Abstract polytope

In mathematics, an abstract polytope is an algebraic partially ordered set or poset which captures the combinatorial properties of a traditional polytope without specifying purely geometric properties such as angles or edge lengths. A polytope is a generalisation of polygons and polyhedra into any number of dimensions.
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.
Introductory concepts.
Traditional versus abstract polytopes.
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").
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 "F""n". 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:
The relation < comprises a set of pairs, which here include
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.
† 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.
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 "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/ø (highlighted green) is the triangle
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.
Facets.
The facet for a given "j"-face "F" is the ("j"−"1")-section "F"/∅, where "F""j" is the greatest face.
For example, in the triangle abc, the facet at ab is ab/b = {∅, 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 "F""n"/"V", where "F""n" 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
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:
An "n"-polytope is a polytope of rank "n".
Notes.
In the case of the "null polytope", the least and greatest faces are the "same single element".
Axiom 2 is equivalent to saying that the poset is a graded poset.
Given the other axioms, Axiom 3 is equivalent to "strong flag-connectedness", which informally means:
Axiom 4 is known as the “diamond property”, since the Hasse Diagram of "a", "b", and the faces between it is diamond-shaped.
It can be shown from the axioms that every section is a polytope, and that Rank("G"/"F") = Rank("G") − Rank("F") − 1.
The abstract polytope associated with a real convex polytope is also referred to as its face lattice.
The simplest polytopes.
Rank < 1.
There is just one poset for each 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 {ø, 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" < formula_1, 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, ...
For "p" = 2, we have the digon, and "p" = formula_1 we get the apeirogon.
The digon.
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. {ø, a, b, c, ab, ac, bc, abc} for the triangle abc. This method has the advantage of "implying" the < 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 {ø, a, b, E', E", G} with the relation < given by
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 realisation 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.
For a regular abstract polytope, if the combinatorial automorphisms of the abstract polytope are realised by geometric symmetries then the geometric figure will be a regular polytope.
Moduli space.
The group "G" of symmetries of a realization "V" of an abstract apeirogon "P" is generated by two reflections, the product of which translates each vertex of "P" to the next. The product of the two reflections can be decomposed as a product of a non-zero translation, finitely many rotations, and possibly trivial reflection.
Generally, the moduli space of realizations of an abstract polytope is a convex cone of infinite dimension. The realization cone of the abstract apeirogon has uncountably infinite algebraic dimension and cannot be closed in the Euclidean topology.
The amalgamation problem and universal polytopes.
An important question in the theory of abstract polytopes is the "amalgamation problem". This is a series of questions such as
For example, if "K" is the square, and "L" is the triangle, the answers to these questions are
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
"Q"="P"/"N" is called 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:
These two problems are, in general, very difficult.
Returning to the example above, if "K" is the square, and "L" is the triangle, the universal polytope {"K","L"} is the cube (also written {4,3}). The hemicube is the quotient {4,3}/"N", where "N" is a group of symmetries (automorphisms) of the cube with just two elements - the identity, and the symmetry that maps each corner (or edge or face) to its opposite.
If "L" is, instead, also a square, the universal polytope {"K","L"} (that is, {4,4}) is the tessellation of the Euclidean plane by squares. This tessellation has infinitely many quotients with square faces, four per vertex, some regular and some not. Except for the universal polytope itself, they all correspond to various ways to tessellate either a torus or an infinitely long cylinder with squares.
The 11-cell and the 57-cell.
The 11-cell, discovered independently by H. S. M. Coxeter and Branko Grünbaum, 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 Schläfli type {5,3,5}. 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.
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 
Exchange maps.
Let "Ψ" 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 "Ψ" by a rank "i" element, and the same otherwise. If we call this flag "Ψ"("i"), then this defines a collection of maps on the polytopes flags, say "φ""i". These maps are called exchange maps, since they swap pairs of flags : ("Ψφ""i")"φ""i" = "Ψ" always. Some other properties of the exchange maps :
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.
Incidence matrices.
A polytope can also be represented by tabulating its incidences. 
The following incidence matrix is that of a triangle:
The table shows a 1 wherever a face is a subface of another, "or vice versa" (so the table is symmetric about the diagonal)- so in fact, the table has "redundant information"; it would suffice to show only a 1 when the row face ≤ the column face.
Since both the body and the empty set are incident with all other elements, the first row and column as well as the last row and column are trivial and can conveniently be omitted.
Square pyramid.
Further information is gained by counting each occurrence. This numerative usage enables a symmetry grouping, as in the Hasse Diagram of the square pyramid: If vertices B, C, D, and E are considered symmetrically equivalent within the abstract polytope, then edges f, g, h, and j will be grouped together, and also edges k, l, m, and n, And finally also the triangles P, Q, R, and S. Thus the corresponding incidence matrix of this abstract polytope may be shown as:
In this accumulated incidence matrix representation the diagonal entries represent the total counts of either element type.
Elements of different type of the same rank clearly are never incident so the value will always be 0, however to help distinguish such relationships, an asterisk (*) is used instead of 0.
The sub-diagonal entries of each row represent the incidence counts of the relevant sub-elements, while the super-diagonal entries represent the respective element counts of the vertex-, edge- or whatever -figure.
Already this simple square pyramid shows that the symmetry-accumulated incidence matrices are no longer symmetrical. But there is still a simple entity-relation (beside the generalised Euler formulae for the diagonal, respectively the sub-diagonal entities of each row, respectively the super-diagonal elements of each row - those at least whenever no holes or stars etc. are considered), as for any such incidence matrix formula_3 holds:
formula_4
History.
In the 1960s Branko Grünbaum issued a call to the geometric community to consider generalizations of the concept of regular polytopes that he called "polystromata". He developed a theory of polystromata, showing examples of new objects including the 11-cell.
The 11-cell is a self-dual 4-polytope whose facets are not icosahedra, but are "hemi-icosahedra" — that is, they are the shape one gets if one considers opposite faces of the icosahedra to be actually the "same" face (Grünbaum, 1977). A few years after Grünbaum's discovery of the 11-cell, H.S.M. Coxeter discovered a similar polytope, the 57-cell (Coxeter 1982, 1984), and then independently rediscovered the 11-cell.
With the earlier work by Branko Grünbaum, H. S. M. Coxeter and Jacques Tits having laid the groundwork, the basic theory of the combinatorial structures now known as abstract polytopes was first described by Egon Schulte in his 1980 PhD dissertation. In it he defined "regular incidence complexes" and "regular incidence polytopes". Subsequently, he and Peter McMullen developed the basics of the theory in a series of research articles that were later collected into a book. Numerous other researchers have since made their own contributions, and the early pioneers (including Grünbaum) have also accepted Schulte's definition as the "correct" one.
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.