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Abstract polytope
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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 concept of space, such as Euclidean space, is required to contain it.
The term polytope is a generalisation of polygons and polyhedra into any number of dimensions.
A formal definition of an abstract polytope is given below.
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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 concept of space, such as Euclidean space, is required to contain it.
The term polytope is a generalisation of polygons and polyhedra into any number of dimensions.
A formal definition of an abstract polytope is given below. In fact, this definition is more general than the traditional concept of a polytope, and allows many new objects that have no counterpart in traditional theory.
Throughout this article, polytope means abstract polytope, unless stated otherwise. The term traditional will be used, without being formally defined here, to refer to what is generally understood by polytope, excluding abstract polytopes. Some authors also use the terms classical or geometric.
Traditional versus abstract polytopes
In Euclidean geometry, the five 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 five quadrilaterals are all the “same”. More rigorously, they are said to be isomorphic or “structure preserving”.
Properties 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.
Polytopes as posets
A graph, i.e. just “dots and lines”, suffices to represent railway maps or electrical circuits. Polytopes, however, have a dimensional hierarchy; for example, the vertices, edges and faces of a cube have dimension 0, 1, and 2 respectively. To represent such a structure, an order structure is needed.
The term face refers to elements - vertices, edges, faces etc. - of any dimension, not just 2-faces. A k-dimensional element is called a k-face.
Given two faces F and G, the notation F < G (or G > F) denotes that F is (strictly) contained in G. For example,
Vertex a < Edge ab < 2-face abcd.
We shall also require that every polytope has a least face, which is contained in all others, and a greatest face, which contains all the others.
The set of faces of a polytope, with this containment relation <, is a (strict)partially ordered set, or poset. An abstact polytope, then, will be defined as a poset satisying certain axioms, which are given shortly. Posets are often best visualised graphically.
Graphical representation
It is often useful to have a diagram of a polytope, and for the simplest polytopes, the (undirected) graph or 1-skeleton may suffice. However, this graph shows only vertices and edges; in general, it is not possible to deduce all the faces of higher rank, and indeed, different polytopes can have the same graph.
To show all the faces of a polytope and their relationships, another (directed) graph, known as a Hasse diagram is used. This fully describes the poset, and therefore defines the polytope. By convention, for clarity, all faces of the same rank are placed on the same vertical level (though given that the graph is directed this isn't formally necessary). The lowest level has rank -1, the highest rank n for an n-polytope.
The above example may be tabulated:
| Element | Rank (k) | Cardinality | k-faces |
|---|
| Null | −1 | 1 | ø (empty set) | | Vertex | 0 | 5 | A,B,C,D,E | | Edge | 1 | 8 | f,g,h,j,k,l,m,n | | Face | 2 | 5 | P,Q,R,S,T | | Body | 3 | 1 | ? (capital 'pi') |
Dimension or 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) where F' < F" < ... < F.
The rank of a poset P is the maximum rank of any face, i.e. that of the
greatest face (given that we require that there is one).
The rank of an abstract face usually corresponds to the dimension of the equivalent geometrical element. For example a face of rank 1 corresponds to a (1-dimensional) edge.
It follows that the least face, and no other, has dimension −1. This is called the null face, and is the empty set, ø.
Flags
A flag is a maximal chain of faces, i.e. comprising one face from each rank such that each face is connected to the faces above and/or below it. For example, is a flag in the triangle ABC having edges d,e,f and body ?.
The edge or line segment
A line segment or edge is a poset that has a least face, precisely two 0-faces, and a greatest face, for example . It follows easily that the vertices a and b have dimension 0, and that the greatest face ab, and therefore the poset, both have dimension 1. This lends credibility to the definition of rank/dimension.
Sections
Clearly, any subset of a poset P is a poset (with the same relation <).
In particular, given any two faces F, H of P with F = H, the set 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).
A k-section is a section of rank k.
