Regular polytopes in ordinary space

The three aims of this paper are to obtain the proof by Dress of the completeness of the enumeration of the Grünbaum–Dress polyhedra (that is, the regular apeirohedra, or apeirotopes of rank 3) in ordinary space 𝔼³ in a quicker and more perspicuous way, to give presentations of those of their symmetry groups which are affinely irreducible, and to describe all the discrete regular apeirotopes of rank 4 in 𝔼³. The paper gives a complete classification of the discrete regular polytopes in ordinary space.