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	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.

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

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

(0, ±ωμ, -±ων, ±ωλ) (-±ωμ, 0, ±ων, ±ωλ) (±ωμ, -±ων, 0, ±ωλ) (-±ωλ, -±ωμ, -±ων, 0)

(±ωλ√3, 0, 0, 0) (0, ±ωλ√3, 0, 0) (0, 0, ±ωλ√3, 0) (0, 0, 0, ±ωλ√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. The 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(3,2²), consisting of 85 points, 357 lines, and 85 planes.

Related real polytope

Its 240 vertices are shared with the real 8-dimensional polytope 4₂₁, . Its 2160 3-edges are sometimes drawn as 6480 simple edges, slightly less than the 6720 edges of 4₂₁. The 240 difference is accounted by 40 central hexagons in 4₂₁ 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 5₂₁, .

References

• Coxeter, H. S. M. and Moser, W. O. J.; Generators and Relations for Discrete Groups (1965), esp pp 67–80.
• Coxeter, H. S. M.; Regular Complex Polytopes, Cambridge University Press, second edition (1991). pp. 132–5, 143, 146, 152.
• Coxeter, H. S. M. and Shephard, G.C.; Portraits of a family of complex polytopes, Leonardo Vol 25, No 3/4, (1992), pp 239–244 [1]