Abstract polytope

Isomorphic quadrilaterals.

In Euclidean geometry, two shapes that are not similar can nonetheless share a common structure. For example, a square and a trapezoid both comprise an alternating chain of four vertices and four sides, which makes them quadrilaterals. 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]

A traditional polytope is said to be a realization of the associated abstract polytope. A realization 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 realizations 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 realizations. A conventional polytope is a faithful realization.

In an abstract polytope, each structural element (vertex, edge, cell, etc.) is associated with a corresponding member of the set. The term face is used to refer 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 ranks, 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.

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 <. Formally, P (with <) will be a (strict) partially ordered set, or poset.

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.[2] 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.[3]

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

Smaller posets, and polytopes in particular, are often best visualized 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.

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.

In geometry, 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.

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.

P is thus a section of itself.

This concept of section does not have the same meaning as in traditional geometry.

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/∅ = {∅, 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.

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.

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, ... ,H_k

such that F = H₁, G = H_k, and each H_i, 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.

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.

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.

For each p, 3 ≤ p < ${\displaystyle \infty }$, 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 = ${\displaystyle \infty }$ we get the apeirogon.

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.

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.

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 cannot be used to define a polytope - all the 2-faces and the 3-face have the same vertex set.

The group G of symmetries of a realization V of an abstract polytope P is generated by two reflections, the product of which translates each vertex of P to the next.[9][10] The product of the two reflections can be decomposed as a product of a non-zero translation, finitely many rotations, and possibly trivial reflection.[11][10]

Generally, the moduli space of realizations of an abstract polytope is a convex cone of infinite dimension.[12][13] The realization cone of the abstract polytope has uncountably infinite algebraic dimension and cannot be closed in the Euclidean topology.[11][14]

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. 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.

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 [15]

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 ${\displaystyle I=(I_{ij})}$ holds:

${\displaystyle I_{ii}\cdot I_{ij}=I_{ji}\cdot I_{jj}\ \ (i<j).}$