Seven cubes and ten 24-cells (Q1380771)

The two infinite sequences of regular polytopes such that each polytope is the vertex figure of the next one in the sequence are the regular simplices and the regular crosspolytopes in dimensions \(2,3,4,\dots\). They were discovered by Schlaefli. A similar sequence containing the uniform polytopes \(2_{21}, 3_{21}, 4_{21}, 5_{21}\) was discovered by Gosset. In this sequence the 45 diagonal triangles of \(2_{21}\) are the vertex figures of 45 cubes \(\{4,3\}\) inscribed in \(3_{21}\), the 315 cubes of \(3_{21}\) are the vertex figures of 315 24-cells \(\{3,4,3\}\) incribed in \(4_{21}\), the 3150 24-cells of \(4_{21}\) are the vertex figures of 3150 \(\{3,3,4,3\}\)'s incribed in \(5_{21}\). The author analyses the relationships between the polytopes of this sequence in detail and, as the vertices of \(\{3,3,4,3\}\) constitute the \(\widetilde{D}_4\) lattice and the vertices of \(5_{21}\) constitute the \(\widetilde{E}_8\) lattice, he concludes that each point of the \(\widetilde{E}_8\) lattice belongs to 3150 incribed \(\widetilde{D}_4\) lattices of minimal size.