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D8 polytope

{{Short description|Uniform polytopes with D8 symmetry}}

{{DISPLAYTITLE:D8 polytope}}

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|+ Orthographic projections in the D8 Coxeter plane

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|160px
8-demicube
{{CDD|nodes_10ru|split2|node|3|node|3|node|3|node|3|node|3|node}}
= {{CDD|node_h|4|node|3|node|3|node|3|node|3|node|3|node|3|node}}

|160px
8-orthoplex
{{CDD|nodes|split2|node|3|node|3|node|3|node|3|node|3|node_1}}
= {{CDD|node|4|node|3|node|3|node|3|node|3|node|3|node|3|node_1}}

In 8-dimensional geometry, there are 191 uniform polytopes with D8 symmetry, 64 are unique, and 127 are shared with the B8 symmetry. There are two regular forms, the 8-orthoplex, and 8-demicube with 16 and 128 vertices respectively.

They can be visualized as symmetric orthographic projections in Coxeter planes of the D8 Coxeter group, and other subgroups.

__TOC__

Graphs

Symmetric orthographic projections of these 64 polytopes can be made in the D8, D7, D6, D5, D4, D3, A5, A3, Coxeter planes. Ak has [k+1] symmetry, Dk has [2(k-1)] symmetry. B8 is also included although only half of its [16] symmetry exists in these polytopes.

These 64 polytopes are each shown in these 10 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=10|Coxeter plane graphs

!rowspan=2|Coxeter diagram
Names

B8
[16/2]||D8
[14]|| D7
[12]|| D6
[10]|| D5
[8]|| D4
[6]|| D3
[4]|| A7
[8]|| A5
[6]|| A3
[4]
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!1

60px60px60px60px60px60px60px60px60px60px{{CDD|nodes_10ru|split2|node|3|node|3|node|3|node|3|node|3|node}} = {{CDD|node_h1|4|node|3|node|3|node|3|node|3|node|3|node|3|node}}
8-demicube (hocto)
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!2

60px60px60px60px60px60px60px60px60px60px{{CDD|nodes_10ru|split2|node_1|3|node|3|node|3|node|3|node|3|node}} = {{CDD|node_h1|4|node|3|node_1|3|node|3|node|3|node|3|node|3|node}}
cantic 8-cube (thocto)
align=center

!3

60px60px60px60px60px60px60px60px60px60px{{CDD|nodes_10ru|split2|node|3|node_1|3|node|3|node|3|node|3|node}} = {{CDD|node_h1|4|node|3|node|3|node_1|3|node|3|node|3|node|3|node}}
runcic 8-cube
align=center

!4

60px60px60px60px60px60px60px60px60px60px{{CDD|nodes_10ru|split2|node|3|node|3|node_1|3|node|3|node|3|node}} = {{CDD|node_h1|4|node|3|node|3|node|3|node_1|3|node|3|node|3|node}}
steric 8-cube
align=center

!5

60px60px60px60px60px60px60px60px60px60px{{CDD|nodes_10ru|split2|node|3|node|3|node|3|node_1|3|node|3|node}} = {{CDD|node_h1|4|node|3|node|3|node|3|node|3|node_1|3|node|3|node}}
pentic 8-cube
align=center

!6

60px60px60px60px60px60px60px60px60px60px{{CDD|nodes_10ru|split2|node|3|node|3|node|3|node|3|node_1|3|node}} = {{CDD|node_h1|4|node|3|node|3|node|3|node|3|node_1|3|node|3|node}}
hexic 8-cube
align=center

!7

60px60px60px60px60px60px60px60px60px60px{{CDD|nodes_10ru|split2|node|3|node|3|node|3|node|3|node|3|node_1}} = {{CDD|node_h1|4|node|3|node|3|node|3|node|3|node|3|node|3|node_1}}
heptic 8-cube
align=center

!8

60px60px60px60px60px60px60px60px60px60px{{CDD|nodes_10ru|split2|node_1|3|node_1|3|node|3|node|3|node|3|node}} = {{CDD|node_h1|4|node|3|node_1|3|node_1|3|node|3|node|3|node|3|node}}
align=center

