# 6-demicube

Last updated
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

In geometry, a 6-demicube or demihexeract 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.

## Contents

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 ${\displaystyle \left\{3{\begin{array}{l}3,3,3\\3\end{array}}\right\}}$ 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( )f03215602060153066 r{3,3,3,3} D6/A4 = 32*6!/5! = 32
A3A1A1{ }f1224084126842 {}x{3,3} D6/A3A1A1 = 32*6!/4!/2/2 = 240
A3A2 {3} f233640133331 {3}v( ) D6/A3A2 = 32*6!/4!/3! = 640
A3A1 h{4,3} f3464160*3030 {3} D6/A3A1 = 32*6!/4!/2 = 160
A3A2 {3,3} 464*4801221 {}v( ) D6/A3A2 = 32*6!/4!/3! = 480
D4A1 h{4,3,3} f4824328860*20{ }D6/D4A1 = 32*6!/8/4!/2 = 60
A4 {3,3,3} 5101005*19211D6/A4 = 32*6!/5! = 192
D5 h{4,3,3,3} f516801604080101612*( )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 planeD6D5
Graph
Dihedral symmetry[10][8]
Coxeter planeD4D3
Graph
Dihedral symmetry[6][4]
Coxeter planeA5A3
Graph
Dihedral symmetry[6][4]

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}

h2,3{4,34}

h2,4{4,34}

h2,5{4,34}

h3,4{4,34}

h3,5{4,34}

h4,5{4,34}

h2,3,4{4,34}

h2,3,5{4,34}

h2,4,5{4,34}

h3,4,5{4,34}

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
A3A1A5D6 E7 ${\displaystyle {\tilde {E}}_{7}}$ = E7+${\displaystyle {\bar {T}}_{8}}$=E7++
Coxeter
diagram
Symmetry [3−1,3,1][30,3,1][31,3,1][32,3,1][33,3,1][34,3,1]
Order 4872023,0402,903,040
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 fourth figure is the Euclidean honeycomb 133 and the final is a noncompact hyperbolic honeycomb, 134.

13k dimensional figures
SpaceFiniteEuclideanHyperbolic
n 4 5 6 7 8 9
Coxeter
group
A3A1A5D6 E7 ${\displaystyle {\tilde {E}}_{7}}$=E7+${\displaystyle {\bar {T}}_{8}}$=E7++
Coxeter
diagram
Symmetry [3−1,3,1][30,3,1][31,3,1][32,3,1][[33,3,1]][34,3,1]
Order 4872023,0402,903,040
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]

## Related Research Articles

In six-dimensional geometry, a uniform 6-polytope is a six-dimensional uniform polytope. A uniform polypeton is vertex-transitive, and all facets are uniform 5-polytopes.

In five-dimensional geometry, a 5-cube is a name for a five-dimensional hypercube with 32 vertices, 80 edges, 80 square faces, 40 cubic cells, and 10 tesseract 4-faces.

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

In geometry, a 6-orthoplex, or 6-cross polytope, is a regular 6-polytope with 12 vertices, 60 edges, 160 triangle faces, 240 tetrahedron cells, 192 5-cell 4-faces, and 64 5-faces.

In geometry, a demihepteract or 7-demicube is a uniform 7-polytope, constructed from the 7-hypercube (hepteract) with alternated vertices removed. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes.

In five-dimensional geometry, a rectified 5-simplex is a convex uniform 5-polytope, being a rectification of the regular 5-simplex.

In geometry, a 10-demicube or demidekeract is a uniform 10-polytope, constructed from the 10-cube with alternated vertices removed. It is part of a dimensionally infinite family of uniform polytopes called demihypercubes.

In 7-dimensional geometry, 132 is a uniform polytope, constructed from the E7 group.

In geometry, the regular skew polyhedra are generalizations to the set of regular polyhedra which include the possibility of nonplanar faces or vertex figures. Coxeter looked at skew vertex figures which created new 4-dimensional regular polyhedra, and much later Branko Grünbaum looked at regular skew faces.

In 7-dimensional geometry, there are 128 uniform polytopes with B7 symmetry. There are two regular forms, the 7-orthoplex, and 8-cube with 14 and 128 vertices respectively. The 7-demicube is added with half of the symmetry.

In six-dimensional geometry, a runcic 5-cube or is a convex uniform 5-polytope. There are 2 runcic forms for the 5-cube. Runcic 5-cubes have half the vertices of runcinated 5-cubes.

In seven-dimensional geometry, a cantic 7-cube or truncated 7-demicube as a uniform 7-polytope, being a truncation of the 7-demicube.

In nine-dimensional geometry, a rectified 9-simplex is a convex uniform 9-polytope, being a rectification of the regular 9-simplex.

In six-dimensional geometry, a runcic 6-cube is a convex uniform 6-polytope. There are 2 unique runcic for the 6-cube.

In six-dimensional geometry, a steric 6-cube is a convex uniform 6-polytope. There are unique 4 steric forms of the 6-cube.

In six-dimensional geometry, a stericated 6-orthoplex is a convex uniform 6-polytope, constructed as a sterication of the regular 6-orthoplex.

In six-dimensional geometry, a pentic 6-cube is a convex uniform 6-polytope.

In seven-dimensional geometry, a runcic 7-cube is a convex uniform 7-polytope, related to the uniform 7-demicube. There are 2 unique forms.

In five-dimensional Euclidean geometry, the quarter 5-cubic honeycomb is a uniform space-filling tessellation. It has half the vertices of the 5-demicubic honeycomb, and a quarter of the vertices of a 5-cube honeycomb. Its facets are 5-demicubes and runcinated 5-demicubes.

In seven-dimensional geometry, a stericated 7-cube is a convex uniform 7-polytope, being a runcination of the uniform 7-demicube. There are 4 unique runcinations for the 7-demicube including truncation and cantellation.

## 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:. Retrieved 4 April 2020.
Family An Bn I2(p) / Dn E6 / E7 / E8 / / Hn
Regular polygon Triangle Square p-gon Hexagon Pentagon
Uniform polyhedron Tetrahedron OctahedronCube Demicube DodecahedronIcosahedron
Uniform polychoron Pentachoron 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