# Abstract polytope

In mathematics, an abstract polytope is an algebraic partially ordered set or poset which captures the combinatorial properties of a traditional polytope, but not any purely geometric properties such as angles, edge lengths, etc.

An ordinary geometric polytope is said to be a realization in some real N-dimensional space, typically Euclidean, of the corresponding abstract polytope.

The abstract definition allows some more general combinatorial structures than traditional definitions of a polytope, thus allowing many new objects that have no counterpart in traditional theory.

The term polytope is a generalisation of polygons and polyhedra into any number of dimensions.

## Introductory concepts

In Euclidean geometry, the six quadrilaterals illustrated are all different. Yet they have a common structure in the alternating chain of four vertices and four sides which gives them their name. They are said to be isomorphic or “structure preserving”.

This common structure may be represented in an underlying abstract polytope, a purely algebraic partially-ordered set which captures the pattern of connections or incidences between the various structural elements. The measurable properties of traditional polytopes such as angles, edge-lengths, skewness, straightness and convexity have no meaning for an abstract polytope.

What is true for traditional polytopes (also called classical or geometric polytopes) may not be so for abstract ones, and vice versa. For example, a traditional polytope is regular if all its facets and vertex figures are regular, but this is not necessarily so for an abstract polytope.

#### Realisations

A traditional geometric polytope is said to be a realisation of the associated abstract polytope. A realisation is a mapping or injection of the abstract object into a real space, typically Euclidean, to construct a traditional polytope as a real geometric figure.

The six quadrilaterals shown are all distinct realisations of the abstract quadrilateral, each with different geometric properties. Some of them do not conform to traditional definitions of a quadrilateral and are said to be unfaithful realisations. A conventional polytope is a faithful realisation.

### Faces, ranks and ordering

In an abstract polytope, each structural element - vertex, edge, cell, etc. is associated with a corresponding member or element of the set. The term face often refers to any such element e.g. a vertex (0-face), edge (1-face) or a general k-face, and not just a polygonal 2-face.

The faces are ranked according to their associated real dimension: vertices have rank = 0, edges rank = 1 and so on.

Incident faces of different rank, for example a vertex F of an edge G, are ordered by the relation F < G. F is said to be a subface of G, or G has subface F.

F, G are said to be incident if either F = G or F < G or G < F. This usage of "incidence" also occurs in Finite geometry, although it differs from traditional geometry and some other areas of mathematics. For example in the square abcd, edges ab and bc are not abstractly incident (although they are both incident with vertex b).[citation needed]

A polytope is then defined as a set of faces P with an order relation <, and which satisfies certain additional axioms. Formally, P (with <) will be a (strict) partially ordered set, or poset.

### Least and greatest faces

Just as the number zero is necessary in mathematics, so also every set has the empty set ∅ as a subset. In an abstract polytope ∅ is by convention identified as the least or null face and is a subface of all the others. Since the least face is one level below the vertices or 0-faces, its rank is −1 and it may be denoted as F−1. Thus F−1 ≡ ∅ and the abstract polytope also contains the empty set as an element. It is not usually realized.

There is also a single face of which all the others are subfaces. This is called the greatest face. In an n-dimensional polytope, the greatest face has rank = n and may be denoted as Fn. It is sometimes realized as the interior of the geometric figure.

These least and greatest faces are sometimes called improper faces, with all others being proper faces.

### A simple example

The faces of the abstract quadrilateral or square are shown in the table below:

Face type Rank (k) Count k-faces
Least −1 1 F−1
Vertex 0 4 a, b, c, d
Edge 1 4 W, X, Y, Z
Greatest 2 1 G

The relation < comprises a set of pairs, which here include

F−1<a, ... , F−1<X, ... , F−1<G, ... , b<Y, ... , c<G, ... , Z<G.

Order relations are transitive, i.e. F < G and G < H implies that F < H. Therefore, to specify the hierarchy of faces, it is not necessary to give every case of F < H, only the pairs where one is the successor of the other, i.e. where F < H and no G satisfies F < G < H.

