Abstract polytope
Encyclopedia
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 space, such as Euclidean space
, is required to contain it.
The term polytope is a generalisation of polygon
s and polyhedra
into any number of dimensions.
We shall present a precise, formal definition of an abstract polytope 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.
s 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 six quadrilaterals are all the “same”. More rigorously, they are said to be isomorphic or “structure preserving”.
Properties, particularly measurable ones, 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.
Throughout this article, polytope means abstract polytope - unless stated otherwise. The term traditional will be used, somewhat loosely, to refer to what is generally understood by polytope, excluding our abstract polytopes. Some authors also use the terms classical or geometric.
. Polytopes, however, have a dimensional hierarchy. For example, the vertices, edges and faces of a cube have dimension 0, 1, and 2 respectively; the cube itself is 3-dimensional.
In our abstract theory, the concept of rank replaces that of dimension; we formally define it below.
We use the term face to refer to an element of any rank, e.g. vertices (rank 0) or edges (rank 1), and not just faces of rank 2. An element of rank k is called a k-face.
We shall define a polytope, then, 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.
When F < G, we say that F is a subface of G (or G has subface F).
We say F, G are incident if either F = G or F < G or G < F. This meaning differs from its usage in traditional geometry and other areas of mathematics. For example, in the square abcd, edges ab and bc are not incident.
In fact, a polytope can have just one face; in this case the least and greatest faces are one and the same.
The least and greatest faces are called improper faces; all others faces are proper faces.
The least face is called the null face, since it has no vertices (or any other faces) as subfaces. Since the least face is one level below the vertices or 0-faces, its rank is −1; we often denote it as F−1. If this seems strange at first, the feeling is quickly dispelled on seeing the elegant symmetry which this concept brings to our theory. (Historically, mathematicians resisted such commonplace concepts as negative, fractional, irrational and complex numbers - and even zero!)
The relation < is defined as set of pairs, which (for this example) would include
In this particular example, we could have written the edges W, X, Y and Z as ab, ad, bc, and cd respectively, and we often will use this vertex notation. But as we shall shortly see, such notation is not always appropriate.
We have called this a square rather than a quadrilateral (or tetragon) because, in our abstract world, there are no angles, and edges do not have lengths. All four edges are identical, and the "geometry" at each vertex is the same.
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.
, as shown. By convention, faces of the equal rank are placed on the same vertical level. Each "line" between faces indicates a pair F, G such that F < G where F is below G in the diagram.
A polytope is often depicted informally by its graph
, but the two cannot be equated. A graph has vertices and edges, but no other faces. Furthermore, for most polytopes, it is not possible to deduce all the other faces from the graph, and, in general, different polytopes can have the same graph.
A Hasse diagram, on the other hand, fully describes any poset - all the structure of the polytope is captured in the Hasse diagram. Isomorphic polytopes give rise to isomorphic Hasse diagrams, and vice-versa.
The rank of a poset P is the maximum rank n of any face, i.e. that of the greatest face (given that we require that there is one). Throughout this article, we shall always use n to denote the rank of the poset or polytope under discussion.
It follows that the least face, and no other, has rank −1; and that the greatest face has rank n. We often denote these as F−1 and Fn respectively.
The rank of a face or polytope usually corresponds to the dimension of its counterpart in traditional theory - but not always. For example, a face of rank 1 corresponds to an edge, which is 1-dimensional. But a skew polygon in traditional geometry is 3-dimensional, since it is not flat (planar); while its abstract equivalent, and indeed all abstract polygons, have rank 2.
For some ranks, we have names for their face-types, as in the table.
† Although traditionally "face" has meant a rank 2 face, we shall always write "2-face" to avoid ambiguity, reserving the term "face" to mean a face of any rank.
For example, {ø, a, ab, abc} is a flag in the triangle abc.
We shall additionally require that, for a given polytope, all flags contain the same number of faces. Posets do not, in general, satisfy this requirement; the poset {ø, a, b, bc, abc} has 2 flags of unequal size, and is not therefore a polytope.
Clearly, given any two distinct faces F, G in a flag, either F < G or F > G.
In particular, 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 terminology, a section is called a closed interval of the poset and denoted [F, H], but the concepts are identical).
P is thus a section of itself.
For example, in the prism abcxyz (see Figure) 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. The square abcd is a subset of the tetrahedron abcd, but is not a section of it.
