Nature

Nature is a concept created by Jonathan Bowers to differentiate polytopes that have co-spatial elements. There are three categories: Tame, Feral, and Wild that a d-dimensional (d being the rank of one of its elements) polytope can have, determined from two properties:


 * There are three (or more) d-elements that share a (d+1) space and a (d-1) element
 * 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 vertecies, but no face coplanar to said edges. A polyhedron with at least one of its faces being feral will automaticly also be 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 vertex figures with three colinear points, but there is no actual line there.

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