Ever since I've come across it, I have been puzzled by the sequence $(a_n)_{n\in\mathbb{N}_0}=(1,1,\infty,5,6,3,3,3,\cdots)$, describing the number of regular $n$-topes in $n$-dimensions, where $a(k)=3$ for $k\geq 5$.
E.g; $a(3)=5$, we have our familiar friends, the 5 platonic solids, etc.
I am curious, is there generalization of this for fractional dimensions? I understand that this is a quite is abstract question.
I.e; what is the most canonical (or natural) function $f(n):\mathbb{R}\to\mathbb{R}$ such that $f(n)=a_n$ when $a\in\mathbb{N}_0$? Of course, there are an uncountably infinite number of functions which would fulfill this, but perhaps there is one that encapsulates what it means to be a regular $n$-tope.
Perhaps a function which would look something like this?