This concept of section does not have the same meaning as in traditional geometry.
For example, in the triangle abc, the vertex figure at b, abc/b, is {b, ab, bc, abc}, which is a line segment. The vertex figures of a cube are triangles.
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.
, whose elements we call faces, satisfying the 4 axioms:
An n-polytope is a polytope of rank n.
For n ≤ 1, all n-sections of a polytope are the (unique) n-polytope. However, faces of rank 0 and 1 of a polytope are called vertices and edges respectively.
, 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
.
Until now, we have defined face sets using "vertex notation" - e.g. {ø, a, b, c, ab, ac, bc, abc} for the triangle abc. This method has the decided advantage of implying the < relation.
With the digon, and many other abstract polytopes, vertex notation cannot be used. We are forced to give the faces individual names and specify the subface pairs F < G.
Thus a digon must be defined as a set {ø, a, b, E', E", G} with the relation < given by
where E' and E" are the two edges, and G the greatest face.
To summarise, a polytope can only be fully described using vertex notation if every face has a unique set of vertices. A polytope having this property is called atomistic.
In general, the set of j-faces (−1 ≤ j ≤ n) of a traditional n-polytope form an abstract n-polytope.
The digon is generalized by the hosohedron and higher dimensional hosotopes, which can all be realized as spherical polyhedra – they tessellate the sphere.
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 solid
s, and can be realized as (globally) projective polyhedra – they tessellate the real projective plane
.
The hemicube is another example of where vertex notation can't be used to define a polytope - all the 2-faces and the 3-face have the same vertex set.
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 in the example above is self-dual.
The vertex figure at a vertex V is the dual of the facet to which V maps in the dual polytope.
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
.
All polytopes of rank ≤ 2 are regular. The most famous regular polyhedra are the five Platonic solids. The hemicube (shown) is also regular.
Informally, for each rank k, this means that there is no way to distinguish any k-face from any other - the faces must be identical, and must have identical neighbors, and so forth. For example, a cube is regular because all the faces are squares, each square's vertices are attached to three squares, and each of these squares is attached to identical arrangements of other faces, edges and vertices, and so on.
This condition alone is sufficient to ensure that any regular abstract polytope has isomorphic regular (n−1)-faces and isomorphic regular vertex figures.
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 each rank.
The simplest irregular polytope is the square pyramid
, though this still has many symmetries.
An example of a polyhedron with no symmetries is shown - no pair of vertices, edges, or 2-faces are "the same", as defined above. This is possibly the simplest such polytope.
s 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 {(0,0), (0,1), (1,0), (1,1)} 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.
doctoral dissertation, although earlier work by Branko Grünbaum
, 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 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 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 tesselation of the Euclidean plane by squares. This tesselation 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 tesselate either a torus
or an infinitely long cylinder
with squares.
, 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 (defined below): 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.
, 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 solid
s 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 plane
s. 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)
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.
The table shows a dot 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 dot 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.
Further information is gained by counting each occurrence of an incidence as 1 (and hence non-incidence as 0). 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:
and Petrie of the three infinite structures {4, 6}, {6, 4} and {6, 6}, which they called regular skew polyhedra
.
In the 1960s Branko Grünbaum
issued a call to the geometric community to consider more abstract types of regular polytope
s that he called polystromata. He developed the theory of polystromata, showing examples of new objects he called regular apeirotopes
, that is, regular polytopes with infinitely
many faces
.
Grünbaum also discovered the 11-cell
, 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 independently discovered the same shape. He had earlier discovered a similar polytope, the 57-cell (Coxeter 1982, 1984).
Egon Schulte
defined "regular incidence complexes" and "regular incidence polytopes" in his PhD dissertation in the 1980s - the first time the modern definition was introduced. 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) had also accepted Schulte's definition as the "correct" one.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, an abstract polytope, informally speaking, is a structure which considers only the combinatorial
Combinatorics
Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Aspects of combinatorics include counting the structures of a given kind and size , deciding when certain criteria can be met, and constructing and analyzing objects meeting the criteria ,...
properties of a traditional polytope
Polytope
In elementary geometry, a polytope is a geometric object with flat sides, which exists in any general number of dimensions. A polygon is a polytope in two dimensions, a polyhedron in three dimensions, and so on in higher dimensions...
