7-demicube

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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 diagram 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 2c.pngCDel node h.pngCDel 2c.pngCDel node h.pngCDel 2c.pngCDel node h.pngCDel 2c.pngCDel node h.pngCDel 2c.pngCDel node h.pngCDel 2c.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, [36,1,1] = [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 truncated. 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 \left\{3 \begin{array}{l}3, 3, 3, 3\\3\end{array}\right\} or {3,34,1}.

Cartesian coordinates[edit]

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[edit]

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]

Related polytopes[edit]

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

References[edit]

  • H.S.M. Coxeter:
    • Coxeter, Regular Polytopes, (3rd edition, 1973), Dover edition, ISBN 0-486-61480-8, p.296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
    • H.S.M. Coxeter, Regular Polytopes, 3rd Edition, Dover New York, 1973, p.296, Table I (iii): Regular Polytopes, three regular polytopes in n-dimensions (n≥5)
    • 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 [1]
      • (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]
  • John H. Conway, Heidi Burgiel, Chaim Goodman-Strass, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26. pp. 409: Hemicubes: 1n1)
  • Richard Klitzing, 7D uniform polytopes (polyexa), x3o3o *b3o3o3o3o - hesa


External links[edit]