In mathematics, an abstract polytope is an algebraic partially ordered set which captures certain combinatorial properties of a traditional polytope without specifying purely geometric properties such as the position of vertices.
A geometric polytope is said to be a realization of an abstract polytope in some real n-dimensional space, typically Euclidean. This abstract definition allows more general combinatorial structures than traditional definitions of a polytope, thus allowing new objects that have no counterpart in traditional theory.
Principles
Traditional versus abstract polytopes
In Euclidean geometry, two shapes that are not similar can nonetheless share a common structure. For example, a square and a trapezoid both comprise an alternating chain of four vertices and four sides, which makes them quadrilaterals. 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.
Realizations A traditional polytope is said to be a realization of the associated abstract polytope. A realization 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 realizations 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 realizations. A conventional polytope is a faithful realization.
Faces, ranks and ordering In an abstract polytope, each structural element (vertex, edge, cell, etc.) is associated with a corresponding member of the set. The term face is used to refer 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 ranks, 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.
Two faces F and 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. It is understood in terms of the nodes and vertices in the graph of the Hasse diagram of the polytope, not the geometric drawing of the polytope. For example, in the square ABCD, edges AB and BC are not abstractly incident as can be seen in the Hasse diagram above (although they are both incident, in the geometric sense, with vertex B in the geometric drawing).
A polytope is then defined as a set of faces P with an order relation <. 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 included as a face. It is the least or null face and is a subface of all the others, because the empty set is a subset of all other sets. 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 usually not realized, though the lack of its realization could be interpreted as it being realized as the set containing no points, the empty set.
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)Countk-facesLeast −1 1F−1 Vertex 0 4a, b, c, dEdge 1 4W, X, Y, ZGreatest 2 1G
The relation < comprises a set of pairs.
Order relations are transitive, i.e. F < G and G < H implies F < H. Therefore, to specify the hierarchy of faces, it is not necessary to give every case of F < H, but 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 visualized in a Hasse diagram, as shown. By convention, faces of equal rank are placed on the same vertical level. Each connection between faces F and 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−1, F0, …, F) satisfying F−1 < F0 < … < F. Note that F−1 is always the least face.
The rank of an abstract polytope P is the maximum rank of any face. It is always the rank of the greatest face.
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 Face term −1 Least 0 Vertex 1 Edge 2 Traditionally face means a rank-2 face or 2-face. In abstract theory the term it denotes a face of any rank. 3 Cell ⋯ ⋯ n − 2 Subfacet or ridge n − 1 Facet n Greatest
Flags
In geometry, 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 and G in a flag, either F < G or F > G.
For example, {∅, 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
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 and H of P with F ≤ H, the set {G | F ≤ G ≤ H} 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/∅ (highlighted green) is the triangle
{∅, x, y, z, xy, xz, yz, xyz}.
A k-section is a section of rank k.
P is thus a section of itself.
This concept of section does not have the same meaning as in traditional geometry.
Facets The facet for a given k-face F is the (k − 1)-section F/∅.
For example, in the triangle abc, the facet at ab is ab/∅ = {∅, a, b, ab}, which is a line segment.
The distinction between F and F/∅ is not usually significant and the two are often treated as identical.
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 F1 and Fk, there is a sequence of proper faces
F1, F2, …,Fk
such that each Fi, i < k, is incident with its successor.
The above condition ensures that a pair of disjoint triangles abc and xyz is not a 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 can, for example, be glued at their square faces, giving an octahedron, but the square is not a face of the octahedron.
Formal definition An abstract polytope is a partially ordered set, whose elements we call faces, satisfying these axioms:
It has exactly one least face and exactly one greatest face. All flags contain the same number of faces. It is strongly connected. If the ranks of two faces a > b differ by exactly 2, then there are exactly 2 faces that lie strictly between a and b.
An n-polytope is a polytope of rank n. The abstract polytope associated with a real convex polytope is also referred to as its face lattice.
The simplest polytopes
Ranks −1 and 0 There is just one poset for rank −1 and 0. These are, respectively, the null face and the point. These are not always considered to be valid abstract polytopes.
Rank 1: the line segment
There is only one polytope of rank 1, which is the line segment. It has a least face, just two 0-faces and a greatest face, for example {∅, a, b, ab}. It follows that the vertices a and b have rank 0, and that the greatest face ab, and therefore the poset, both have rank 1.
Rank 2: polygons For each p, 3 ≤ p < ∞, we have (the abstract equivalent of) the traditional polygon with p vertices and p edges, or a p-gon. For p = 3, 4, 5, ... we have the triangle, square, pentagon, etc.
For p = 2, we have the digon, and p = ∞ we get the apeirogon.
A digon is a polygon with just 2 edges. Unlike any other polygon, both edges have the same two vertices. For this reason, it is degenerate in the Euclidean plane.
Faces are sometimes described using vertex notation, e.g. {∅, a, b, c, ab, ac, bc, abc} for the triangle abc. This method has the advantage of implying the ordering relation.
With the digon this vertex notation cannot be used. It is necessary to give the faces individual symbols and specify the subface pairs F < G.
Thus, a digon is defined as a set {∅, a, b, E’, E”, G} with the ordering relation given by
∅ < a, b and a, b < E’, E” and E’, E” < G
where E’ and E” are the two edges, and G the greatest face.
This need to identify each element of the polytope with a unique symbol applies to many other abstract polytopes and is therefore common practice.
A polytope can only be fully described using vertex notation if every face is incident with a unique set of vertices. A polytope having this property is said to be atomistic.
Examples of higher rank
The set of k-faces (−1 ≤ k ≤ n) of a traditional n-polytope form an abstract n-polytope.
The concept of an abstract polytope is more general and also includes: Apeirotopes or infinite polytopes, which include tessellations (tilings) Proper decompositions of unbounded manifolds such as the torus or real projective plane. Many other objects, such as the 11-cell and the 57-cell, that cannot be faithfully realized in Euclidean spaces.
Hosohedra and hosotopes
The digon is generalized by the hosohedron and higher dimensional hosotopes, which can all be realized as spherical polyhedra – they tessellate the sphere.
Projective polytopes