# 6-demicube

Demihexeract (6-demicube) :
| Petrie polygon projection |||
| Type | Uniform 6-polytope | Uniform 6-polytope |
| Family | demihypercube | demihypercube |
| Schläfli symbol | {3,3$^{3,1}$} = h{4,3$^4$} s{2$^{1,1,1,1,1}$} | {3,3$^{3,1}$} = h{4,3$^4$} s{2$^{1,1,1,1,1}$} |
| Coxeter diagrams | = = | = = |
| Coxeter symbol | 1$_{31}$ | 1$_{31}$ |
| 5-faces | 44 | 12 {3$^{1,2,1}$} 32 {3$^4$} |
| 4-faces | 252 | 60 {3$^{1,1,1}$} 192 {3$^3$} |
| Cells | 640 | 160 {3$^{1,0,1}$} 480 {3,3} |
| Faces | 640 | {3} |
| Edges | 240 | 240 |
| Vertices | 32 | 32 |
| Vertex figure | Rectified 5-simplex | Rectified 5-simplex |
| Symmetry group | D$_6$, [3$^{3,1,1}$] = [1$^+$,4,3$^4$] [2$^5$]$^+$ | D$_6$, [3$^{3,1,1}$] = [1$^+$,4,3$^4$] [2$^5$]$^+$ |
| Petrie polygon | decagon | decagon |
| Properties | convex | 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.
E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as HM$_6$ for a 6-dimensional half measure polytope.
Coxeter named this polytope as 1$_{31}$ 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 $\left\{3{\begin{array}{l}3,3,3\\3\end{array}}\right\}$ or {3,3$^{3,1}$}.

## 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.
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.
| D$_6$ | k-face | f$_k$ | f$_0$ | f$_1$ | f$_2$ | f$_3$ | f$_3$ | f$_4$ | f$_4$ | f$_5$ | f$_5$ | k-figure | notes |
| A$_4$ | ( ) | f$_0$ | 32 | 15 | 60 | 20 | 60 | 15 | 30 | 6 | 6 | r{3,3,3,3} | D$_6$/A$_4$ = 32*6!/5! = 32 |
| A$_3$A$_1$A$_1$ | { } | f$_1$ | 2 | 240 | 8 | 4 | 12 | 6 | 8 | 4 | 2 |x{3,3} | D$_6$/A$_3$A$_1$A$_1$ = 32*6!/4!/2/2 = 240 |
| A$_3$A$_2$ | {3} | f$_2$ | 3 | 3 | 640 | 1 | 3 | 3 | 3 | 3 | 1 | {3}v( ) | D$_6$/A$_3$A$_2$ = 32*6!/4!/3! = 640 |
| A$_3$A$_1$ | h{4,3} | f$_3$ | 4 | 6 | 4 | 160 | * | 3 | 0 | 3 | 0 | {3} | D$_6$/A$_3$A$_1$ = 32*6!/4!/2 = 160 |
| A$_3$A$_2$ | {3,3} | f$_3$ | 4 | 6 | 4 | * | 480 | 1 | 2 | 2 | 1 |v( ) | D$_6$/A$_3$A$_2$ = 32*6!/4!/3! = 480 |
| D$_4$A$_1$ | h{4,3,3} | f$_4$ | 8 | 24 | 32 | 8 | 8 | 60 | * | 2 | 0 | { } | D$_6$/D$_4$A$_1$ = 32*6!/8/4!/2 = 60 |
| A$_4$ | {3,3,3} | f$_4$ | 5 | 10 | 10 | 0 | 5 | * | 192 | 1 | 1 | { } | D$_6$/A$_4$ = 32*6!/5! = 192 |
| D$_5$ | h{4,3,3,3} | f$_5$ | 16 | 80 | 160 | 40 | 80 | 10 | 16 | 12 | * | ( ) | D$_6$/D$_5$ = 32*6!/16/5! = 12 |
| A$_5$ | {3,3,3,3} | f$_5$ | 6 | 15 | 20 | 0 | 15 | 0 | 6 | * | 32 | ( ) | D$_6$/A$_5$ = 32*6!/6! = 32 |

## Images

orthographic projections :
| Coxeter plane | B$_6$ |
| Graph | |
| Dihedral symmetry | [12/2] |
| Coxeter plane | D$_6$ | D$_5$ |
| Graph | | |
| Dihedral symmetry | [10] | [8] |
| Coxeter plane | D$_4$ | D$_3$ |
| Graph | | |
| Dihedral symmetry | [6] | [4] |
| Coxeter plane | A$_5$ | A$_3$ |
| Graph | | |
| Dihedral symmetry | [6] | [4] |

## Related polytopes

There are 47 uniform polytopes with D$_6$ symmetry, 31 are shared by the B$_6$ symmetry, and 16 are unique:
D6 polytopes :
| h{4,3$^4$} | h$_2${4,3$^4$} | h$_3${4,3$^4$} | h$_4${4,3$^4$} | h$_5${4,3$^4$} | h$_{2,3}${4,3$^4$} | h$_{2,4}${4,3$^4$} | h$_{2,5}${4,3$^4$} |
| h$_{3,4}${4,3$^4$} | h$_{3,5}${4,3$^4$} | h$_{4,5}${4,3$^4$} | h$_{2,3,4}${4,3$^4$} | h$_{2,3,5}${4,3$^4$} | h$_{2,4,5}${4,3$^4$} | h$_{3,4,5}${4,3$^4$} | h$_{2,3,4,5}${4,3$^4$} |
The 6-demicube, 1$_{31}$ is third in a dimensional series of uniform polytopes, expressed by Coxeter as k$_{31}$ series. The fifth figure is a Euclidean honeycomb, 3$_{31}$, and the final is a noncompact hyperbolic honeycomb, 4$_{31}$. Each progressive uniform polytope is constructed from the previous as its vertex figure.
k$_{31}$ dimensional figures :
| n | 4 | 5 | 6 | 7 | 8 | 9 |
| Coxeter group | A$_3$A$_1$ | A$_5$ | D$_6$ | E$_7$ | ${\tilde {E}}_{7}$ = E$_7$$^+$ | ${\bar {T}}_{8}$=E$_7$$^{++}$ |
| 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$_{31}$ | 0$_{31}$ | 1$_{31}$ | 2$_{31}$ | 3$_{31}$ | 4$_{31}$ |
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 1$_{33}$ and the final is a noncompact hyperbolic honeycomb, 1$_{34}$.
1$_{3k}$ dimensional figures :
| Space | Finite | Finite | Finite | Finite | Euclidean | Hyperbolic |
| n | 4 | 5 | 6 | 7 | 8 | 9 |
| Coxeter group | A$_3$A$_1$ | A$_5$ | D$_6$ | E$_7$ | ${\tilde {E}}_{7}$=E$_7$$^+$ | ${\bar {T}}_{8}$=E$_7$$^{++}$ |
| 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$_{3,-1}$ | 1$_{30}$ | 1$_{31}$ | 1$_{32}$ | 1$_{33}$ | 1$_{34}$ |

### 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.