Description of a highly symmetric polytope observed in Thomson’s problem of charges on a hypersphere

In a recent paper, Altschuler and Pérez-Garrido [Phys. Rev. E 76, 016705 (2007)] have presented a four-dimensional polytope with 80 vertices. We demonstrate how this polytope can be derived from the regular four-dimensional 600-cell with 120 vertices if two orthogonal positive disclinations are created. Some related polytopes are also described.

Figure 1

Transformation of a icosahedron (left) into a gyro-elongated square antiprism (gesa, right). The shaded wedge is cut out and the left and right lips are glued together at the dashed line in the gesa.

Figure 2

Cut through the 600-cell in toroidal orientation. The numbers indicated the number of vertices on the chains and tori.