!9

60px60px60px60px60px60px60px60px60px{{CDD|nodes_10ru|split2|node_1|3|node|3|node_1|3|node|3|node|3|node}} = {{CDD|node_h1|4|node|3|node_1|3|node|3|node_1|3|node|3|node|3|node}}
align=center

!10

60px60px60px60px60px60px60px60px60px{{CDD|nodes_10ru|split2|node|3|node_1|3|node_1|3|node|3|node|3|node}} = {{CDD|node_h1|4|node|3|node|3|node_1|3|node_1|3|node|3|node|3|node}}
align=center

!11

60px60px60px60px60px60px60px60px60px60px{{CDD|nodes_10ru|split2|node_1|3|node|3|node|3|node_1|3|node|3|node}} = {{CDD|node_h1|4|node|3|node_1|3|node|3|node|3|node_1|3|node|3|node}}
align=center

!12

60px60px60px60px60px60px60px60px60px{{CDD|nodes_10ru|split2|node|3|node_1|3|node|3|node_1|3|node|3|node}} = {{CDD|node_h1|4|node|3|node|3|node_1|3|node|3|node_1|3|node|3|node}}
align=center

!13

60px60px60px60px60px60px60px60px60px{{CDD|nodes_10ru|split2|node|3|node|3|node_1|3|node_1|3|node|3|node}} = {{CDD|node_h1|4|node|3|node|3|node|3|node_1|3|node_1|3|node|3|node}}
align=center

!14

60px60px60px60px60px60px60px60px60px{{CDD|nodes_10ru|split2|node_1|3|node|3|node|3|node|3|node_1|3|node}} = {{CDD|node_h1|4|node|3|node_1|3|node|3|node|3|node|3|node_1|3|node}}
align=center

!15

60px60px60px60px60px60px60px60px60px{{CDD|nodes_10ru|split2|node|3|node_1|3|node|3|node|3|node_1|3|node}} = {{CDD|node_h1|4|node|3|node|3|node_1|3|node|3|node|3|node_1|3|node}}
align=center

!16

60px60px60px60px60px60px60px60px60px{{CDD|nodes_10ru|split2|node|3|node|3|node_1|3|node|3|node_1|3|node}} = {{CDD|node_h1|4|node|3|node|3|node|3|node_1|3|node|3|node_1|3|node}}
align=center

!17

60px60px60px60px60px60px60px60px60px{{CDD|nodes_10ru|split2|node|3|node|3|node|3|node_1|3|node_1|3|node}} = {{CDD|node_h1|4|node|3|node|3|node|3|node|3|node_1|3|node_1|3|node}}
align=center

!18

60px60px60px60px60px60px60px60px60px{{CDD|nodes_10ru|split2|node_1|3|node|3|node|3|node|3|node|3|node_1}} = {{CDD|node_h1|4|node|3|node_1|3|node|3|node|3|node|3|node|3|node_1}}
align=center

!19

60px60px60px60px60px60px60px60px60px{{CDD|nodes_10ru|split2|node|3|node_1|3|node|3|node|3|node|3|node_1}} = {{CDD|node_h1|4|node|3|node|3|node_1|3|node|3|node|3|node|3|node_1}}
align=center

!20

60px60px60px60px60px60px60px60px60px{{CDD|nodes_10ru|split2|node|3|node|3|node_1|3|node|3|node|3|node_1}} = {{CDD|node_h1|4|node|3|node|3|node|3|node_1|3|node|3|node|3|node_1}}
align=center

!21

60px60px60px60px60px60px60px60px60px{{CDD|nodes_10ru|split2|node|3|node|3|node|3|node_1|3|node|3|node_1}} = {{CDD|node_h1|4|node|3|node|3|node|3|node|3|node_1|3|node|3|node_1}}
align=center

!22

60px60px60px60px60px60px60px60px60px60px{{CDD|nodes_10ru|split2|node|3|node|3|node|3|node|3|node_1|3|node_1}} = {{CDD|node_h1|4|node|3|node|3|node|3|node|3|node|3|node_1|3|node_1}}
align=center

!23

60px60px60px60px60px60px60px60px60px{{CDD|nodes_10ru|split2|node_1|3|node_1|3|node_1|3|node|3|node|3|node}} = {{CDD|node_h1|4|node|3|node_1|3|node_1|3|node_1|3|node|3|node|3|node}}
align=center

