Abstract polytope

Traditional versus abstract polytopes

Six geometric quadrilaterals.

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.^[1]

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 set theory requires an empty set which, technically, every set contains. In an abstract polytope this is known 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 may be denoted as F₋₁. 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

F₋₁<a, ... , F₋₁<X, ... , F₋₁<G, ... , b<Y, ... , c<G, ... , Z<G.

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

The graph (left) and Hasse diagram of a quadrilateral, showing ranks (right)

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 F_n.

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

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 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

{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. 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.

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:

For any section of the polytope (including the polytope itself), any flag can be changed into any other by changing just one face at a time.

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.

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

The graph (left) and Hasse Diagram of a 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 < , 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 = we get the apeirogon.

The digon

The graph (left) and Hasse Diagram of a 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

{ø<a, ø<b, a<E', a<E", b<E', b<E", E'<G, 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.

Hosohedra and hosotopes

A hexagonal hosohedron, realized as a spherical polyhedron.

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

The Hemicube may be derived from a cube by identifying opposite vertices, edges, and faces. It has 4 vertices, 6 edges, and 3 faces.

Four examples of non-traditional abstract polyhedra are the Hemicube (shown), Hemi-octahedron, Hemi-dodecahedron, and the Hemi-icosahedron. These are the projective counterparts of the Platonic solids, and can be realized as (globally) projective polyhedra – they tessellate the real projective plane.

The hemicube is another example of where vertex notation can not be used to define a polytope - all the 2-faces and the 3-face have the same vertex set.

An irregular example

An irregular polyhedron which has no automorphisms at all.

Given the amount of attention lavished on regular polytopes, one might almost think that all polytopes are regular. In reality, regular polytopes are just very special cases.

The simplest irregular polytope is the square pyramid, though this still has many symmetries.

An example of a polyhedron with no nontrivial symmetries is shown - no pair of vertices, edges, or 2-faces are "the same", as defined above. This is possibly the simplest such polytope.

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 (McMullen & Schulte, 2002)

Square pyramid

A square pyramid and the associated abstract polytope.

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 holds: