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6-demicube - Wikipedia

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Uniform 6-polytope

Demihexeract
(6-demicube)
Petrie polygon projection Type Uniform 6-polytope Family demihypercube Schläfli symbol {3,3^3,1} = h{4,3⁴}
s{2^1,1,1,1,1} Coxeter diagrams =
=

Coxeter symbol 1₃₁ 5-faces 44 12 {3^1,2,1}
32 {3⁴} 4-faces 252 60 {3^1,1,1}
192 {3³} Cells 640 160 {3^1,0,1}
480 {3,3} Faces 640 {3} Edges 240 Vertices 32 Vertex figure Rectified 5-simplex
Symmetry group D₆, [3^3,1,1] = [1⁺,4,3⁴]
[2⁵]⁺ Petrie polygon decagon Properties convex

In geometry, a 6-demicube, demihexeract or hemihexeract 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. Acronym: hax.^[1]

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 { 3 3 , 3 , 3 3 } {\displaystyle \left\{3{\begin{array}{l}3,3,3\\3\end{array}}\right\}} or {3,3^3,1}.

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.

As a configuration[edit]

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.^[2]^[3]

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.^[1]

D₆ k-face f_k f₀ f₁ f₂ f₃ f₄ f₅ k-figure notes A₄ ( ) f₀ 32 15 60 20 60 15 30 6 6 r{3,3,3,3} D₆/A₄ = 32*6!/5! = 32 A₃A₁A₁ { } f₁ 2 240 8 4 12 6 8 4 2 {}x{3,3} D₆/A₃A₁A₁ = 32*6!/4!/2/2 = 240 A₃A₂ {3} f₂ 3 3 640 1 3 3 3 3 1 {3}v( ) D₆/A₃A₂ = 32*6!/4!/3! = 640 A₃A₁ h{4,3} f₃ 4 6 4 160 * 3 0 3 0 {3} D₆/A₃A₁ = 32*6!/4!/2 = 160 A₃A₂ {3,3} 4 6 4 * 480 1 2 2 1 {}v( ) D₆/A₃A₂ = 32*6!/4!/3! = 480 D₄A₁ h{4,3,3} f₄ 8 24 32 8 8 60 * 2 0 { } D₆/D₄A₁ = 32*6!/8/4!/2 = 60 A₄ {3,3,3} 5 10 10 0 5 * 192 1 1 D₆/A₄ = 32*6!/5! = 192 D₅ h{4,3,3,3} f₅ 16 80 160 40 80 10 16 12 * ( ) D₆/D₅ = 32*6!/16/5! = 12 A₅ {3,3,3,3} 6 15 20 0 15 0 6 * 32 D₆/A₅ = 32*6!/6! = 32

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

D6 polytopes
h{4,3⁴}
h₂{4,3⁴}
h₃{4,3⁴}
h₄{4,3⁴}
h₅{4,3⁴}
h_2,3{4,3⁴}
h_2,4{4,3⁴}
h_2,5{4,3⁴}
h_3,4{4,3⁴}
h_3,5{4,3⁴}
h_4,5{4,3⁴}
h_2,3,4{4,3⁴}
h_2,3,5{4,3⁴}
h_2,4,5{4,3⁴}
h_3,4,5{4,3⁴}
h_2,3,4,5{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 A₃A₁ A₅ D₆ E₇ E ~ 7 {\displaystyle {\tilde {E}}_{7}} = E₇⁺ T ¯ 8 {\displaystyle {\bar {T}}_{8}} =E₇⁺⁺ Coxeter
diagram Symmetry [3^−1,3,1] [3^0,3,1] [3^1,3,1] [3^2,3,1] [3^3,3,1] [3^4,3,1] 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 1_3k series. The fourth figure is the Euclidean honeycomb 1₃₃ and the final is a noncompact hyperbolic honeycomb, 1₃₄.

1_3k dimensional figures Space Finite Euclidean Hyperbolic n 4 5 6 7 8 9 Coxeter
group A₃A₁ A₅ D₆ E₇ E ~ 7 {\displaystyle {\tilde {E}}_{7}} =E₇⁺ T ¯ 8 {\displaystyle {\bar {T}}_{8}} =E₇⁺⁺ Coxeter
diagram Symmetry [3^−1,3,1] [3^0,3,1] [3^1,3,1] [3^2,3,1] [[3^3,3,1]] [3^4,3,1] Order 48 720 23,040 2,903,040 ∞ Graph - - Name 1_3,-1 1₃₀ 1₃₁ 1₃₂ 1₃₃ 1₃₄

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]

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, wiley.com, 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-Strauss, The Symmetries of Things 2008, ISBN 978-1-56881-220-5 (Chapter 26. pp. 409: Hemicubes: 1_n1)
Klitzing, Richard. "6D uniform polytopes (polypeta) with acronyms x3o3o *b3o3o3o – hax".

Olshevsky, George. "Demihexeract". Glossary for Hyperspace. Archived from the original on 4 February 2007.
Multi-dimensional Glossary

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