6-demicube

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

Demihexeract
(6-demicube)

Petrie polygon projection
Type Uniform 6-polytope
Family demihypercube
Schläfli symbol {3,33,1} = h{4,34}
s{21,1,1,1,1}
Coxeter diagrams =
=





Coxeter symbol 131
5-faces4412 {31,2,1}
32 {34}
4-faces25260 {31,1,1}
192 {33}
Cells640160 {31,0,1}
480 {3,3}
Faces640{3}
Edges240
Vertices32
Vertex figure Rectified 5-simplex
Symmetry group D6, [33,1,1] = [1+,4,34]
[25]+
Petrie polygon decagon
Properties convex

E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as HM6 for a 6-dimensional half measure polytope.

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

Cartesian coordinates

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.

As a configuration

This configuration matrix represents the 6-demicube. The rows and columns correspond to vertices, edges, faces, cells, 4-faces and 5-faces. The diagonal numbers say how many of each element occur in the whole 6-demicube. The nondiagonal numbers say how many of the column's element occur in or at the row's 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]

D6k-facefkf0f1f2f3f4f5k-figurenotes
A4( ) f0 3215602060153066r{3,3,3,3}D6/A4 = 32*6!/5! = 32
A3A1A1{ } f1 224084126842{}x{3,3}D6/A3A1A1 = 32*6!/4!/2/2 = 240
A3A2{3} f2 33640133331{3}v( )D6/A3A2 = 32*6!/4!/3! = 640
A3A1h{4,3} f3 464160*3030{3}D6/A3A1 = 32*6!/4!/2 = 160
A3A2{3,3} 464*4801221{}v( )D6/A3A2 = 32*6!/4!/3! = 480
D4A1h{4,3,3} f4 824328860*20{ }D6/D4A1 = 32*6!/8/4!/2 = 60
A4{3,3,3} 5101005*19211D6/A4 = 32*6!/5! = 192
D5h{4,3,3,3} f5 16801604080101612*( )D6/D5 = 32*6!/16/5! = 12
A5{3,3,3,3} 6152001506*32D6/A5 = 32*6!/6! = 32

Images
orthographic projections
Coxeter plane B6
Graph
Dihedral symmetry [12/2]
Coxeter plane D6 D5
Graph
Dihedral symmetry [10] [8]
Coxeter plane D4 D3
Graph
Dihedral symmetry [6] [4]
Coxeter plane A5 A3
Graph
Dihedral symmetry [6] [4]

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

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		 A3A1 A5 D6 E7 = E7+ =E7++
Coxeter
diagram
Symmetry [3−1,3,1] [30,3,1] [31,3,1] [32,3,1] [33,3,1] [34,3,1]
Order 48 720 23,040 2,903,040 ∞
Graph - -
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
Space Finite Euclidean Hyperbolic
n 4 5 6 7 8 9
Coxeter
group		 A3A1 A5 D6 E7 =E7+ =E7++
Coxeter
diagram
Symmetry [3−1,3,1] [30,3,1] [31,3,1] [32,3,1] [[3<sup>3,3,1</sup>]] [34,3,1]
Order 48 720 23,040 2,903,040 ∞
Graph - -
Name 13,-1 130 131 132 133 134

Skew icosahedron

Coxeter identified a subset of 12 vertices that form a regular skew icosahedron {3, 5} with the same symmetries as the icosahedron itself, but at different angles. He dubbed this the regular skew icosahedron.[4][5]

References
  1. Coxeter, Regular Polytopes, sec 1.8 Configurations
  2. Coxeter, Complex Regular Polytopes, p.117
  3. Klitzing, Richard. "x3o3o *b3o3o3o - hax".
  4. Coxeter, H. S. M. The beauty of geometry : twelve essays (Dover ed.). Dover Publications. pp. 450–451. ISBN 9780486409191.
  5. Deza, Michael; Shtogrin, Mikhael (2000). "Embedding the graphs of regular tilings and star-honeycombs into the graphs of hypercubes and cubic lattices". Advanced Studies in Pure Mathematics: 77. doi:10.2969/aspm/02710073. Retrieved 4 April 2020.
  • 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
      • (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)
  • Klitzing, Richard. "6D uniform polytopes (polypeta) x3o3o *b3o3o3o – hax".
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 Octahedron • Cube Demicube Dodecahedron • Icosahedron
Uniform 4-polytope 5-cell 16-cell • Tesseract Demitesseract 24-cell 120-cell • 600-cell
Uniform 5-polytope 5-simplex 5-orthoplex • 5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex • 6-cube 6-demicube 122 • 221
Uniform 7-polytope 7-simplex 7-orthoplex • 7-cube 7-demicube 132 • 231 • 321
Uniform 8-polytope 8-simplex 8-orthoplex • 8-cube 8-demicube 142 • 241 • 421
Uniform 9-polytope 9-simplex 9-orthoplex • 9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex • 10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplex • n-cube n-demicube 1k2 • 2k1 • k21 n-pentagonal polytope
Topics: Polytope families • Regular polytope • List of regular polytopes and compounds
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