Seven Cubes and Ten 24-Cells

Abstract.

The 45 diagonal triangles of the six-dimensional polytope 2 ₂₁ (representing 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 altogether 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 inscribed in the eight-dimensional polytope 4 ₂₁ . They are the vertex figures of 3150 four-dimensional honeycombs { 3,3,4,3 } inscribed in the eight-dimensional honeycomb 5 ₂₁ . In other words, each point of the $\tilde{E}$ ₈ lattice belongs to 3150 inscribed $\tilde{D}$ ₄ lattices of minimal size.

Analogously, in unitary 4 -space there are 3150 regular complex polygons 3 { 4 } 3 inscribed in the Witting polytope 3 { 3 } 3 { 3 } 3 { 3 } 3 .