The smallest regular polytopes of given rank

An abstract polytope is called regular   if its automorphism group has a single orbit on flags (maximal chains). In this paper, the regular n$n$-polytopes with the smallest number of flags are found, for every rank n>1$n>1$. With a few small exceptions, the smallest regular n$n$-polytopes come from a family of ‘tight’ polytopes with 2⋅4ⁿ⁻¹$2\cdot 4^{n-1}$ flags, one for each n$n$, with Schläfli symbol {4∣4∣?∣4}$\{4\mid 4\mid ?\mid 4\}$. Also with few exceptions, these have both the smallest number of elements, and the smallest number of edges in their Hasse diagram.