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# Abstract polytope

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 Title: Abstract polytope Author: World Heritage Encyclopedia Language: English Subject: Collection: Publisher: World Heritage Encyclopedia Publication Date:

### Abstract polytope

In mathematics, an abstract polytope, informally speaking, is a structure which considers only the combinatorial properties of a traditional polytope, ignoring many of its other properties, such as angles, edge lengths, etc. No space, such as Euclidean space, is required to contain it. The abstract formulation embodies the combinatorial properties as a partially ordered set or poset.

The abstract definition allows some more general combinatorial structures than the traditional concept of a polytope, and allows 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.

## Contents

• Traditional versus abstract polytopes 1
• Introductory concepts 2
• Polytopes as posets 2.1
• Least and greatest faces 2.2
• A simple example 2.3
• The Hasse diagram 2.4
• Rank 2.5
• The line segment 2.6
• Flags 2.7
• Sections 2.8
• Vertex figures 2.8.1
• Connectedness 2.8.2
• Formal definition 3
• Notes 3.1
• The simplest polytopes 4
• Rank < 2 4.1
• Rank 2 4.2
• The digon 4.2.1
• Examples of higher rank 5
• Hosohedra and hosotopes 5.1
• Projective polytopes 5.2
• Duality 6
• Abstract regular polytopes 7
• An irregular example 7.1
• Realisations 8
• The amalgamation problem and universal polytopes 9
• The 11-cell and the 57-cell 9.1
• Local topology 9.2
• Exchange maps 10
• Incidence matrices 11
• History 12
• See also 13
• Notes 14
• References 15

## Traditional versus abstract polytopes

In Euclidean geometry, the six quadrilaterals above are all different. Yet they have something in common that is not shared by a triangle or a cube, for example.

The elegant, but geographically inaccurate, London Tube map provides all the relevant information to go from A to B. An even better example is an electrical circuit diagram or schematic; the final layout of wires and parts is often unrecognisable at first glance.

In each of these examples, the connections between elements are the same, regardless of the physical layout. The objects are said to be combinatorially equivalent. This equivalence is what is encapsulated in the concept of an abstract polytope. So, combinatorially, our six quadrilaterals are all the “same”. More rigorously, they are said to be isomorphic or “structure preserving”.

Properties, particularly measurable ones, of traditional polytopes such as angles, edge-lengths, skewness, and convexity have no meaning for an abstract polytope. Other traditional concepts may carry over, but not always identically. Care must be exercised, for what is true for traditional 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 so for abstract polytopes.

## Introductory concepts

To define an abstract polytope, a few preliminary concepts are needed.

Throughout this article, polytope means abstract polytope - unless stated otherwise. The term traditional will be used, somewhat loosely, to refer to what is generally understood by polytope, excluding our abstract polytopes. Some authors also use the terms classical or geometric.

### Polytopes as posets

The connections on a railway map or electrical circuit can be represented quite satisfactorily with just “dots and lines” - i.e. a graph. Polytopes, however, have a dimensional hierarchy. For example, the vertices, edges and faces of a cube have dimension 0, 1, and 2 respectively; the cube itself is 3-dimensional.

In our abstract theory, the concept of rank replaces that of dimension; we formally define it below.

We use the term face to refer to an element of any rank, e.g. vertices (rank 0) or edges (rank 1), and not just faces of rank 2. An element of rank k is called a k-face.

We shall define a polytope, then, 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.

When F < G, we say that F is a subface of G (or G has subface F).

We say F, G are incident if either F = G or F < G or G < F. This meaning differs from its usage in traditional geometry and some other areas of mathematics. For example, in the square abcd, edges ab and bc are not incident. But it is the same notion of incidence as used in Finite geometry.

### Least and greatest faces

Just as the concepts of zero and infinity are indispensable in mathematics, it turns out to be extremely useful and elegant, indeed essential, to insist that every polytope also has a least face, which is a subface of all the others, and a greatest face, of which all the others are subfaces.

In fact, a polytope can have just one face; in this case the least and greatest faces are one and the same.

The least and greatest faces are called improper faces; all others faces are proper faces.

The least face is called the null face, since it has no vertices (or any other faces) as subfaces. Since the least face is one level below the vertices or 0-faces, its rank is −1; we often denote it as F−1. If this seems strange at first, the feeling is quickly dispelled on seeing the elegant symmetry which this concept brings to our theory. (Historically, mathematicians resisted such commonplace concepts as negative, fractional, irrational and complex numbers - and even zero!)

### A simple example

As an example, we now create an abstract square, which has faces as 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 < is defined as set of pairs, which (for this example) would include

F−1<a, ... , F−1F−1bc

In this particular example, we could have written the edges W, X, Y and Z as ab, ad, bc, and cd respectively, and we often will use this vertex notation. But as we shall shortly see, such notation is not always appropriate.

We have called this a square rather than a quadrilateral (or tetragon) because, in our abstract world, there are no angles, and edges do not have lengths. All four edges are identical, and the "geometry" at each vertex is the same.

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 Hasse diagram

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

A polytope is often depicted informally by its graph, but the two cannot be equated. A graph has vertices and edges, but no other faces. Furthermore, for most polytopes, it is not possible to deduce all the other faces from the graph, and, in general, different polytopes can have the same graph.

A Hasse diagram, on the other hand, fully describes any poset - all the structure of the polytope is captured in the Hasse diagram. Isomorphic polytopes give rise to isomorphic Hasse diagrams, and vice versa.

### Rank

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

The rank of a poset P is the maximum rank n of any face, i.e. that of the greatest face (given that we require that there is one). Throughout this article, we shall always use n to denote the rank of the poset or polytope under discussion.

It follows that the least face, and no other, has rank −1; and that the greatest face has rank n. We often denote these as F−1 and Fn respectively.

The rank of a face or polytope usually corresponds to the dimension of its counterpart in traditional theory - but not always. For example, a face of rank 1 corresponds to an edge, which is 1-dimensional. But a skew polygon in traditional geometry is 3-dimensional, since it is not flat (planar); while its abstract equivalent, and indeed all abstract polygons, have rank 2.

For some ranks, we have names for their face-types, as in the table.

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

† Although traditionally "face" has meant a rank 2 face, we shall always write "2-face" to avoid ambiguity, reserving the term "face" to mean a face of any rank.

### The line segment

A line segment is a poset that has a least face, precisely two 0-faces, and a greatest face, for example {ø, a, b, ab}. It follows easily that the vertices a and b have rank 0, and that the greatest face ab, and therefore the poset, both have rank 1.

### 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.

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

We shall additionally require that, for a given polytope, all flags contain the same number of faces. Posets do not, in general, satisfy this requirement; the poset {øabbcabc} has 2 flags of unequal size, and is not therefore a polytope.

Clearly, given any two distinct faces F, G in a flag, either F < G or F > G.

### 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 particular, 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 terminology, a section is called a closed interval of the poset and denoted [F, H], but the concepts are identical).

P is thus a section of itself.

For example, in the prism abcxyz (see Figure) 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. The square abcd is a subset of the tetrahedron abcd, but is not a section of it.

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, abc/b, is {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. Every 1-section is a line segment.

An n-polytope is a polytope of rank n.

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