Element

An element of a polytope is any of its fundamental building blocks. These include vertices, edges, faces, and so on. Elements are categorized by their rank, so that a vertex is a rank 0 element, an edge is a rank 1 element, and so on.

In formal mathematical writing, the term face is almost exclusively used instead, though here it is used with a different meaning (an element of rank 2 specifically).

Though they're often ignored and are more of a theoretical construct, every polytope of rank n has an element of rank &minus;1 called the minimal, null, or least element, and an element of rank n called the maximal or greatest element. These two elements together are called the improper elements. The minimal element is often denoted by $$\varnothing$$ and is identified with the nullitope. The maximal element is often identified by the polytope itself.

Naming
Elements have special names based on their rank. Some of these are in professional use, while others have been coined within the amateur community and don't have wider usage.

Definition
The precise definition of an element varies depending on the precise notion of a polytope being worked with. When dealing with abstract polytopes, elements are precisely the elements (in the set-theoretic sense) of the polytope. For convex polytopes, the definition is more elaborate.

Elements as polytopes
Elements are often identified with polytopes. For instance, an edge in a polytope might be thought of as a dyad. When dealing with convex polytopes, this identification is literal. When dealing with abstract polytopes however, these are distinct notions. Rather, from any element $$P$$ a section $$P/\varnothing$$ may be built, which is then identified with the element.

Related notions
A pseudoelement of a polytope P is a polytope whose lower-dimensional elements are all elements of P, but which is not an element of P. For example, the octahedron has three pseudo-squares, while the tetrahemihexahedron (which has the same edges) has the same squares as faces. However, four of the octahedron's faces are pseudo-faces of the tetrahemihexahedron.