Abstract polytope

An abstract polytope is a combinatorial structure (as opposed to an inherently geometric structure) that encodes the incidences between element s in a polytope. It circumvents many of the issues that often arise when trying to define polytopes, such as the definition of their interior, or whether any of the many degenerate cases are to be considered, by completely forgoing any consideration of space and treating polytopes in purely structural terms.

In the context of abstract polytopes, "ordinary" geometrical polytopes can be treated as concrete polytopes. Any concrete polytope with the same incidence relations as a given abstract polytope is called a realization of it. Abstract polytopes may have none or many realizations with any desired properties, be it regularity, uniformity, planarity, or any others.

The study of abstract polytopes is useful because it allows for much simpler formal treatment of a wide variety of shapes. Many constructions, like vertex figures, pyramid products, or antiprisms that are tricky to define in geometric terms, can be very easily stated in terms of abstract polytopes. Any of these constructions can then be made concrete if we have a mapping from the vertices of the abstract polytope into points in a space. All other elements will be automatically defined, which allows us to consider conditions such as planarity as secondary.

General terminology
Before we define an abstract polytope, we need to define some more general mathematical terminology.

Partially ordered sets
A partially ordered set (Wikipedia), or poset for short, is a set $$S$$ with a binary relation $$\le$$ such that for any $$a,b,c\in S$$:


 * $$a\le a$$. (Reflexivity)
 * If $$a\le b$$ and $$b\le a$$, then $$a=b$$. (Antisymmetry)
 * If $$a\le b$$ and $$b\le c$$, then $$a\le c$$ (Transitivity)

If we use $$\le$$ to denote the relation of a poset, we may define $$a$$ are then defined in the obvious way.

If either $$a\le b$$ or $$b\le a$$, the elements are said to be comparable. Note that there's no requirement that all pairs of elements are comparable, hence the name "partial". If this condition were to hold, the order would be called total.

If $$a\le c$$ and there exists no $$b$$ such that $$a<b<c$$, then $$c$$ is said to cover $$a$$.

Bounded posets
An element $$a$$ in a poset is said to be a minimum or bottom element if $$a\le b$$ for any $$b$$. Likewise, $$a$$ is said to be a maximum or top element if $$b\le a$$ for any $$b$$. If a poset has a minimum or maximum element, they must be unique. A poset with both a minimum and maximum is said to be bounded. In the context of abstract polytopes, these two elements are called the improper elements. All other elements are the proper ones.

Note that the words "minimal" and "maximal" have slightly different meanings in the context of partial orders, and should thus be avoided when describing these elements.

Chains and flags
A chain of a poset $$(S,\le)$$ is any subset $$T\subseteq S$$ such that $$(T,\le)$$ forms a total order. That is, any two elements of the chain may be compared. Often chains may be extended by adding elements to them. If a chain can't be extended, it's called a maximal chain or flag.

A poset is called ranked or pure if all flags have the same size. Any such poset can be given a rank function $$r:S\to\mathbb Z$$ such that for any $$a,b\in S$$,


 * If $$a\le b$$, then $$r(a)\le r(b)$$. (Monotonicity)
 * If $$b$$ covers $$a$$, then $$r(a)+1=r(b)$$.

Rank functions are unique up to addition by a constant. By convention, when talking about abstract polytopes, we set the rank function of a bounded ranked poset such that the rank of the minimum element is &minus;1.

If two flags differ in exactly one element of rank j, they are said to be j-adjacent.

Sections
A section of a partially ordered set (called an interval in more general mathematical contexts) is the set of elements between two others. In the context of abstract polytopes, a section between $$F$$ and $$G$$ may be denoted as $$G/F$$, so that


 * $$G/F=\{H\in S:F\le H,H\le G\}.$$

Any section of a ranked poset is ranked, and so may be given a rank too.

