Nature

Nature is a concept created by Jonathan Bowers to differentiate polytopes that have co-spatial elements. An n-polytope's nature is either tame, feral, or wild, depending on two properties:

• There are three or more d-elements that share both a (d+1)-space and a (d-1)-element for some d
• Some of those d-elements also share a (d+1)-element

The category can be determined from looking at the (d-3) element figures of a d-dimensional polytope.

For instance, a feral polyhedron will have at least three coplanar edges meeting at at least one of its vertices, but no face coplanar to said edges. A polyhedron with at least one feral face is automatically feral (or wild), as d could be the rank of any of its elements, not just the rank one less than its own.

Tame

A polytope is said to be tame if it satisfies neither of the two properties.

For example, a polyhedron where no three edges are in the same plane and meet at a vertex.

A tame polyhedron will have no vertex figures with 3 collinear points. All convex polytopes are tame, and every uniform polyhedron is tame.

Feral

A polytope is said to be feral if it satisfies the first case, but not the second. In other words, there are coplanar elements, but they have no (d+1) elements between them.

For example, a feral polyhedron has 3 coplanar edges meeting at a vertex, but no two of them are in the same face. And its vertex figure has three collinear points, but there is no actual line there.

Wild

This triangular toroprism is wild. An edge of its triangular face lies in the same plane as its square face, and shares a vertex with that square face too.

Any polytope that satisfies both properties is said to be wild.

For example, a polyhedron with three coplanar edges meeting at a vertex, with two of them belonging to the same face. (So a face and an edge are in the same plane and share a vertex, but that edge isn't part of that face.)

Wild polyhedra have vertex figures with three collinear points and an edge connecting two of them.