# Polytope (EntityClass, 7)

A polytope is a shape with no curved elements. All elements of a polytope are lower-dimensional polytopes. Polytopes are grouped by their dimension into the polygons, polyhedra, polychora, polytera, polypeta, etc.

## Uniform polytope

A uniform polytope is a polytope which is vertex-transitive and whose facets are lower-dimensional uniform polytopes.

The uniform polytopes in three and four dimensions can be indexed by their Bowers acronym or Kana mnemonic. See also Table of regular polytopes by Tamfang name and List of uniform polychora.

### Regular polytope

Main article: Regular polytope

A regular polytope is a uniform polytope whose elements within each dimension are all congruent (that is, all edges are congruent to all other edges, all faces are congruent to all other faces, and so forth).

## Convex regular-faced polytope

A convex regular-faced polytope (abbreviated as CRF polytope) is an n-polytope which is strictly convex (i.e. for any two points in the shape, the line segment between them is also entirely inside the shape, and there are no two n-1-cells which are in the same n-1-plane) and all its faces are regular.

Excluding the infinite sets of prisms and antiprisms, there are precisely 110 CRF polyhedra. These (exhaustively) include the five Platonic solids, the 13 Archimedean solids and the 92 Johnson solids.

It is currently unknown how many CRF polychora there are. The ongoing CRF polychora discovery project proposes at least 1,912 outside of infinite families. These are:

• all 64 convex uniform polychora,
• 92 prisms of the Johnson solids,
• 50 pyramidal forms (4 x 13 polyhedra, minus 2 uniform results),
• 34 cupolae of the regular polyhedra,
• 9 bicupolic rings,
• 6 ursachora,
• 4 convex non-uniform scaliform figures,
• 20 rotundae, and
• 1,633 duoprisms augmented with pyramids in various arrangements.

The infinite families found so far are the obvious m,n-duoprisms (for mn ≥ 3), the n-gonal antiprismatic prisms, and an interesting family of biantiprismatic rings - a certain subset of ringed forms discovered by Mrrl.

## Kanitope

The term kanitope is convenient shorthand for "convex non-prismatic Wythoffian uniform polytope". As such, all kanitopes are CRF polytopes. There are tables of 3D and 4D kanitopes available.

## Zonotope

Main article: Zonotope

A zonohedron is a convex polyhedron whose faces all have point symmetry. A zonotope is either a zonohedron or a convex n-polytope (with n ≥ 4) whose facets are all zonotopes themselves. Zonotopes can be constructed as the Minkowski sum of a set of vectors.