, ignoring many of its other properties, such as angles, edge lengths, etc. No space, such as Euclidean space
Euclidean space
In mathematics, Euclidean space is the Euclidean plane and three-dimensional space of Euclidean geometry, as well as the generalizations of these notions to higher dimensions...
, is required to contain it.
The term polytope is a generalisation of polygon
Polygon
In geometry a polygon is a flat shape consisting of straight lines that are joined to form a closed chain orcircuit.A polygon is traditionally a plane figure that is bounded by a closed path, composed of a finite sequence of straight line segments...
s and polyhedra
Polyhedron
In elementary geometry a polyhedron is a geometric solid in three dimensions with flat faces and straight edges...
into any number of dimensions.
We shall present a precise, formal definition of an abstract polytope 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.
Traditional versus abstract polytopes
In Euclidean geometry, the six quadrilateralQuadrilateral
In Euclidean plane geometry, a quadrilateral is a polygon with four sides and four vertices or corners. Sometimes, the term quadrangle is used, by analogy with triangle, and sometimes tetragon for consistency with pentagon , hexagon and so on...
s 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
Tube map
The Tube map is a schematic transit map representing the lines and stations of London's rapid transit railway systems, namely the London Underground , the Docklands Light Railway and London Overground....
provides all the relevant information to go from A to B. An even better example is an electrical circuit diagram
Circuit diagram
A circuit diagram is a simplified conventional graphical representation of an electrical circuit...
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 six quadrilaterals are all the “same”. More rigorously, they are said to be isomorphic or “structure preserving”.
Properties, particularly measurable ones, 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.Throughout this article, polytope means abstract polytope - unless stated otherwise. The term traditional will be used, somewhat loosely, to refer to what is generally understood by polytope, excluding our abstract polytopes. Some authors also use the terms classical or geometric.
Polytopes as posets
The connections on a railway map or electrical circuit can be represented quite satisfactorily with just “dots and lines” - i.e. a graphGraph (mathematics)
In mathematics, a graph is an abstract representation of a set of objects where some pairs of the objects are connected by links. The interconnected objects are represented by mathematical abstractions called vertices, and the links that connect some pairs of vertices are called edges...
. Polytopes, however, have a dimensional hierarchy. For example, the vertices, edges and faces of a cube have dimension 0, 1, and 2 respectively; the cube itself is 3-dimensional.
In our abstract theory, the concept of rank replaces that of dimension; we formally define it below.
We use the term face to refer to an element of any rank, e.g. vertices (rank 0) or edges (rank 1), and not just faces of rank 2. An element of rank k is called a k-face.
We shall define a polytope, then, 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
Partially ordered set
In mathematics, especially order theory, a partially ordered set formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation that indicates that, for certain pairs of elements in the...
, or poset.
When F < G, we say that F is a subface of G (or G has subface F).
We say F, G are incident if either F = G or F < G or G < F. This meaning differs from its usage in traditional geometry and other areas of mathematics. For example, in the square abcd, edges ab and bc are not incident.
Least and greatest faces
Just as the concepts of zero and infinity are indispensable in mathematics, it turns out to be extremely useful and elegant, indeed essential, to insist that every polytope also has a least face, which is a subface of all the others, and a greatest face, of which all the others are subfaces.In fact, a polytope can have just one face; in this case the least and greatest faces are one and the same.
The least and greatest faces are called improper faces; all others faces are proper faces.
The least face is called the null face, since it has no vertices (or any other faces) as subfaces. Since the least face is one level below the vertices or 0-faces, its rank is −1; we often denote it as F−1. If this seems strange at first, the feeling is quickly dispelled on seeing the elegant symmetry which this concept brings to our theory. (Historically, mathematicians resisted such commonplace concepts as negative, fractional, irrational and complex numbers - and even zero!)
A simple example
As an example, we now create an abstract square, which has faces as in the table below:| Face type | Rank (k) | Count | k-faces |
|---|---|---|---|
| Least | −1 | 1 | F−1 |
| Vertex | 0 | 4 | a, b, c, d |
| Edge | 1 | 4 | W, X, Y, Z |
| Greatest | 2 | 1 | G |
The relation < is defined as set of pairs, which (for this example) would include
- F−1
In this particular example, we could have written the edges W, X, Y and Z as ab, ad, bc, and cd respectively, and we often will use this vertex notation. But as we shall shortly see, such notation is not always appropriate.
