Regular 9-polytopes can be represented by the Schläfli symbol, with w 8-polytope facets around each peak. There are exactly three such convexregular 9-polytopes:
The topology of any given 9-polytope is defined by its Betti numbers and torsion coefficients. The value of the Euler characteristic used to characterise polyhedra does not generalize usefully to higher dimensions, whatever their underlying topology. 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. 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.
Uniform 9-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 9-polytopes from each family include:
* 383 uniform 9-polytope as permutations of rings in the group diagram, including:
*# - 9-demicube or demienneract, 161 - ; also as h.
*# - 9-orthoplex, 611 -
The A9 family
The A9 family has symmetry of order 3628800. There are 256+16-1=271 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings. These are all enumerated below. Bowers-style acronym names are given in parentheses for cross-referencing.
The B9 family
There are 511 forms based on all permutations of the Coxeter-Dynkin diagrams with one or more rings. Eleven cases are shown below: Ninerectified forms and 2 truncations. Bowers-style acronym names are given in parentheses for cross-referencing. Bowers-style acronym names are given in parentheses for cross-referencing.
The D9 family
The D9 family has symmetry of order 92,897,280. This family has 3×128−1=383 Wythoffian uniform polytopes, generated by marking one or more nodes of the D9 Coxeter-Dynkin diagram. Of these, 255 are repeated from the B9 family and 128 are unique to this family, with the eight 1 or 2 ringed forms listed below. Bowers-style acronym names are given in parentheses for cross-referencing.
Regular and uniform honeycombs
There are five fundamental affine Coxeter groups that generate regular and uniform tessellations in 8-space: Regular and uniform tessellations include:
45 uniquely ringed forms
*8-simplex honeycomb:
271 uniquely ringed forms
* Regular 8-cube honeycomb:,
: 383 uniquely ringed forms, 255 shared with, 128 new
* 8-demicube honeycomb: h or, or
, : 155 unique ring permutations, and 15 are new, the first,, Coxeter called a quarter 8-cubic honeycomb, representing as q, or qδ9.
There are no compact hyperbolic Coxeter groups of rank 9, groups that can generate honeycombs with all finite facets, and a finite vertex figure. However, there are 4 noncompact hyperbolic Coxeter groups of rank 9, each generating uniform honeycombs in 8-space as permutations of rings of the Coxeter diagrams.
