# 7-demicube

Demihepteract (7-demicube) :
| Petrie polygon projection |||
| Type | Uniform 7-polytope | Uniform 7-polytope |
| Family | demihypercube | demihypercube |
| Coxeter symbol | 1$_{41}$ | 1$_{41}$ |
| Schläfli symbol | {3,3$^{4,1}$} = h{4,3$^5$} s{2$^{1,1,1,1,1,1}$} | {3,3$^{4,1}$} = h{4,3$^5$} s{2$^{1,1,1,1,1,1}$} |
| Coxeter diagrams | = | = |
| 6-faces | 78 | 14 {3$^{1,3,1}$} 64 {3$^5$} |
| 5-faces | 532 | 84 {3$^{1,2,1}$} 448 {3$^4$} |
| 4-faces | 1624 | 280 {3$^{1,1,1}$} 1344 {3$^3$} |
| Cells | 2800 | 560 {3$^{1,0,1}$} 2240 {3,3} |
| Faces | 2240 | {3} |
| Edges | 672 | 672 |
| Vertices | 64 | 64 |
| Vertex figure | Rectified 6-simplex | Rectified 6-simplex |
| Symmetry group | D$_7$, [3$^{4,1,1}$] = [1$^+$,4,3$^5$] [2$^6$]$^+$ | D$_7$, [3$^{4,1,1}$] = [1$^+$,4,3$^5$] [2$^6$]$^+$ |
| Dual | ? | ? |
| Properties | convex | convex |
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.
E. L. Elte identified it in 1912 as a semiregular polytope, labeling it as HM$_7$ for a 7-dimensional half measure polytope.
Coxeter named this polytope as 1$_{41}$ from its Coxeter diagram, with a ring on one of the 1-length branches, and Schläfli symbol $\left\{3{\begin{array}{l}3,3,3,3\\3\end{array}}\right\}$ or {3,3$^{4,1}$}.

## Cartesian coordinates

Cartesian coordinates for the vertices of a demihepteract centered at the origin are alternate halves of the hepteract:
    (±1,±1,±1,±1,±1,±1,±1)
with an odd number of plus signs.

## Images

orthographic projections :
| Coxeter plane | B$_7$ | D$_7$ | D$_6$ |
| Graph | | | |
| Dihedral symmetry | [14/2] | [12] | [10] |
| Coxeter plane | D$_5$ | D$_4$ | D$_3$ |
| Graph | | | |
| Dihedral symmetry | [8] | [6] | [4] |
| Coxeter plane | A$_5$ | A$_3$ |
| Graph | | |
| Dihedral symmetry | [6] | [4] |