!24

 60px60px60px60px60px60px60px60px{{CDD|nodes_10ru|split2|node_1|3|node_1|3|node|3|node_1|3|node|3|node}} = {{CDD|node_h1|4|node|3|node_1|3|node_1|3|node|3|node_1|3|node|3|node}}
align=center

!25

 60px60px60px60px60px60px60px60px{{CDD|nodes_10ru|split2|node_1|3|node|3|node_1|3|node_1|3|node|3|node}} = {{CDD|node_h1|4|node|3|node_1|3|node|3|node_1|3|node_1|3|node|3|node}}
align=center

!26

 60px60px60px60px60px60px60px60px{{CDD|nodes_10ru|split2|node|3|node_1|3|node_1|3|node_1|3|node|3|node}} = {{CDD|node_h1|4|node|3|node|3|node_1|3|node_1|3|node_1|3|node|3|node}}
align=center

!27

 60px60px60px60px60px60px60px60px{{CDD|nodes_10ru|split2|node_1|3|node_1|3|node|3|node|3|node_1|3|node}} = {{CDD|node_h1|4|node|3|node_1|3|node_1|3|node|3|node|3|node_1|3|node}}
align=center

!28

 60px60px60px60px60px60px60px60px{{CDD|nodes_10ru|split2|node_1|3|node|3|node_1|3|node|3|node_1|3|node}} = {{CDD|node_h1|4|node|3|node_1|3|node|3|node_1|3|node|3|node_1|3|node}}
align=center

!29

 60px60px60px60px60px60px60px60px{{CDD|nodes_10ru|split2|node|3|node_1|3|node_1|3|node|3|node_1|3|node}} = {{CDD|node_h1|4|node|3|node|3|node_1|3|node_1|3|node|3|node_1|3|node}}
align=center

!30

 60px60px60px60px60px60px60px60px{{CDD|nodes_10ru|split2|node_1|3|node|3|node|3|node_1|3|node_1|3|node}} = {{CDD|node_h1|4|node|3|node_1|3|node|3|node|3|node_1|3|node_1|3|node}}
align=center

!31

 60px60px60px60px60px60px60px60px{{CDD|nodes_10ru|split2|node|3|node_1|3|node|3|node_1|3|node_1|3|node}} = {{CDD|node_h1|4|node|3|node|3|node_1|3|node|3|node_1|3|node_1|3|node}}
align=center

!32

 60px60px60px60px60px60px60px60px60px{{CDD|nodes_10ru|split2|node|3|node|3|node_1|3|node_1|3|node_1|3|node}} = {{CDD|node_h1|4|node|3|node|3|node|3|node_1|3|node_1|3|node_1|3|node}}
align=center

!33

 60px60px60px60px60px60px60px60px60px{{CDD|nodes_10ru|split2|node_1|3|node_1|3|node|3|node|3|node|3|node_1}} = {{CDD|node_h1|4|node|3|node_1|3|node_1|3|node|3|node|3|node|3|node_1}}
align=center

!34

 60px60px60px60px60px60px60px60px{{CDD|nodes_10ru|split2|node_1|3|node|3|node_1|3|node|3|node|3|node_1}} = {{CDD|node_h1|4|node|3|node_1|3|node|3|node_1|3|node|3|node|3|node_1}}
align=center

!35

 60px60px60px60px60px60px60px60px60px{{CDD|nodes_10ru|split2|node|3|node_1|3|node_1|3|node|3|node|3|node_1}} = {{CDD|node_h1|4|node|3|node|3|node_1|3|node_1|3|node|3|node|3|node_1}}
align=center

!36

 60px60px60px60px60px60px60px60px{{CDD|nodes_10ru|split2|node_1|3|node|3|node|3|node_1|3|node|3|node_1}} = {{CDD|node_h1|4|node|3|node_1|3|node|3|node|3|node_1|3|node|3|node_1}}
align=center

!37

 60px60px60px60px60px60px60px60px{{CDD|nodes_10ru|split2|node|3|node_1|3|node|3|node_1|3|node|3|node_1}} = {{CDD|node_h1|4|node|3|node|3|node_1|3|node|3|node_1|3|node|3|node_1}}
align=center

