The focus now moves to the regular polytopes and apeirotopes of nearly full rank; this chapter treats those that occur in every dimension. The role played by blended polytopes is discussed first. Next considered is the part played by twisting certain diagrams. There are four infinite families of finite regular polytopes, three related (as one would expect) to the simplices, staurotopes and cubes and one related to half-cubes. Surprisingly, the cubic tiling leads to many families, while yet other families are connected with certain non-string reflexion groups. At this stage, the classification is incomplete, since it relies on that in smaller dimensions. In particular, the 4-dimensional cases are needed to tackle the ‘gateway’ dimension five.