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 10-polytopes with reflective symmetry can be generated by these three Coxeter groups, represented by permutations of rings of the Coxeter-Dynkin diagrams: Selected regular and uniform 10-polytopes from each family include: The A10 family has symmetry of order 39,916,800 (11 factorial).
t0{3,3,3,3,3,3,3,3,3}10-simplex (ux) t1{3,3,3,3,3,3,3,3,3}Rectified 10-simplex (ru) t2{3,3,3,3,3,3,3,3,3}Birectified 10-simplex (bru) t3{3,3,3,3,3,3,3,3,3}Trirectified 10-simplex (tru) t4{3,3,3,3,3,3,3,3,3}Quadrirectified 10-simplex (teru) t0,1{3,3,3,3,3,3,3,3,3}Truncated 10-simplex (tu) t0,2{3,3,3,3,3,3,3,3,3}Cantellated 10-simplex t1,2{3,3,3,3,3,3,3,3,3}Bitruncated 10-simplex t0,3{3,3,3,3,3,3,3,3,3}Runcinated 10-simplex t1,3{3,3,3,3,3,3,3,3,3}Bicantellated 10-simplex t2,3{3,3,3,3,3,3,3,3,3}Tritruncated 10-simplex t0,4{3,3,3,3,3,3,3,3,3}Stericated 10-simplex t1,4{3,3,3,3,3,3,3,3,3}Biruncinated 10-simplex t2,4{3,3,3,3,3,3,3,3,3}Tricantellated 10-simplex t3,4{3,3,3,3,3,3,3,3,3}Quadritruncated 10-simplex t0,5{3,3,3,3,3,3,3,3,3}Pentellated 10-simplex t1,5{3,3,3,3,3,3,3,3,3}Bistericated 10-simplex t2,5{3,3,3,3,3,3,3,3,3}Triruncinated 10-simplex t3,5{3,3,3,3,3,3,3,3,3}Quadricantellated 10-simplex t4,5{3,3,3,3,3,3,3,3,3}Quintitruncated 10-simplex t0,6{3,3,3,3,3,3,3,3,3}Hexicated 10-simplex t1,6{3,3,3,3,3,3,3,3,3}Bipentellated 10-simplex t2,6{3,3,3,3,3,3,3,3,3}Tristericated 10-simplex t3,6{3,3,3,3,3,3,3,3,3}Quadriruncinated 10-simplex t0,7{3,3,3,3,3,3,3,3,3}Heptellated 10-simplex t1,7{3,3,3,3,3,3,3,3,3}Bihexicated 10-simplex t2,7{3,3,3,3,3,3,3,3,3}Tripentellated 10-simplex t0,8{3,3,3,3,3,3,3,3,3}Octellated 10-simplex t1,8{3,3,3,3,3,3,3,3,3}Biheptellated 10-simplex t0,9{3,3,3,3,3,3,3,3,3}Ennecated 10-simplex There are 1023 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings.
This family has 3×256−1=767 Wythoffian uniform polytopes, generated by marking one or more nodes of the D10 Coxeter-Dynkin diagram.