Witting polytope
# Witting polytope

Witting polytope :

| Schläfli symbol | $_3{3}_3{3}_3{3}_3$ |
| - | - |
| Coxeter diagram | |
| Cells | 240 $_3{3}_3{3}_3$ |
| Faces | 2160 $_3{3}_3$ |
| Edges | 2160 $_3$ |
| Vertices | 240 |
| Petrie polygon | 30-gon |
| van Oss polygon | 90 $_3{4}_3$ |
| Shephard group | $L_4$ = $_3[3]_3[3]_3[3]_3,$ order 155,520 |
| Dual polyhedron | Self-dual |
| Properties | Regular |

In 4-dimensional complex geometry, the Witting polytope is a regular complex polytope, named as: $_3{3}_3{3}_3{3}_3,$ and Coxeter diagram . It has 240 vertices, 2160 $_3$ edges, 2160 $_3{3}_3$ faces, and 240 $_3{3}_3{3}_3$ cells. It is self-dual. Each vertex belongs to 27 edges, 72 faces, and 27 cells, corresponding to the Hessian polyhedron vertex figure.

## Symmetry

Its symmetry by $_3[3]_3[3]_3[3]_3$ or , order 155,520. It has 240 copies of , order 648 at each cell.

## Structure

The configuration matrix is: $\left[{\begin{smallmatrix}240&27&72&27\\3&2160&8&8\\8&8&2160&3\\27&72&27&240\end{smallmatrix}}\right]$

The number of vertices, edges, faces, and cells are seen in the diagonal of the matrix. These are computed by the order of the group divided by the order of the subgroup, by removing certain complex reflections, shown with X below. The number of elements of the k-faces are seen in rows below the diagonal. The number of elements in the vertex figure, etc., are given in rows above the digonal.

| $L_4$ | k-face | $f_k$ | $f_0$ | $f_1$ | $f_2$ | $f_3$ | k-figure | Notes |
| - | - | - | - | - | - | - | - | - |
| $L_3$ | ( ) | $f_0$ | 240 | 27 | 72 | 27 | $_3{3}_3{3}_3$ | $L_4/L_3$ = 216*6!/27/4! = 240 |
| $L_2L_1$ | $_3{$ } | $f_1$ | 3 | 2160 | 8 | 8 | $_3{3}_3$ | $L_4/L_2L_1$ = 216*6!/4!/3 = 2160 |
| $L_2L_1$ | $_3{3}_3$ | $f_2$ | 8 | 8 | 2160 | 3 | $_3{$ } | $L_4/L_2L_1$ = 216*6!/4!/3 = 2160 |
| $L_3$ | $_3{3}_3{3}_3$ | $f_3$ | 27 | 72 | 27 | 240 | ( ) | $L_4/L_3$ = 216*6!/27/4! = 240 |

## Coordinates

Its 240 vertices are given coordinates in $\mathbb {C} ^{4}$:

| (0, $±ω^μ,$ $-±ω^ν,$ $±ω^λ)$ $(-±ω^μ,$ 0, $±ω^ν,$ $±ω^λ)$ $(±ω^μ,$ $-±ω^ν,$ 0, $±ω^λ)$ $(-±ω^λ,$ $-±ω^μ,$ $-±ω^ν,$ 0) | $(±iω^λ√3,$ 0, 0, 0) (0, $±iω^λ√3,$ 0, 0) (0, 0, $±iω^λ√3,$ 0) (0, 0, 0, $±iω^λ√3)$ |
| - | - |

where $\omega ={\tfrac {-1+i{\sqrt {3}}}{2}},\lambda ,\nu ,\mu =0,1,2$.

The last 6 points form hexagonal holes on one of its 40 diameters. There are 40 hyperplanes contain central $_3{3}_3{4}_2,$ figures, with 72 vertices.

## Witting configuration

Coxeter named it after Alexander Witting for being a Witting configuration in complex projective 3-space:

    $\left[{\begin{smallmatrix}40&12&12\\2&240&2\\12&12&40\end{smallmatrix}}\right]$ or $\left[{\begin{smallmatrix}40&9&12\\4&90&4\\12&9&40\end{smallmatrix}}\right]$

The Witting configuration is related to the finite space $PG(3,2^2),$ consisting of 85 points, 357 lines, and 85 planes.

## Related real polytope

Its 240 vertices are shared with the real 8-dimensional polytope $4_{21},$ . Its 2160 3-edges are sometimes drawn as 6480 simple edges, slightly less than the 6720 edges of $4_{21}.$ The 240 difference is accounted by 40 central hexagons in $4_{21}$ whose edges are not included in $_3{3}_3{3}_3{3}_3.$

## The honeycomb of Witting polytopes

The regular Witting polytope has one further stage as a 4-dimensional honeycomb, . It has the Witting polytope as both its facets, and vertex figure. It is self-dual, and its dual coincides with itself.

Hyperplane sections of this honeycomb include 3-dimensional honeycombs .

The honeycomb of Witting polytopes has a real representation as the 8-dimensional polytope $5_{21},$ .

Its f-vector element counts are in proportion: 1, 80, 270, 80, 1. The configuration matrix for the honeycomb is:

| $L_5$ | k-face | $f_k$ | $f_0$ | $f_1$ | $f_2$ | $f_3$ | $f_4$ | k-figure | Notes |
| - | - | - | - | - | - | - | - | - | - |
| $L_4$ | ( ) | $f_0$ | N | 240 | 2160 | 2160 | 240 | $_3{3}_3{3}_3{3}_3$ | $L_5/L_4$ = N |
| $L_3L_1$ | $_3{$ } | $f_1$ | 3 | 80N | 27 | 72 | 27 | $_3{3}_3{3}_3$ | $L_5/L_3L_1$ = 80N |
| $L_2L_2$ | $_3{3}_3$ | $f_2$ | 8 | 8 | 270N | 8 | 8 | $_3{3}_3$ | $L_5/L_2L_2$ = 270N |
| $L_3L_1$ | $_3{3}_3{3}_3$ | $f_3$ | 27 | 72 | 27 | 80N | 3 | $_3$ | $L_5/L_3L_1$ = 80N |
| $L_4$ | $_3{3}_3{3}_3{3}_3$ | $f_4$ | 240 | 2160 | 2160 | 240 | N | ( ) | $L_5/L_4$ = N |
