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A cube with a highlighted vertex, edge, and face.

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[edit | edit source]

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

Abstract polytopes[edit | edit source]

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.

Convex polytopes[edit | edit source]

Terminology[edit | edit source]

Elements have special names based on their rank. Common names are:

  • vertex (pl. vertices), rank 0
  • edge, rank 1
  • face, rank 2
  • cell, rank 3
  • facet, rank n – 1 where n is the rank of the entire polytope
  • ridge, rank n – 2

In general, an element of rank r may be called an r-face or an r-element.

Uncommon names developed in the amateur community include "teron," "peton," "exon," "zetton," "yotton," and "xennon" for ranks 4-9, and "spire" and "peak" for ranks n – 4 and n – 3 respectively.

A basic property of polytopes is their element counts in various ranks, such as the number of faces, edges, and vertices of a polyhedron. The number of proper elements in each rank may be aggregated into a sequence of positive integers known as the face vector[1] or f-vector.[2] For example, as the cube as 8 vertices, 12 edges, and 6 faces, its face vector is (8, 12, 6). The improper elements are conventionally excluded.

A set of elements that are all mutually incident on each other is a chain, which under standard definitions of polytopes, may not contain two elements of different ranks. A chain that has one element of every rank, including the improper elements, is known as a flag.

Elements as polytopes[edit | edit source]

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 a section may be built, which is then identified with the element.

See also[edit | edit source]