The edges W, X, Y and Z are sometimes written as ab, ad, bc, and cd respectively, but such notation is not always appropriate.

All four edges are structurally similar and the same is true of the vertices. The figure therefore has the symmetries of a square and is usually referred to as the square.

### The Hasse diagram

Smaller posets, and polytopes in particular, are often best visualised in a Hasse diagram, as shown. By convention, faces of equal rank are placed on the same vertical level. Each "line" between faces, say F, G, indicates an ordering relation < such that F < G where F is below G in the diagram.

The Hasse diagram defines the unique poset and therefore fully captures the structure of the polytope. Isomorphic polytopes give rise to isomorphic Hasse diagrams, and vice versa. The same is not generally true for the graph representation of polytopes.

### Rank

The rank of a face F is defined as (m − 2), where m is the maximum number of faces in any chain (F', F", ... , F) satisfying F' < F" < ... < F. F' is always the least face, F−1.

The rank of an abstract polytope P is the maximum rank n of any face. It is always the rank of the greatest face Fn.

The rank of a face or polytope usually corresponds to the dimension of its counterpart in traditional theory.

For some ranks, their face-types are named in the following table.

Rank -1 0 1 2 3 ... n - 2 n - 1 n
Face Type Least Vertex Edge Cell Subfacet or ridge Facet Greatest

† Traditionally "face" has meant a rank 2 face or 2-face. In abstract theory the term "face" denotes a face of any rank.

### Flags

A flag is a maximal chain of faces, i.e. a (totally) ordered set Ψ of faces, each a subface of the next (if any), and such that Ψ is not a subset of any larger chain. Given any two distinct faces F, G in a flag, either F < G or F > G.

For example, {ø, a, ab, abc} is a flag in the triangle abc.

For a given polytope, all flags contain the same number of faces. Other posets do not, in general, satisfy this requirement.

### Sections The graph (left) and Hasse Diagram of a triangular prism, showing a 1-section (red), and a 2-section (green).

Any subset P' of a poset P is a poset (with the same relation <, restricted to P').

In an abstract polytope, given any two faces F, H of P with FH, the set {G | FGH} is called a section of P, and denoted H/F. (In order theory, a section is called a closed interval of the poset and denoted [F, H].

For example, in the prism abcxyz (see diagram) the section xyz/ø (highlighted green) is the triangle

{ø, x, y, z, xy, xz, yz, xyz}.

A k-section is a section of rank k.

A polytope that is the subset of another polytope is not necessarily a section. In the diagram, the square abcd is a subset of the tetrahedron abcd, but is not a section of it.

P is thus a section of itself.

This concept of section does not have the same meaning as in traditional geometry.

#### Vertex figures

The vertex figure at a given vertex V is the (n−1)-section Fn/V, where Fn is the greatest face.

For example, in the triangle abc, the vertex figure at b is abc/b = {b, ab, bc, abc}, which is a line segment. The vertex figures of a cube are triangles.

#### Connectedness

A poset P is connected if P has rank ≤ 1, or, given any two proper faces F and G, there is a sequence of proper faces

H1, H2, ... ,Hk

such that F = H1, G = Hk, and each Hi, i < k, is incident with its successor.

The above condition ensures that a pair of disjoint triangles abc and xyz is not a (single) polytope.

A poset P is strongly connected if every section of P (including P itself) is connected.

With this additional requirement, two pyramids that share just a vertex are also excluded. However, two square pyramids, for example, can, be "glued" at their square faces - giving an octahedron. The "common face" is not then a face of the octahedron.

## Formal definition

An abstract polytope is a partially ordered set, whose elements we call faces, satisfying the 4 axioms:

1. It has a least face and a greatest face.
2. All flags contain the same number of faces.
3. It is strongly connected.
4. If the ranks of two faces a > b differ by 2, then there are exactly 2 faces that lie strictly between a and b.

An n-polytope is a polytope of rank n.