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 .
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.
The last 6 points form hexagonal holes on one of its 40 diameters.
Coxeter named it after Alexander Witting for being a Witting configuration in complex projective 3-space:[4] The Witting configuration is related to the finite space PG(3,22), consisting of 85 points, 357 lines, and 85 planes.
The 240 difference is accounted by 40 central hexagons in 421 whose edges are not included in 3{3}3{3}3{3}3.
[6] 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.