7-demicube

Demihepteract
(7-demicube)
Demihepteract ortho petrie.svg
Petrie polygon projection
Type Uniform 7-polytope
Family demihypercube
Coxeter symbol 141
Schläfli symbol {3,34,1} = h{4,35}
s{21,1,1,1,1,1}
Coxeter diagrams CDel nodes 10ru.pngCDel split2.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png = CDel node h1.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png

CDel node h.pngCDel 2x.pngCDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 4.pngCDel node.png
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2x.pngCDel node h.png

6-faces 78 14 {31,3,1}Demihexeract ortho petrie.svg
64 {35}6-simplex t0.svg
5-faces 532 84 {31,2,1}Demipenteract graph ortho.svg
448 {34}5-simplex t0.svg
4-faces 1624 280 {31,1,1}4-orthoplex.svg
1344 {33}4-simplex t0.svg
Cells 2800 560 {31,0,1}3-simplex t0.svg
2240 {3,3}3-simplex t0.svg
Faces 2240 {3}2-simplex t0.svg
Edges 672
Vertices 64
Vertex figure Rectified 6-simplex
6-simplex t1.svg
Symmetry group D7, [34,1,1] = [1+,4,35]
[26]+
Dual ?
Properties convex

In geometry, a demihepteract or 7-demicube is a uniform 7-polytope, constructed from the 7-hypercube (hepteract) with alternated vertices removed. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes.

E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as HM7 for a 7-dimensional half measure polytope.

Coxeter named this polytope as 141 from its Coxeter diagram, with a ring on one of the 1-length branches, CDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png and Schläfli symbol or {3,34,1}.

Cartesian coordinates[]

Cartesian coordinates for the vertices of a demihepteract centered at the origin are alternate halves of the hepteract:

(±1,±1,±1,±1,±1,±1,±1)

with an odd number of plus signs.

Images[]

orthographic projections
Coxeter
plane
B7 D7 D6
Graph 7-demicube t0 B7.svg 7-demicube t0 D7.svg 7-demicube t0 D6.svg
Dihedral
symmetry
[14/2] [12] [10]
Coxeter plane D5 D4 D3
Graph 7-demicube t0 D5.svg 7-demicube t0 D4.svg 7-demicube t0 D3.svg
Dihedral
symmetry
[8] [6] [4]
Coxeter
plane
A5 A3
Graph 7-demicube t0 A5.svg 7-demicube t0 A3.svg
Dihedral
symmetry
[6] [4]

Construction[]

The elements of the regular polytopes can be expressed in a configuration matrix. Rows and columns reference vertices, edges, faces, and cells, with diagonal element their counts (f-vectors). The nondiagonal elements represent the number of row elements are incident to the column element.[1][2]

The diagonal f-vector numbers are derived through the Wythoff construction, dividing the full group order of a subgroup order by removing one mirror at a time.[3]

D7 CDel nodea 1.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png k-face fk f0 f1 f2 f3 f4 f5 f6 k-figures notes
A6 CDel nodea x.pngCDel 2.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png ( ) f0 64 21 105 35 140 35 105 21 42 7 7 041 D7/A6 = 64*7!/7! = 64
A4A1A1 CDel nodea 1.pngCDel 2.pngCDel nodes x0.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png { } f1 2 672 10 5 20 10 20 10 10 5 2 { }×{3,3,3} D7/A4A1A1 = 64*7!/5!/2/2 = 672
A3A2 CDel nodea 1.pngCDel 3a.pngCDel nodes 0x.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png 100 f2 3 3 2240 1 4 4 6 6 4 4 1 {3,3}v( ) D7/A3A2 = 64*7!/4!/3! = 2240
A3A3 CDel nodea 1.pngCDel 3a.pngCDel branch.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png 101 f3 4 6 4 560 * 4 0 6 0 4 0 {3,3} D7/A3A3 = 64*7!/4!/4! = 560
A3A2 CDel nodea 1.pngCDel 3a.pngCDel nodes 0x.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel nodea.png 110 4 6 4 * 2240 1 3 3 3 3 1 {3}v( ) D7/A3A2 = 64*7!/4!/3! = 2240
D4A2 CDel nodea 1.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.pngCDel 3a.pngCDel nodea.png 111 f4 8 24 32 8 8 280 * 3 0 3 0 {3} D7/D4A2 = 64*7!/8/4!/2 = 280
A4A1 CDel nodea 1.pngCDel 3a.pngCDel nodes 0x.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.png 120 5 10 10 0 5 * 1344 1 2 2 1 { }v( ) D7/A4A1 = 64*7!/5!/2 = 1344
D5A1 CDel nodea 1.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.pngCDel 2.pngCDel nodea.png 121 f5 16 80 160 40 80 10 16 84 * 2 0 { } D7/D5A1 = 64*7!/16/5!/2 = 84
A5 CDel nodea 1.pngCDel 3a.pngCDel nodes 0x.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.png 130 6 15 20 0 15 0 6 * 448 1 1 D7/A5 = 64*7!/6! = 448
D6 CDel nodea 1.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 2.pngCDel nodea x.png 131 f6 32 240 640 160 480 60 192 12 32 14 * ( ) D7/D6 = 64*7!/32/6! = 14
A6 CDel nodea 1.pngCDel 3a.pngCDel nodes 0x.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png 140 7 21 35 0 35 0 21 0 7 * 64 D7/A6 = 64*7!/7! = 64

Related polytopes[]

There are 95 uniform polytopes with D6 symmetry, 63 are shared by the B6 symmetry, and 32 are unique:

References[]

  1. ^ Coxeter, Regular Polytopes, sec 1.8 Configurations
  2. ^ Coxeter, Complex Regular Polytopes, p.117
  3. ^ Klitzing, Richard. "x3o3o *b3o3o3o - hax".

External links[]

Fundamental convex regular and uniform polytopes in dimensions 2–10
Family An Bn I2(p) / Dn E6 / E7 / E8 / F4 / G2 Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform 4-polytope 5-cell 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds