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 in this wiki 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 −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 Ø and is identified with the nullitope. The maximal element is often identified by the polytope itself.

Definition

Different notions of polytopes have different notions of elements. Here, we focus on the two main ones.

Abstract polytopes

Elements of an abstract polytope are precisely the elements in a set-theoretic sense. To be more precise, recall that abstract polytopes are defined as partially ordered sets. The elements of this set then form the elements of the polytope. As such, the elements of an abstract polytope can effectively be anything.

Properties like rank or subelements are thus not intrinsic, and rely entirely on the incidence structure.

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.

Rank Name
0 Vertex
1 Edge
2 Face
3 Cell
4 Teron
5 Peton
6 Exon
7 Zetton
8 Yotton
9 Xennon
...
n–4 Spire
n–3 Peak
n–2 Ridge
n–1 Facet

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 ${\displaystyle P}$ a section ${\displaystyle P/\varnothing }$ may be built, which is then identified with the element.