6-demicube


6-demicube
Demihexeract
(6-demicube)
Demihexeract ortho petrie.svg
Petrie polygon projection
Type Uniform 6-polytope
Family demihypercube
Schläfli symbol {3,33,1}
h{4,3,3,3,3}
s{2,2,2,2,2}
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 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 2.pngCDel node h.pngCDel 2.pngCDel node h.pngCDel 2.pngCDel node h.pngCDel 2.pngCDel node h.pngCDel 2.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 alternate vertices deleted. 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}.

Contents

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.

Images

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

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

6-demicube t0 D6.svg
t0(131)
6-demicube t01 D6.svg
t0,1(131)
6-demicube t02 D6.svg
t0,2(131)
6-demicube t03 D6.svg
t0,3(131)
6-demicube t04 D6.svg
t0,4(131)
6-demicube t012 D6.svg
t0,1,2(131)
6-demicube t013 D6.svg
t0,1,3(131)
6-demicube t014 D6.svg
t0,1,4(131)
6-demicube t023 D6.svg
t0,2,3(131)
6-demicube t024 D6.svg
t0,2,4(131)
6-demicube t034 D6.svg
t0,3,4(131)
6-demicube t0123 D6.svg
t0,1,2,3(131)
6-demicube t0124 D6.svg
t0,1,2,4(131)
6-demicube t0134 D6.svg
t0,1,3,4(131)
6-demicube t0234 D6.svg
t0,2,3,4(131)
6-demicube t01234 D6.svg
t0,1,2,3,4(131)

References

  • 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, editied 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


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • 5-demicube — Demipenteract (5 demicube) Petrie polygon projection Type Uniform 5 polytope Family (Dn) 5 demicube Families (En) k21 polytope 1k2 poly …   Wikipedia

  • 7-demicube — Demihepteract (7 demicube) Petrie polygon projection Type Uniform 7 polytope Family demihypercube Coxeter symbol 141 Schläfli symbol …   Wikipedia

  • 9-demicube — Demienneract (9 demicube) Petrie polygon Type Uniform 9 polytope Family demihypercube Coxeter symbol 161 Schläfli symbol …   Wikipedia

  • 8-demicube — Demiocteract (8 demicube) Petrie polygon projection Type Uniform 8 polytope Family demihypercube Coxeter symbol 151 …   Wikipedia

  • 10-demicube — Demidekeract (10 demicube) Petrie polygon projection Type Uniform 10 polytope Family demihypercube Coxeter symbol 171 Schläfli symbol …   Wikipedia

  • Demihypercube — Not to be confused with Hemicube (geometry). Alternation of the n cube yields one of two n demicubes, as in this 3 dimensional illustration of the two tetrahedra that arise as the 3 demicubes of the 3 cube. In geometry, demihypercubes (also… …   Wikipedia

  • Coxeter group — In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry …   Wikipedia

  • 16-cell — Regular hexadecachoron (16 cell) (4 orthoplex) Schlegel diagram (vertices and edges) Type Convex regular 4 polytope Schläfli symbo …   Wikipedia

  • Cube — This article is about the geometric shape. For other uses, see Cube (disambiguation). Regular Hexahedron (Click here for rotating model) Type Platonic solid Elements F = 6, E = 12 V = 8 (χ = 2) …   Wikipedia

  • Tetrahedron — For the academic journal, see Tetrahedron (journal). Regular Tetrahedron (Click here for rotating model) Type Platonic solid Elements F = 4, E = 6 V = 4 (χ = 2) Faces by s …   Wikipedia


Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”

We are using cookies for the best presentation of our site. Continuing to use this site, you agree with this.