For example, in a triangular prism abcxyz with a 2-face F' = abc and 0-face (vertex) V = b, the section F/V is , which is the poset of an edge.
This concept of section does not have the same meaning as in traditional geometry.
Formal definition
An abstract n-polytope is a partially ordered set satisfying:
- It has a least face and a greatest face.
- Every flag has exactly n + 2 faces.
- It is strongly connected. That is, any flag can be changed into any other by changing just one face at a time, and the same is true for any two flags of every section of the polytope.
- Every 1-section is an edge.
In the case of the null polytope, which has rank −1, the least and greatest faces are the same single element.
The reason for requiring polytopes to be strongly connected is to exclude objects such as a pair of disjoint triangles, or two cubes that share only a vertex.
The 4th axiom of the definition is known as the “diamond property”, since the Hasse Diagram of an edge is diamond-shaped.
It can be shown that every section is a polytope, and clearly
Rank(G/F) = Rank(G) − Rank(F) − 1.
Examples
As stated above, this concept of Abstract Polytope is very general, and includes:
- Degenerate polytopes, such as the digon
- Apeirotopes, i.e. infinite polytopes or tesselations (tilings)
- Decompositions of other manifolds such as the torus or real projective plane
- Many other objects, such as the 11-cell and the 57-cell, that don't fit well into "normal" geometric spaces.
There is just one polytope in each of the dimensions -1, 0 and 1, which are, respectively, the null polytope, the point, and the line segment. The line segment is usually an edge [section] of a higher polytope, so it often loosely referred to as an edge even in isolation.
All polygons, including the digon and the apeirogon, are 2-polytopes.
In general, the set of j-faces (-1 = j = n) of a traditional n-polytope form an abstract n-polytope.
Notable abstract polytopes
The simplest non-traditional abstract polytope is a digon — both its edges have the same two vertices.
Four examples of non-traditional abstract polyhedra are the Hemicube (shown), Hemi-octahedron, Hemi-dodecahedron, and the Hemi-icosahedron.
The 11-cell, discovered independently by H. S. M. Coxeter and Branko Grόnbaum, is an abstract 4-polytope. Its facets are hemi-icosahedra. Since its facets are, topologically, projective planes instead of spheres, the 11-cell is not a tessellation of any manifold in the usual sense. Instead, the 11-cell is a locally projective polytope. The 11-cell is not only beautiful in the mathematical sense, it is also historically important as one of the first non-traditional abstract polytopes discovered. It is self-dual and universal: it is the only polytope with hemi-icosahedral facets and hemi-dodecahedral vertex figures.
The 57-cell is also self-dual, with hemi-dodecahedral facets. It was discovered by H. S. M. Coxeter shortly after the discovery of the 11-cell. Like the 11-cell, it is also universal, being the only polytope with hemi-dodecahedral facets and hemi-icosahedral vertex figures. On the other hand, there are many other polytopes with hemi-dodecahedral facets and Schlδfli type . The universal polytope with hemi-dodecahedral facets and icosahedral (not hemi-icosahedral) vertex figures is finite, but very large, with 10006920 facets and half as many vertices.
Other representations
A given polytope can be also be represented definitively in other ways.
Set representation
The most informal way to specify a polytope is by listing all its faces in terms of the vertices it contains. Thus the square abcd would be
.
However, not all polytopes can be defined by this method - only those whose faces all have unique vertex sets. The hemicube, for example, can't be - its 2-faces and the 3-face all have the same vertex set.
Incidence matrices
Two given faces are said to be incident if one contains the other. This meaning differs from that in traditional geometry. For example, the edge ab of a cube is incident with the face abcd, but not with the edge ae.
