Polytope

A polytope is an object that generalizes the intuitive notions of "flat" shapes like polygons and polyhedra to any amount of dimensions. An n-dimensional polytope consists of various (n–1)-dimensional facets. Each of these facets has itself various (n–2)-dimensional facets called ridges, so that at each of the polytope's ridges, two facets meet. This generalizes the condition that two edges must meet at a polygon's vertex, and that two faces must meet at a polyhedron's edge.

The term "polytope" can have many different and often contradictory meanings, depending on the context.

Types of polytopes
If a polytope is isogonal and has (geometrically) one type of edge, and all of its elements are realizable as such, then it is called a uniform polytope. If the requirement of uniform elements is removed, allowing for Johnson solids as cells, then it is called a scaliform polytope, and if the requirement of vertex-transitivity is removed (or rather negated), then it is called a convex regular-faced polytope.