## As a configuration

This configuration matrix represents the 7-demicube. The rows and columns correspond to vertices, edges, faces, cells, 4-faces, 5-faces and 6-faces. The diagonal numbers say how many of each element occur in the whole 7-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$_7$ | k-face | f$_k$ | f$_0$ | f$_1$ | f$_2$ | f$_3$ | f$_3$ | f$_4$ | f$_4$ | f$_5$ | f$_5$ | f$_6$ | f$_6$ | k-figures | notes |
| A$_6$ | ( ) | f$_0$ | 64 | 21 | 105 | 35 | 140 | 35 | 105 | 21 | 42 | 7 | 7 | 0$_{41}$ | D$_7$/A$_6$ = 64*7!/7! = 64 |
| A$_4$A$_1$A$_1$ | { } | f$_1$ | 2 | 672 | 10 | 5 | 20 | 10 | 20 | 10 | 10 | 5 | 2 | { }×{3,3,3} | D$_7$/A$_4$A$_1$A$_1$ = 64*7!/5!/2/2 = 672 |
| A$_3$A$_2$ | 1$_{00}$ | f$_2$ | 3 | 3 | 2240 | 1 | 4 | 4 | 6 | 6 | 4 | 4 | 1 | {3,3}v( ) | D$_7$/A$_3$A$_2$ = 64*7!/4!/3! = 2240 |
| A$_3$A$_3$ | 1$_{01}$ | f$_3$ | 4 | 6 | 4 | 560 | * | 4 | 0 | 6 | 0 | 4 | 0 | {3,3} | D$_7$/A$_3$A$_3$ = 64*7!/4!/4! = 560 |
| A$_3$A$_2$ | 1$_{10}$ | f$_3$ | 4 | 6 | 4 | * | 2240 | 1 | 3 | 3 | 3 | 3 | 1 | {3}v( ) | D$_7$/A$_3$A$_2$ = 64*7!/4!/3! = 2240 |
| D$_4$A$_2$ | 1$_{11}$ | f$_4$ | 8 | 24 | 32 | 8 | 8 | 280 | * | 3 | 0 | 3 | 0 | {3} | D$_7$/D$_4$A$_2$ = 64*7!/8/4!/2 = 280 |
| A$_4$A$_1$ | 1$_{20}$ | f$_4$ | 5 | 10 | 10 | 0 | 5 | * | 1344 | 1 | 2 | 2 | 1 | { }v( ) | D$_7$/A$_4$A$_1$ = 64*7!/5!/2 = 1344 |
| D$_5$A$_1$ | 1$_{21}$ | f$_5$ | 16 | 80 | 160 | 40 | 80 | 10 | 16 | 84 | * | 2 | 0 | { } | D$_7$/D$_5$A$_1$ = 64*7!/16/5!/2 = 84 |
| A$_5$ | 1$_{30}$ | f$_5$ | 6 | 15 | 20 | 0 | 15 | 0 | 6 | * | 448 | 1 | 1 | { } | D$_7$/A$_5$ = 64*7!/6! = 448 |
| D$_6$ | 1$_{31}$ | f$_6$ | 32 | 240 | 640 | 160 | 480 | 60 | 192 | 12 | 32 | 14 | * | ( ) | D$_7$/D$_6$ = 64*7!/32/6! = 14 |
| A$_6$ | 1$_{40}$ | f$_6$ | 7 | 21 | 35 | 0 | 35 | 0 | 21 | 0 | 7 | * | 64 | ( ) | D$_7$/A$_6$ = 64*7!/7! = 64 |

## Related polytopes

There are 95 uniform polytopes with D$_6$ symmetry, 63 are shared by the B$_6$ symmetry, and 32 are unique:
D7 polytopes :
| t$_0$(1$_{41}$) | t$_{0,1}$(1$_{41}$) | t$_{0,2}$(1$_{41}$) | t$_{0,3}$(1$_{41}$) | t$_{0,4}$(1$_{41}$) | t$_{0,5}$(1$_{41}$) | t$_{0,1,2}$(1$_{41}$) | t$_{0,1,3}$(1$_{41}$) |
| t$_{0,1,4}$(1$_{41}$) | t$_{0,1,5}$(1$_{41}$) | t$_{0,2,3}$(1$_{41}$) | t$_{0,2,4}$(1$_{41}$) | t$_{0,2,5}$(1$_{41}$) | t$_{0,3,4}$(1$_{41}$) | t$_{0,3,5}$(1$_{41}$) | t$_{0,4,5}$(1$_{41}$) |
| t$_{0,1,2,3}$(1$_{41}$) | t$_{0,1,2,4}$(1$_{41}$) | t$_{0,1,2,5}$(1$_{41}$) | t$_{0,1,3,4}$(1$_{41}$) | t$_{0,1,3,5}$(1$_{41}$) | t$_{0,1,4,5}$(1$_{41}$) | t$_{0,2,3,4}$(1$_{41}$) | t$_{0,2,3,5}$(1$_{41}$) |
| t$_{0,2,4,5}$(1$_{41}$) | t$_{0,3,4,5}$(1$_{41}$) | t$_{0,1,2,3,4}$(1$_{41}$) | t$_{0,1,2,3,5}$(1$_{41}$) | t$_{0,1,2,4,5}$(1$_{41}$) | t$_{0,1,3,4,5}$(1$_{41}$) | t$_{0,2,3,4,5}$(1$_{41}$) | t$_{0,1,2,3,4,5}$(1$_{41}$) |