# 5-demicube

Demipenteract (5-demicube) :
| Petrie polygon projection |||
| Type | Uniform 5-polytope | Uniform 5-polytope |
| Family (D$_n$) | 5-demicube | 5-demicube |
| Families (E$_n$) | k$_{21}$ polytope 1$_{k2}$ polytope | k$_{21}$ polytope 1$_{k2}$ polytope |
| Coxeter symbol | 1$_{21}$ | 1$_{21}$ |
| Schläfli symbols | {3,3$^{2,1}$} = h{4,3$^3$} 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{2$^{1,1,1,1}$} or h{2}h{2}h{2}s{2} | {3,3$^{2,1}$} = h{4,3$^3$} 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{2$^{1,1,1,1}$} or h{2}h{2}h{2}s{2} |
| Coxeter diagrams | = | = |
| 4-faces | 26 | 10 {3$^{1,1,1}$} 16 {3,3,3} |
| Cells | 120 | 40 {3$^{1,0,1}$} 80 {3,3} |
| Faces | 160 | {3} |
| Edges | 80 | 80 |
| Vertices | 16 | 16 |
| Vertex figure | rectified 5-cell | rectified 5-cell |
| Petrie polygon | Octagon | Octagon |
| Symmetry | D$_5$, [3$^{2,1,1}$] = [1$^+$,4,3$^3$] [2$^4$]$^+$ | D$_5$, [3$^{2,1,1}$] = [1$^+$,4,3$^3$] [2$^4$]$^+$ |
| Properties | convex | 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 HM$_5$ for a 5-dimensional half measure polytope.
Coxeter named this polytope as 1$_{21}$ from its Coxeter diagram, which has branches of length 2, 1 and 1 with a ringed node on one of the short branches, and Schläfli symbol $\left\{3{\begin{array}{l}3,3\\3\end{array}}\right\}$ or {3,3$^{2,1}$}.
It exists in the k$_{21}$ polytope family as 1$_{21}$ with the Gosset polytopes: 2$_{21}$, 3$_{21}$, and 4$_{21}$.
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 coordinates

Cartesian coordinates for the vertices of a demipenteract centered at the origin and edge length 2√2 are alternate halves of the penteract:
    (±1,±1,±1,±1,±1)
with an odd number of plus signs.

## As a configuration

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.
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$_5$ | k-face | f$_k$ | f$_0$ | f$_1$ | f$_2$ | f$_3$ | f$_3$ | f$_4$ | f$_4$ | k-figure | notes(*) |
| A$_4$ | ( ) | f$_0$ | 16 | 10 | 30 | 10 | 20 | 5 | 5 | rectified 5-cell | D$_5$/A$_4$ = 16*5!/5! = 16 |
| A$_2$A$_1$A$_1$ | { } | f$_1$ | 2 | 80 | 6 | 3 | 6 | 3 | 2 | triangular prism | D$_5$/A$_2$A$_1$A$_1$ = 16*5!/3!/2/2 = 80 |
| A$_2$A$_1$ | {3} | f$_2$ | 3 | 3 | 160 | 1 | 2 | 2 | 1 | Isosceles triangle | D$_5$/A$_2$A$_1$ = 16*5!/3!/2 = 160 |
| A$_3$A$_1$ | h{4,3} | f$_3$ | 4 | 6 | 4 | 40 | * | 2 | 0 | Segment { } | D$_5$/A$_3$A$_1$ = 16*5!/4!/2 = 40 |
| A$_3$ | {3,3} | f$_3$ | 4 | 6 | 4 | * | 80 | 1 | 1 | Segment { } | D$_5$/A$_3$ = 16*5!/4! = 80 |
| D$_4$ | h{4,3,3} | f$_4$ | 8 | 24 | 32 | 8 | 8 | 10 | * | Point ( ) | D$_5$/D$_4$ = 16*5!/8/4! = 10 |
| A$_4$ | {3,3,3} | f$_4$ | 5 | 10 | 10 | 0 | 5 | * | 16 | Point ( ) | D$_5$/A$_4$ = 16*5!/5! = 16 |
* = The number of elements (diagonal values) can be computed by the symmetry order D$_5$ divided by the symmetry order of the subgroup with selected mirrors removed.

## Projected images

* Perspective projection.

## Images

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

## Related polytopes

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 D$_5$ symmetry of the demipenteract, 8 of which are unique to this family, and 15 are shared within the penteractic family.
D5 polytopes :
| h{4,3,3,3} | h$_2${4,3,3,3} | h$_3${4,3,3,3} | h$_4${4,3,3,3} | h$_{2,3}${4,3,3,3} | h$_{2,4}${4,3,3,3} | h$_{3,4}${4,3,3,3} | h$_{2,3,4}${4,3,3,3} |
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 1$_{21}$.
k$_{21}$ figures in n dimensions :
| Space | Finite | Finite | Finite | Finite | Finite | Finite | Euclidean | Hyperbolic |
| E$_n$ | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| Coxeter group | E$_3$=A$_2$A$_1$ | E$_4$=A$_4$ | E$_5$=D$_5$ | E$_6$ | E$_7$ | E$_8$ | E$_9$ = ${\tilde {E}}_{8}$ = E$_8$$^+$ | E$_{10}$ = ${\bar {T}}_{8}$ = E$_8$$^{++}$ |
| Coxeter diagram | | | | | | | | |
| Symmetry | [3$^{−1,2,1}$] | [3$^{0,2,1}$] | [3$^{1,2,1}$] | [3$^{2,2,1}$] | [3$^{3,2,1}$] | [3$^{4,2,1}$] | [3$^{5,2,1}$] | [3$^{6,2,1}$] |
| Order | 12 | 120 | 1,920 | 51,840 | 2,903,040 | 696,729,600 | ∞ | ∞ |
| Graph | | | | | | | - | - |
| Name | −1$_{21}$ | 0$_{21}$ | 1$_{21}$ | 2$_{21}$ | 3$_{21}$ | 4$_{21}$ | 5$_{21}$ | 6$_{21}$ |
1$_{k2}$ figures in n dimensions :
| Space | Finite | Finite | Finite | Finite | Finite | Finite | Euclidean | Hyperbolic |
| n | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
| Coxeter group | E$_3$=A$_2$A$_1$ | E$_4$=A$_4$ | E$_5$=D$_5$ | E$_6$ | E$_7$ | E$_8$ | E$_9$ = ${\tilde {E}}_{8}$ = E$_8$$^+$ | E$_{10}$ = ${\bar {T}}_{8}$ = E$_8$$^{++}$ |
| Coxeter diagram | | | | | | | | |
| Symmetry (order) | [3$^{−1,2,1}$] | [3$^{0,2,1}$] | [3$^{1,2,1}$] | [[3$^{2,2,1}$]] | [3$^{3,2,1}$] | [3$^{4,2,1}$] | [3$^{5,2,1}$] | [3$^{6,2,1}$] |
| Order | 12 | 120 | 1,920 | 103,680 | 2,903,040 | 696,729,600 | ∞ | ∞ |
| Graph | | | | | | | - | - |
| Name | 1$_{−1,2}$ | 1$_{02}$ | 1$_{12}$ | 1$_{22}$ | 1$_{32}$ | 1$_{42}$ | 1$_{52}$ | 1$_{62}$ |