August 28, 2010

Infinity, 5 or 3?

The burden of polyhedralityAfter dinner discussion yesterday: which of the terms in this series stands out?

1, infinity, 5, 6, 3, 3, 3, 3, ...

An extra challenge (below the fold) is of course to tell what generates it.

The series corresponds to the number of regular convex polytopes in different dimensions. In one dimension there is just one (a line segment), in two there are an infinite number of regular polygons, in three there are five platonic solids, in four there are six, and then there are just three (the n-simplex, the n-cube and the n-orthoplex).

In our discussion several views came forth: Infinity is the unusual term, since it is not even a number. 5 and 6 are unusual, since they are non-trivial numbers - 1 and infinity are just a "transient" at the start (it usually takes more bits of information to specify a number "somewhere in the middle" than 1 or infinity). 3 is the unusual number, because it is repeated indefinitely - everything else is just a transient (my position).

If we allow nonconvex regular polytopes, we get the sequence
1, infinity, 9, 16, 3, 3, 3, 3, ...
There are no nonconvex regular polytopes in five dimensions or more. The nonconvex polytope sequence seems to support me - three again!

Of course, the whole question is ill-defined and somewhat pointless, but all the best Oxford postprandial debates are like that. I love starting arguments about
what the river passing the town "really" is named (and of course what kind of object a river name signifies - are rivers physical things, locations, processes, social constructs or something else?)

Overall, it seems to me that what is going on here is that low-dimensional spaces have pretty trivial symmetries only allowing zero, one or an infinity of some class of objects. 3 and 4 dimensions are unusual in that the extra degrees of freedom makes a non-trivial number of objects possible yet do not allow the infinite sets of 2 dimensions - sometimes more is less! In the higher dimensions things tend to smooth out, but I find it odd that the symmetries of space do not end up allowing an infinite number, zero or a single object in this case.

Posted by Anders3 at August 28, 2010 08:03 PM