Quasi-Regular Polytopes of Full Rank

A polytope $${{ \mathsf {P}}}$$ in some euclidean space is called quasi-regular if each facet $${{ \mathsf {F}}}$$ of $${{ \mathsf {P}}}$$ is regular and the symmetry group $${\mathbf {G}}({{ \mathsf {F}}})$$ of $${{ \mathsf {F}}}$$ is a subgroup of the symmetry group $${\mathbf {G}}({{ \mathsf {P}}})$$ of $${{ \mathsf {P}}}$$ . Further, $${{ \mathsf {P}}}$$ is of full rank if its rank and dimension are the same. In this paper, the quasi-regular polytopes of full rank that are not regular are classified. Similarly, an apeirotope of full rank sits in a space of one fewer dimension; the discrete quasi-regular apeirotopes that are not regular are also classified here. One curiosity of the classification is the difference between even and odd dimensions, in that certain families are present in $${\mathbb {E}}^d$$ if d is even, but are absent if d is odd.