We have called this a square rather than a quadrilateral (or tetragon) because, in our abstract world, there are no angles, and edges do not have lengths. All four edges are identical, and the "geometry" at each vertex is the same.
Order relations are transitive
Transitive relation
In mathematics, a binary relation R over a set X is transitive if whenever an element a is related to an element b, and b is in turn related to an element c, then a is also related to c....
, 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
Covering relation
In mathematics, especially order theory, the covering relation of a partially ordered set is the binary relation which holds between comparable elements that are immediate neighbours...
of the other, i.e. where F < H and no G satisfies F < G < H.
The Hasse diagram
Smaller posets, and polytopes in particular, are often best visualised by using a Hasse diagramHasse diagram
In order theory, a branch of mathematics, a Hasse diagram is a type of mathematical diagram used to represent a finite partially ordered set, in the form of a drawing of its transitive reduction...
, as shown. By convention, faces of the equal rank are placed on the same vertical level. Each "line" between faces indicates a pair F, G such that F < G where F is below G in the diagram.
A polytope is often depicted informally by its graph
Graph (mathematics)
In mathematics, a graph is an abstract representation of a set of objects where some pairs of the objects are connected by links. The interconnected objects are represented by mathematical abstractions called vertices, and the links that connect some pairs of vertices are called edges...
, but the two cannot be equated. A graph has vertices and edges, but no other faces. Furthermore, for most polytopes, it is not possible to deduce all the other faces from the graph, and, in general, different polytopes can have the same graph.
A Hasse diagram, on the other hand, fully describes any poset - all the structure of the polytope is captured in the Hasse diagram. Isomorphic polytopes give rise to isomorphic Hasse diagrams, and vice-versa.
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) satisfying F' < F" < ... < F.The rank of a poset P is the maximum rank n of any face, i.e. that of the greatest face (given that we require that there is one). Throughout this article, we shall always use n to denote the rank of the poset or polytope under discussion.
It follows that the least face, and no other, has rank −1; and that the greatest face has rank n. We often denote these as F−1 and Fn respectively.
The rank of a face or polytope usually corresponds to the dimension of its counterpart in traditional theory - but not always. For example, a face of rank 1 corresponds to an edge, which is 1-dimensional. But a skew polygon in traditional geometry is 3-dimensional, since it is not flat (planar); while its abstract equivalent, and indeed all abstract polygons, have rank 2.
For some ranks, we have names for their face-types, as in the table.
| | Rank | | -1 | |0 | |1 | |2 | |3 | | ... | |n - 2 | |n - 1 | n | |
|---|---|---|---|---|---|---|---|---|---|---|
| Face Type | Least | Vertex | Edge | † | Cell | Ridge | Facet | Greatest |
† Although traditionally "face" has meant a rank 2 face, we shall always write "2-face" to avoid ambiguity, reserving the term "face" to mean a face of any rank.
The line segment
A line segment is a poset that has a least face, precisely two 0-faces, and a greatest face, for example {ø, a, b, ab}. It follows easily that the vertices a and b have rank 0, and that the greatest face ab, and therefore the poset, both have rank 1. This lends credibility to the definition of 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.For example, {ø, a, ab, abc} is a flag in the triangle abc.
We shall additionally require that, for a given polytope, all flags contain the same number of faces. Posets do not, in general, satisfy this requirement; the poset {ø, a, b, bc, abc} has 2 flags of unequal size, and is not therefore a polytope.
Clearly, given any two distinct faces F, G in a flag, either F < G or F > G.
Sections
Any subset P' of a poset P is a poset (with the same relation <, restricted to P').In particular, 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 terminology, a section is called a closed interval of the poset and denoted [F, H], but the concepts are identical).
P is thus a section of itself.
For example, in the prism abcxyz (see Figure) 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. The square abcd is a subset of the tetrahedron abcd, but is not a section of it.
This concept of section does not have the same meaning as in traditional geometry.
Vertex figures
The vertex figure at a given vertex V is the (n−1)-section Fn/V, where Fn is the greatest face.For example, in the triangle abc, the vertex figure at b, abc/b, is {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- H1, H2, ... ,Hk
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 setPartially ordered set
In mathematics, especially order theory, a partially ordered set formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation that indicates that, for certain pairs of elements in the...
