6-demicube

From Wikipedia, the free encyclopedia
Jump to: navigation, search
Demihexeract
(6-demicube)
Demihexeract ortho petrie.svg
Petrie polygon projection
Type Uniform 6-polytope
Family demihypercube
Schläfli symbol {3,33,1} = h{4,34}
s{25}
Coxeter-Dynkin diagram CDel nodes 10ru.pngCDel split2.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.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.png
Coxeter symbol 131
5-faces 44 12 {31,2,1}Demipenteract graph ortho.svg
32 {34}5-simplex t0.svg
4-faces 252 60 {31,1,1}Cross graph 4.svg
192 {33}4-simplex t0.svg
Cells 640 160 {31,0,1}3-simplex t0.svg
480 {3,3}3-simplex t0.svg
Faces 640 {3}2-simplex t0.svg
Edges 240
Vertices 32
Vertex figure Rectified 5-simplex
5-simplex t1.svg
Symmetry group D6, [35,1,1] = [1+,4,34]
[25]+
Petrie polygon decagon
Properties convex

In geometry, a 6-demicube or demihexteract is a uniform 6-polytope, constructed from a 6-cube (hexeract) with alternated vertices truncated. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes.

Coxeter named this polytope as 131 from its Coxeter-Dynkin diagram, with a ring on one of the 1-length Coxeter-Dynkin diagram branches. It can named similarly by a 3-dimensional exponential Schläfli symbol, {3,33,1}.

Cartesian coordinates[edit]

Cartesian coordinates for the vertices of a demihexeract centered at the origin are alternate halves of the hexeract:

(±1,±1,±1,±1,±1,±1)

with an odd number of plus signs.

Images[edit]

orthographic projections
Coxeter plane B6
Graph 6-demicube t0 B6.svg
Dihedral symmetry [12/2]
Coxeter plane D6 D5
Graph 6-demicube t0 D6.svg 6-demicube t0 D5.svg
Dihedral symmetry [10] [8]
Coxeter plane D4 D3
Graph 6-demicube t0 D4.svg 6-demicube t0 D3.svg
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph 6-demicube t0 A5.svg 6-demicube t0 A3.svg
Dihedral symmetry [6] [4]

Related polytopes[edit]

There are 47 uniform polytopes with D6 symmetry, 31 are shared by the B6 symmetry, and 16 are unique:

6-demicube t0 D6.svg
h{4,34}
6-demicube t01 D6.svg
h2{4,34}
6-demicube t02 D6.svg
h3{4,34}
6-demicube t03 D6.svg
h4{4,34}
6-demicube t04 D6.svg
h5{4,34}
6-demicube t012 D6.svg
h2,3{4,34}
6-demicube t013 D6.svg
h2,4{4,34}
6-demicube t014 D6.svg
h2,5{4,34}
6-demicube t023 D6.svg
h3,4{4,34}
6-demicube t024 D6.svg
h3,5{4,34}
6-demicube t034 D6.svg
h4,5{4,34}
6-demicube t0123 D6.svg
h2,3,4{4,34}
6-demicube t0124 D6.svg
h2,3,5{4,34}
6-demicube t0134 D6.svg
h2,4,5{4,34}
6-demicube t0234 D6.svg
h3,4,5{4,34}
6-demicube t01234 D6.svg
h2,3,4,5{4,34}

The 6-demicube, 131 is third in a dimensional series of uniform polytopes, expressed by Coxeter as k31 series. The fifth figure is a Euclidean honeycomb, 331, and the final is a noncompact hyperbolic honeycomb, 431. Each progressive uniform polytope is constructed from the previous as its vertex figure.

k31 dimensional figures
n 4 5 6 7 8 9
Coxeter
group
A3×A1 A5 D6 E7 {\tilde{E}}_{7} = E7+ E7++
Coxeter
diagram
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node 1.pngCDel 2.pngCDel node 1.png CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 10.png CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea 1.png CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea 1.png
Symmetry
(order)
[3-1,3,1]
(48)
[30,3,1]
(720)
[31,3,1]
(23,040)
[32,3,1]
(2,903,040)
[33,3,1]
(∞)
[34,3,1]
(∞)
Graph Tetrahedral prism.png 5-simplex t1.svg Demihexeract ortho petrie.svg Up2 2 31 t0 E7.svg
Name −131 031 131 231 331 431

It is also the second in a dimensional series of uniform polytopes and honeycombs, expressed by Coxeter as 13k series. The next figure is the Euclidean honeycomb 133 and the final is a noncompact hyperbolic honeycomb, 134.

13k dimensional figures
n 4 5 6 7 8 9
Coxeter
group
A3×A1 A5 D6 E7 {\tilde{E}}_{7}=E7+ E7++
Coxeter
diagram
CDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.pngCDel 2.pngCDel node 1.png CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01l.png CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.png CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png CDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel branch 01lr.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.pngCDel 3a.pngCDel nodea.png
Symmetry
(order)
[3-1,3,1]
(48)
[30,3,1]
(720)
[31,3,1]
(23,040)
[32,3,1]
(2,903,040)
[[33,3,1]]
(∞)
[34,3,1]
(∞)
Graph 5-simplex t0.svg Demihexeract ortho petrie.svg Up2 1 32 t0 E7.svg
Name 13,-1 130 131 132 133 134

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, 6D uniform polytopes (polypeta), x3o3o *b3o3o3o – hax

External links[edit]