!38

 60px60px60px60px60px60px60px60px60px{{CDD|nodes_10ru|split2|node|3|node|3|node_1|3|node_1|3|node|3|node_1}} = {{CDD|node_h1|4|node|3|node|3|node|3|node_1|3|node_1|3|node|3|node_1}}
align=center

!39

 60px60px60px60px60px60px60px60px60px{{CDD|nodes_10ru|split2|node_1|3|node|3|node|3|node|3|node_1|3|node_1}} = {{CDD|node_h1|4|node|3|node_1|3|node|3|node|3|node|3|node_1|3|node_1}}
align=center

!40

 60px60px60px60px60px60px60px60px60px{{CDD|nodes_10ru|split2|node|3|node_1|3|node|3|node|3|node_1|3|node_1}} = {{CDD|node_h1|4|node|3|node|3|node_1|3|node|3|node|3|node_1|3|node_1}}
align=center

!41

 60px60px60px60px60px60px60px60px60px{{CDD|nodes_10ru|split2|node|3|node|3|node_1|3|node|3|node_1|3|node_1}} = {{CDD|node_h1|4|node|3|node|3|node|3|node_1|3|node|3|node_1|3|node_1}}
align=center

!42

 60px60px60px60px60px60px60px60px60px{{CDD|nodes_10ru|split2|node|3|node|3|node|3|node_1|3|node_1|3|node_1}} = {{CDD|node_h1|4|node|3|node|3|node|3|node|3|node_1|3|node_1|3|node_1}}
align=center

!43

 60px60px60px60px60px60px60px60px{{CDD|nodes_10ru|split2|node_1|3|node_1|3|node_1|3|node_1|3|node|3|node}} = {{CDD|node_h1|4|node|3|node_1|3|node_1|3|node_1|3|node_1|3|node|3|node}}
align=center

!44

 60px60px60px60px60px60px60px60px{{CDD|nodes_10ru|split2|node_1|3|node_1|3|node_1|3|node|3|node_1|3|node}} = {{CDD|node_h1|4|node|3|node_1|3|node_1|3|node_1|3|node|3|node_1|3|node}}
align=center

!45

 60px60px60px60px60px60px60px60px{{CDD|nodes_10ru|split2|node_1|3|node_1|3|node|3|node_1|3|node_1|3|node}} = {{CDD|node_h1|4|node|3|node_1|3|node_1|3|node|3|node_1|3|node_1|3|node}}
align=center

!46

 60px60px60px60px60px60px60px60px{{CDD|nodes_10ru|split2|node_1|3|node|3|node_1|3|node_1|3|node_1|3|node}} = {{CDD|node_h1|4|node|3|node_1|3|node|3|node_1|3|node_1|3|node_1|3|node}}
align=center

!47

 60px60px60px60px60px60px60px60px{{CDD|nodes_10ru|split2|node|3|node_1|3|node_1|3|node_1|3|node_1|3|node}} = {{CDD|node_h1|4|node|3|node|3|node_1|3|node_1|3|node_1|3|node_1|3|node}}
align=center

!48

 60px60px60px60px60px60px60px60px{{CDD|nodes_10ru|split2|node_1|3|node_1|3|node_1|3|node|3|node|3|node_1}} = {{CDD|node_h1|4|node|3|node_1|3|node_1|3|node_1|3|node|3|node|3|node_1}}
align=center

!49

 60px60px60px60px60px60px60px60px{{CDD|nodes_10ru|split2|node_1|3|node_1|3|node|3|node_1|3|node|3|node_1}} = {{CDD|node_h1|4|node|3|node_1|3|node_1|3|node|3|node_1|3|node|3|node_1}}
align=center

!50

 60px60px60px60px60px60px60px60px{{CDD|nodes_10ru|split2|node_1|3|node|3|node_1|3|node_1|3|node|3|node_1}} = {{CDD|node_h1|4|node|3|node_1|3|node|3|node_1|3|node_1|3|node|3|node_1}}
align=center

!51

 60px60px60px60px60px60px60px60px{{CDD|nodes_10ru|split2|node|3|node_1|3|node_1|3|node_1|3|node|3|node_1}} = {{CDD|node_h1|4|node|3|node|3|node_1|3|node_1|3|node_1|3|node|3|node_1}}
align=center

