7 CUBES AND 10 24-CELLS

The 45 diagonal triangles of the six-dimensional polytope 2(21) (repre senting the 45 tritangent planes of the cubic surface) are the vertex figures of 45 cubes {4, 3} inscribed in the seven-dimensional polytope 3(21), which has 56 vertices. Since 45 x 56 = 8 x 315, there are alto gether 315 such cubes. They are the vertex figures of 315 specimens of the four-dimensional polytope {3, 4, 3}, which has 24 vertices. Since 315 x 240 = 24 x 3150, there are altogether 3150 {3, 4, 3}'s inscribe d in the eight-dimensional polytope 4(21). They are the vertex figures of 3150 four-dimensional honeycombs {3, 3, 4, 3} inscribed in the eig ht-dimensional honeycomb 5(21) In other words, each point of the (E) o ver tilde(8) lattice belongs to 3150 inscribed (D) over tilde(4) latti ces of minimal size. Analogously, in unitary 4-space there are 3150 re gular complex polygons 3{4}3 inscribed in the Witting polytope 3{3}3{3 }3{3}3.