Regular polyhedra (polytopes) become boring in 5 or more dimensions. However, things can remain interesting if we expand the list by decreasing the constraints. (Is the reason they become boring because the number of constraints for regularity increases with dimension?)
One way to keep things interesting is uniform polytopes, perhaps with the constraint of convexity.
Are uniform polytopes interesting in 5 or more dimensions? Maybe the threshold is 9 dimensions or more, after the excitement caused by the E6, E7, and E8 simple groups.
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