Connectivity
A bounded poset is said to be connected if it is rank 1 or if for any two proper elements $$F$$ and $$G$$, there exists a sequence $$(F_1,\ldots,F_n)$$ of proper elements such that $$F=F_1$$, $$G=F_n$$, and any $$F_i,F_{i+1}$$ are comparable. The special concession that rank 1 polytopes are connected ensures that the dyad is connected and thus an abstract polytope.

A bounded ranked poset is said to be strongly connected if every section is connected.

Diamond property
A poset is said to satisfy the diamond property whenever any section of rank 1 has exactly four elements. This is so called because the Hasse diagram (see below) of this section looks like a diamond or rhombus.

Definition
An abstract polytope is defined as a poset that's bounded, ranked, strongly connected, and satisfies the diamond condition. In what follows, we justify and contextualize this definition.

The elements of the abstract polytope represent elements of all "dimensions". Rank 0 elements are vertices, rank 1 elements are edges, rank 2 elements are faces, and so on. The relation $$a\le b$$ may be read as "$$a$$ is contained in $$b$$".

The bounded condition is mostly a technicality that makes some definitions and results slightly cleaner. See the article improper element for a discussion of this condition.

The ranked condition makes it so that we can't "skip through dimensions". For instance, there can't be a vertex in a face without being edges that connect both. Moreover, the rank function makes it possible to talk of the "dimension" of an element without a reference to any particular space.

The strong connectivity condition serves to exclude compounds. This is almost universally adopted within the general mathematical community, for two reasons:


 * Many elementary results on polytopes, specifically regular polytopes, hold only when this condition is enforced.
 * Compound polytopes may be studied just by looking at their connected components.

Within this wiki, this condition is often relaxed or removed altogether.

The diamond property is the main property that sets polytopes apart from other more general structures. It generalizes the pattern "two edges join at a vertex in a polygon, two faces join at an edge in a polyhedron, two cells join at a face in a polychoron..." As another immediate consequence of this property, any edge has exactly two vertices.

Hasse diagrams
The Hasse diagram of a partial order is a directed graph that serves as a useful graphical notation. Elements of the poset are drawn as nodes in the graph, and elements that are related in the partial order are connected by an edge, so that the smallest element goes below the largest. To reduce clutter, only direct relations are drawn, so that if a ≤ b ≤ c, the edge between a and c will be omitted.

Since abstract polytopes are partial orders, they also have Hasse diagrams. In their specific case, the nodes are usually grouped by rank. A node will only be connected to elements of the next and the previous rank. For example, a polygonal ("face") element will be linked to several dyad elements (its edges) as well as whatever cells it is a face of, and a dyad ("edge") element will be linked to two point elements as well as whatever faces it is an edge of. Connections only go from d-elements down to (d&minus;1)-elements, where d is any integer between 0 and the rank of P.

The Hasse diagram of a polytope is not to be confused with the abstract polytope itself. For instance, the Hasse diagram to the right has a tesseractic skeleton, but actually represents a tetrahedron.

Abstract regular polytope
An abstract polytope is regular if its automorphism group acts transitively on its flags. Equivalently, an abstract polytope is regular if the order of its automorphism group is equal to the number of flags it has.

Not every abstract regular polytope has a regular realization. For example, there is an infinite family of abstract polytopes that are the comb products of polygons and represent looped portions of the square tiling. They can be realized as non-regular toroids in 3-space or as regular skew polyhedra in 4-space (where they are made of the square faces of duoprisms).

The uniform polyhedra the dodecadodecahedron and ditrigonary dodecadodecahedron as well as their duals the medial rhombic triacontahedron and medial triambic icosahedron are regular when seen as abstract polytopes. There this simply is because an abstract polytope does not distinguish between those 5-fold elements with different geometric realizations, the convex regular pentagons and the regular pentagrams. Along with the ditrigonal icosahedron and some stephanoids, they are called the regular polyhedra of index 2 (where the fully regular polyhedra are of index 1). There are higher indexes that include more polyhedra.