!52

 60px60px60px60px60px60px60px60px{{CDD|nodes_10ru|split2|node_1|3|node_1|3|node|3|node|3|node_1|3|node_1}} = {{CDD|node_h1|4|node|3|node_1|3|node_1|3|node|3|node|3|node_1|3|node_1}}
align=center

!53

 60px60px60px60px60px60px60px60px{{CDD|nodes_10ru|split2|node_1|3|node|3|node_1|3|node|3|node_1|3|node_1}} = {{CDD|node_h1|4|node|3|node_1|3|node|3|node_1|3|node|3|node_1|3|node_1}}
align=center

!54

 60px60px60px60px60px60px60px60px{{CDD|nodes_10ru|split2|node|3|node_1|3|node_1|3|node|3|node_1|3|node_1}} = {{CDD|node_h1|4|node|3|node|3|node_1|3|node_1|3|node|3|node_1|3|node_1}}
align=center

!55

 60px60px60px60px60px60px60px60px{{CDD|nodes_10ru|split2|node_1|3|node|3|node|3|node_1|3|node_1|3|node_1}} = {{CDD|node_h1|4|node|3|node_1|3|node|3|node|3|node_1|3|node_1|3|node_1}}
align=center

!56

 60px60px60px60px60px60px60px60px{{CDD|nodes_10ru|split2|node|3|node_1|3|node|3|node_1|3|node_1|3|node_1}} = {{CDD|node_h1|4|node|3|node|3|node_1|3|node|3|node_1|3|node_1|3|node_1}}
align=center

!57

 60px60px60px60px60px60px60px60px{{CDD|nodes_10ru|split2|node|3|node|3|node_1|3|node_1|3|node_1|3|node_1}} = {{CDD|node_h1|4|node|3|node|3|node|3|node_1|3|node_1|3|node_1|3|node_1}}
align=center

!58

 60px60px60px60px60px60px60px60px{{CDD|nodes_10ru|split2|node_1|3|node_1|3|node_1|3|node_1|3|node_1|3|node}} = {{CDD|node_h1|4|node|3|node_1|3|node_1|3|node_1|3|node_1|3|node_1|3|node}}
align=center

!59

 60px60px60px60px60px60px60px60px{{CDD|nodes_10ru|split2|node_1|3|node_1|3|node_1|3|node_1|3|node|3|node_1}} = {{CDD|node_h1|4|node|3|node_1|3|node_1|3|node_1|3|node_1|3|node|3|node_1}}
align=center

!60

 60px60px60px60px60px60px60px60px{{CDD|nodes_10ru|split2|node_1|3|node_1|3|node_1|3|node|3|node_1|3|node_1}} = {{CDD|node_h1|4|node|3|node_1|3|node_1|3|node_1|3|node|3|node_1|3|node_1}}
align=center

!61

 60px60px60px60px60px60px60px60px{{CDD|nodes_10ru|split2|node_1|3|node_1|3|node|3|node_1|3|node_1|3|node_1}} = {{CDD|node_h1|4|node|3|node_1|3|node_1|3|node|3|node_1|3|node_1|3|node_1}}
align=center

!62

 60px60px60px60px60px60px60px60px{{CDD|nodes_10ru|split2|node_1|3|node|3|node_1|3|node_1|3|node_1|3|node_1}} = {{CDD|node_h1|4|node|3|node_1|3|node|3|node_1|3|node_1|3|node_1|3|node_1}}
align=center

!63

 60px60px60px60px60px60px60px60px{{CDD|nodes_10ru|split2|node|3|node_1|3|node_1|3|node_1|3|node_1|3|node_1}} = {{CDD|node_h1|4|node|3|node|3|node_1|3|node_1|3|node_1|3|node_1|3|node_1}}
align=center

!64

 60px60px60px60px60px60px60px60px{{CDD|nodes_10ru|split2|node_1|3|node_1|3|node_1|3|node_1|3|node_1|3|node_1}} = {{CDD|node_h1|4|node|3|node_1|3|node_1|3|node_1|3|node_1|3|node_1|3|node_1}}

References

{{reflist}}

{{Polytopes}}

Category:8-polytopes