A polytope can also be represented by tabulating its incidences. The following incidence matrix is that of a triangle:
| ø | a | b | c | ab | bc | ca | abc |
|---|
| ø | | | | | | | | |
|---|
| a | | | | | | | | |
|---|
| b | | | | | | | | |
|---|
| c | | | | | | | | |
|---|
| ab | | | | | | | | |
|---|
| bc | | | | | | | | |
|---|
| ca | | | | | | | | |
|---|
| abc | | | | | | | | |
|---|
The table shows a dot wherever a face is contained in another, or vice versa - so in fact, the table has redundant information; it would suffice to show only a dot when the row face = the column face.
Properties
Duality
Every abstract polytope has a dual twin polytope, in which the partial order is reversed: the Hasse diagram of the dual is that of the original turned upside-down. In an n-polytope,
each of the original k-faces maps to an (n − k − 1)-face in the dual. Thus, for example, the n-face maps to the (−1)-face. The dual of a dual is (isomorphic to) the original.
A polytope is self-dual if it is the same as, i.e. isomorphic to, its dual. Hence, the Hasse diagram of a self-dual polytope must be symmetrical about the horizontal axis half-way between the top and bottom. The square pyramid example above is seen to be self-dual.
Vertex figures
The vertex figure at a given vertex V is the section Fn/V, i. e., , where Fn is the maximal face. It is the dual of the facet which corresponds to V in the the dual polytope.
Regularity
The vertices of any abstract tetrahedron are all the “same”, in a sense made precise below; so too are its edges, and its faces. The opposite is true for the square pyramid. In a triangular prism, only the vertices are equivalent.
Formally, an abstract polytope is defined to be regular if its automorphism group acts transitively on the set of its flags. In particular, any two k-faces F, G of an n-polytope are "the same", i.e. that there is an automorphism which maps F to G. When an abstract polytope is regular, its automorphism group is isomorphic to a quotient of a Coxeter group.
This is a weaker condition than regularity for traditional polytopes, in that it refers to the (combinatorial) automorphism group, not the (geometric) symmetry group. For example, any abstract polygon is regular, since angles, edge-lengths, edge curvature, skewness etc. don't exist for abstract polytopes.
There are several other weaker concepts, some not yet fully standardised, such as semi-regular, quasi-regular, uniform, chiral, and Archimedean that apply to polytopes that have some, but not all of their faces equivalent in every dimension.
Much of the research into abstract polytopes has concentrated on regular abstract polytopes, though not all.
Realizations
Any traditional polytope is an example of a realization of its underlying abstract polytope: The traditional pyramid to the left of the Hasse diagram above is a realization of the poset represented. So also are tessellations or tilings of the plane, or other piecewise linear manifolds in two and higher dimensions. The latter include, for example, the projective polytopes. These can be obtained from a polytope with central symmetry by identifying opposite vertices, edges, faces and so forth. In three dimensions, this gives the hemi-cube and the hemi-dodecahedron, and their duals, the hemi-octahedron and the hemi-icosahedron.
More generally, a realization of a regular abstract polytope is a collection of points in space (corresponding to the vertices of the polytope), together with the face structure induced on it by the polytope, which is at least as symmetrical as the original abstract polytope; that is, all combinatorial automorphisms of the abstract polytopes have been realized by geometric symmetries. For example, the set of points is a realisation of the abstract 4-gon (the square). It is not the only realisation, however - one could choose, instead, the set of vertices of a tetrahedron. For every symmetry of the square, there exists a corresponding symmetry of the tetrahedron.
In fact, every abstract polytope with v vertices has at least one realisation, as the vertices of a (v − 1)-dimensional simplex. It is usually desirable to seek lower-dimensional realisations.
If an abstract n-polytope is realized in n-dimensional space, such that the geometrical arrangement does not break any rules for traditional polytopes (such as curved faces, or ridges of zero size), then the realization is said to be faithful. In general, only a restricted set of abstract polytopes of rank n may be realized faithfully in any given n-space. The characterisation of this effect is an outstanding problem.
