Witting Polytope

In 4-dimensional complex geometry, the Witting polytope is a regular complex polytope, named as: ₃₃₃₃, and Coxeter diagram . It has 240 vertices, 2160 ₃ edges, 2160 ₃₃ faces, and 240 ₃₃₃ 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 sub>3 sub>3 sub>3 or , order 155,520. It has 240 copies of , order 648 at each cell.

Structure

The configuration matrix is:
\left begin240&27&72&27\\3&2160&8&8\\8&8&2160&3\\27&72&27&240\end\right /math>

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.

Coordinates

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

where $\omega = \tfrac, \lambda, \nu, \mu = 0,1,2$.

The last 6 points form hexagonal ''holes'' on one of its 40 diameters. There are 40 hyperplanes contain central ₃₃₂, figures, with 72 vertices.

Witting configuration

Coxeter named it after Alexander Witting for being a ''Witting configuration'' in complex projective 3-space:
:\left begin 40&12&12\\2&240&2\\12&12&40 \end\right /math> or \left begin 40&9&12\\4&90&4\\12&9&40 \end\right /math>

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 ₃₃₃₃.

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, .

Its f-vector element counts are in proportion: 1, 80, 270, 80, 1.Coxeter Regular Convex Polytopes, 12.5 The Witting polytope The configuration matrix for the honeycomb is:

Notes

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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 239–24


Polytopes
Complex analysis