www.wiley.com/WileyCDA/WileyTitle/productCd-0471010030.html Wiley::Kaleidoscopes: Selected Writings of H.S.M. Coxeter]
  • (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
  • {{KlitzingPolytopes|polypeta.htm|7D|uniform polytopes (polyexa)}}

Notes

{{reflist}}

{{Polytopes}}

Category:7-polytopes