Uniform 7-polytope

This inadequacy of the Euler characteristic to reliably distinguish between different topologies in higher dimensions led to the discovery of the more sophisticated Betti numbers.

[1] Similarly, the notion of orientability of a polyhedron is insufficient to characterise the surface twistings of toroidal polytopes, and this led to the use of torsion coefficients.

[1] Uniform 7-polytopes with reflective symmetry can be generated by these four Coxeter groups, represented by permutations of rings of the Coxeter-Dynkin diagrams: The A7 family has symmetry of order 40320 (8 factorial).

This family has 3×32−1=95 Wythoffian uniform polytopes, generated by marking one or more nodes of the D7 Coxeter-Dynkin diagram.

Each combination of active mirrors generates a unique uniform polytope.

Some families have two regular constructors and thus may be named in two equally valid ways.

The prismatic forms and bifurcating graphs can use the same truncation indexing notation, but require an explicit numbering system on the nodes for clarity.

Coxeter-Dynkin diagram correspondences between families and higher symmetry within diagrams. Nodes of the same color in each row represent identical mirrors. Black nodes are not active in the correspondence.