The amalgamation problem
The basic theory of the combinatorial structures which are now known as "abstract polytopes" (but were originally called "incidence polytopes"), was first described in Egon Schulte's doctoral dissertation, although earlier work by Branko Grόnbaum, H. S. M. Coxeter and Jacques Tits laid the groundwork. Since then, research in the theory of abstract polytopes has focused mostly on regular polytopes, that is, those whose automorphism groups act transitively on the set of flags of the polytope.
An important question in the theory of abstract polytopes is the amalgamation problem. This is a series of questions such as
- For given abstract polytopes K and L, are there any polytopes P whose facets are K and whose vertex figures are L ?
- If so, are they all finite ?
- What finite ones are there ?
For example, if K is the square, and L is the triangle, the answers to these questions are
- Yes, there are polytopes P with square faces, joined three per vertex (that is, there are polytopes of type ).
- Yes, they are all finite, specifically,
- There is the cube, with six square faces, twelve edges and eight vertices, and the hemi-cube, with three faces, six edges and four vertices.
It is known that if the answer to the first question is 'Yes' for some regular K and L, then there is a unique polytope whose facets are K and whose vertex figures are L, called the universal polytope with these facets and vertex figures, which covers all other such polytopes. That is, suppose P is the universal polytope with facets K and vertex figures L. Then any other polytope Q with these facets and vertex figures can be written Q=P/N, where
- N is a subgroup of the automorphism group of P, and
- P/N is the collection of orbits of elements of P under the action of N, with the partial order induced by that of P.
Q=P/N is a quotient of P, and we say P covers Q.
Given this fact, the search for polytopes with particular facets and vertex figures usually goes as follows:
- Attempt to find the applicable universal polytope
- Attempt to classify its quotients.
These two problems are, in general, very difficult.
If K is the square, and L is the triangle, the universal polytope is the cube. If L is, instead, also a square, the universal polytope is the tesselation of the Euclidean plane by squares.
Local topology
The amalgamation problem has, historically, been pursued according to local topology. That is, rather than restricting K and L to be particular polytopes, they are allowed to be any polytope with a given topology, that is, any polytope tessellating a given manifold. If K and L are spherical (that is, tessellations of a topological sphere), then P is called locally spherical and corresponds itself to a tessellation of some manifold. For example, if K and L are both squares (and so are topologically the same as circles), P will be a tessellation of the plane, torus or Klein bottle by squares. A tessellation of an n-dimensional manifold is actually a rank n + 1 polytope. This is in keeping with the common intuition that the Platonic solids are three dimensional, even though they can be regarded as tessellations of the two-dimensional surface of a ball.
In general, an abstract polytope is called locally X if its facets and vertex figures are, topologically, either spheres or X, but not both spheres. The 11-cell and 57-cell are examples of rank 4 (that is, four-dimensional) locally projective polytopes, since their facets and vertex figures are tessellations of real projective planes. There is a weakness in this terminology however. It does not allow an easy way to describe a polytope whose facets are tori and whose vertex figures are projective planes, for example. Worse still if different facets have different topologies, or no well-defined topology at all. However, much progress has been made on the complete classification of the locally toroidal regular polytopes (McMullen & Schulte, 2002)
Exchange maps
Let ? be a flag of an abstract n-polytope, and let −1 < i < n. From the definition of an abstract polytope, it can be proven that there is a unique flag differing from ? by a rank i element, and the same otherwise. If we call this flag ?(i), then this defines a collection of maps on the polytopes flags, say fi. These maps are called exchange maps, since they swap pairs of flags : (?fi)fi = ? always. Some other properties of the exchange maps :
- fi2 is the identity map
- The fi generate a group. (The action of this group on the flags of the polytope is an example of what is called the
flag action of the group on the polytope) If |i − j| > 1, fifj = fjfi If a is an automorphism of the polytope, then afi = fia If the polytope is regular, the group generated by the fi is isomorphic to the automorphism group, otherwise, it is strictly larger.
The exchange maps and the flag action in particular can be used to prove that any abstract polytope is a quotient of some regular polytope.
See also
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