5-demicube

Demipenteract
(5-demicube)
Demipenteract graph ortho.svg
Petrie polygon projection
Type Uniform 5-polytope
Family (Dn) 5-demicube
Families (En) k21 polytope
1k2 polytope
Coxeter
symbol
121
Schläfli
symbols
{3,32,1} = h{4,33}
s{2,4,3,3} or h{2}h{4,3,3}
sr{2,2,4,3} or h{2}h{2}h{4,3}
h{2}h{2}h{2}h{4}
s{21,1,1,1} or h{2}h{2}h{2}s{2}
Coxeter
diagrams
CDel nodes 10ru.pngCDel split2.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.png
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.pngCDel 3.pngCDel node.png
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 4.pngCDel node.pngCDel 3.pngCDel node.png
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 4.pngCDel node.png
CDel node h.pngCDel 2x.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2x.pngCDel node h.pngCDel 2x.pngCDel node h.png
4-faces 26 10 {31,1,1}Cross graph 4.svg
16 {3,3,3}4-simplex t0.svg
Cells 120 40 {31,0,1}3-simplex t0.svg
80 {3,3}3-simplex t0.svg
Faces 160 {3}2-simplex t0.svg
Edges 80
Vertices 16
Vertex
figure
5-demicube verf.svg
rectified 5-cell
Petrie
polygon
Octagon
Symmetry D5, [32,1,1] = [1+,4,33]
[24]+
Properties convex

In five-dimensional geometry, a demipenteract or 5-demicube is a semiregular 5-polytope, constructed from a 5-hypercube (penteract) with alternated vertices removed.

It was discovered by Thorold Gosset. Since it was the only semiregular 5-polytope (made of more than one type of regular facets), he called it a 5-ic semi-regular. E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as HM5 for a 5-dimensional half measure polytope.

Coxeter named this polytope as 121 from its Coxeter diagram, which has branches of length 2, 1 and 1 with a ringed node on one of the short branches, CDel node 1.pngCDel 3.pngCDel node.pngCDel split1.pngCDel nodes.pngCDel 3a.pngCDel nodea.png and Schläfli symbol or {3,32,1}.

It exists in the k21 polytope family as 121 with the Gosset polytopes: 221, 321, and 421.

The graph formed by the vertices and edges of the demipenteract is sometimes called the Clebsch graph, though that name sometimes refers to the folded cube graph of order five instead.

Cartesian coordinatesEdit

Cartesian coordinates for the vertices of a demipenteract centered at the origin and edge length 22 are alternate halves of the penteract:

(±1,±1,±1,±1,±1)

with an odd number of plus signs.

As a configurationEdit

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

D5         k-face fk f0 f1 f2 f3 f4 k-figure notes
A4         ( ) f0 16 10 30 10 20 5 5 rectified 5-cell D5/A4 = 16*5!/5! = 16
A2A1A1         { } f1 2 80 6 3 6 3 2 triangular prism D5/A2A1A1 = 16*5!/3!/2/2 = 80
A2A1         {3} f2 3 3 160 1 2 2 1 Isosceles triangle D5/A2A1 = 16*5!/3!/2 = 160
A3A1         h{4,3} f3 4 6 4 40 * 2 0 { } D5/A3A1 = 16*5!/4!/2 = 40
A3         {3,3} 4 6 4 * 80 1 1 { } D5/A3 = 16*5!/4! = 80
D4         h{4,3,3} f4 8 24 32 8 8 10 * ( ) D5/D4 = 16*5!/8/4! = 10
A4         {3,3,3} 5 10 10 0 5 * 16 ( ) D5/A4 = 16*5!/5! = 16

Projected imagesEdit

 
Perspective projection.

ImagesEdit

orthographic projections
Coxeter plane B5
Graph  
Dihedral symmetry [10/2]
Coxeter plane D5 D4
Graph    
Dihedral symmetry [8] [6]
Coxeter plane D3 A3
Graph    
Dihedral symmetry [4] [4]

Related polytopesEdit

It is a part of a dimensional family of uniform polytopes called demihypercubes for being alternation of the hypercube family.

There are 23 Uniform 5-polytopes (uniform 5-polytopes) that can be constructed from the D5 symmetry of the demipenteract, 8 of which are unique to this family, and 15 are shared within the penteractic family.

The 5-demicube is third in a dimensional series of semiregular polytopes. Each progressive uniform polytope is constructed vertex figure of the previous polytope. Thorold Gosset identified this series in 1900 as containing all regular polytope facets, containing all simplexes and orthoplexes (5-simplices and 5-orthoplexes in the case of the 5-demicube). In Coxeter's notation the 5-demicube is given the symbol 121.

ReferencesEdit

  1. ^ Coxeter, Regular Polytopes, sec 1.8 Configurations
  2. ^ Coxeter, Complex Regular Polytopes, p.117
  3. ^ Klitzing, Richard. "x3o3o *b3o3o - hin".
  • T. Gosset: On the Regular and Semi-Regular Figures in Space of n Dimensions, Messenger of Mathematics, Macmillan, 1900
  • 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)
  • Klitzing, Richard. "5D uniform polytopes (polytera) x3o3o *b3o3o - hin".

External linksEdit

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 OctahedronCube Demicube DodecahedronIcosahedron
Uniform 4-polytope 5-cell 16-cellTesseract Demitesseract 24-cell 120-cell600-cell
Uniform 5-polytope 5-simplex 5-orthoplex5-cube 5-demicube
Uniform 6-polytope 6-simplex 6-orthoplex6-cube 6-demicube 122221
Uniform 7-polytope 7-simplex 7-orthoplex7-cube 7-demicube 132231321
Uniform 8-polytope 8-simplex 8-orthoplex8-cube 8-demicube 142241421
Uniform 9-polytope 9-simplex 9-orthoplex9-cube 9-demicube
Uniform 10-polytope 10-simplex 10-orthoplex10-cube 10-demicube
Uniform n-polytope n-simplex n-orthoplexn-cube n-demicube 1k22k1k21 n-pentagonal polytope
Topics: Polytope familiesRegular polytopeList of regular polytopes and compounds