, whose elements we call faces, satisfying the 4 axioms:
- It has a least face and a greatest face.
- All flags contain the same number of faces.
- It is strongly connected.
- Every 1-section is a line segment.
An n-polytope is a polytope of rank n.
Rank < 2
There is just one polytope for each rank -1, 0 and 1, and these are, respectively, the null polytope, the point, and the line segment.For n ≤ 1, all n-sections of a polytope are the (unique) n-polytope. However, faces of rank 0 and 1 of a polytope are called vertices and edges respectively.
Rank 2
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
Digon
In geometry, a digon is a polygon with two sides and two vertices. It is degenerate in a Euclidean space, but may be non-degenerate in a spherical space.A digon must be regular because its two edges are the same length...
, and p =
we get the apeirogonApeirogon
An apeirogon is a degenerate polygon with a countably infinite number of sides. It is the limit of a sequence of polygons with more and more sides.Like any polygon, it is a sequence of line segments and angles...
.
The digon
A digon, as its name implies, is a polygon of 2 edges. Unlike any other polygon, both edges have the same two vertices. For this reason, it is regarded as degenerate.Until now, we have defined face sets using "vertex notation" - e.g. {ø, a, b, c, ab, ac, bc, abc} for the triangle abc. This method has the decided advantage of implying the < relation.
With the digon, and many other abstract polytopes, vertex notation cannot be used. We are forced to give the faces individual names and specify the subface pairs F < G.
Thus a digon must be defined as a set {ø, a, b, E', E", G} with the relation < given by
-
-
- {ø
- {ø
-
where E' and E" are the two edges, and G the greatest face.
To summarise, a polytope can only be fully described using vertex notation if every face has a unique set of vertices. A polytope having this property is called atomistic.
Examples of higher rank
As stated above, this concept of an abstract polytope is very general, and includes:- ApeirotopesApeirohedronAn apeirohedron is a polyhedron having infinitely many faces. Like an ordinary polyhedron it forms a surface with no border. But where an ordinary polyhedral surface has no border because it folds round to close back on itself, an apeirohedron has no border because its surface is unbounded.Two main...
, i.e. infinite polytopes or tessellationTessellationA tessellation or tiling of the plane is a pattern of plane figures that fills the plane with no overlaps and no gaps. One may also speak of tessellations of parts of the plane or of other surfaces. Generalizations to higher dimensions are also possible. Tessellations frequently appeared in the art...
s (tilings) - Decompositions of other manifolds such as the torus or real projective plane
- Many other objects, such as the 11-cell11-cellIn mathematics, the 11-cell is a self-dual abstract regular 4-polytope . Its 11 cells are hemi-icosahedral. It has 11 vertices, 55 edges and 55 faces. Its symmetry group is the projective special linear group L2, so it has660 symmetries...
and the 57-cell, that don't fit well into "normal" geometric spaces.
In general, the set of j-faces (−1 ≤ j ≤ n) of a traditional n-polytope form an abstract n-polytope.
Hosohedra and hosotopes
Projective polytopes
Hemi-cube (geometry)
In abstract geometry, a hemi-cube is an abstract regular polyhedron, containing half the faces of a cube.It can be realized as a projective polyhedron , which can be visualized by constructing the projective plane as a hemisphere where opposite points along the boundary are connected and dividing...
(shown), Hemi-octahedron
Hemi-octahedron
A hemi-octahedron is an abstract regular polyhedron, containing half the faces of a regular octahedron.It has 4 triangular faces, 6 edges, and 3 vertices...
, Hemi-dodecahedron
Hemi-dodecahedron
A hemi-dodecahedron is an abstract regular polyhedron, containing half the faces of a regular dodecahedron. It can be realized as a projective polyhedron , which can be visualized by constructing the projective plane as a hemisphere where opposite points along the boundary are connected and...
, and the Hemi-icosahedron
Hemi-icosahedron
A hemi-icosahedron is an abstract regular polyhedron, containing half the faces of a regular icosahedron. It can be realized as a projective polyhedron , which can be visualized by constructing the projective plane as a hemisphere where opposite points along the boundary are connected and dividing...
. These are the projective counterparts of the Platonic solid
Platonic solid
In geometry, a Platonic solid is a convex polyhedron that is regular, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruent regular polygons, with the same number of faces meeting at each vertex; thus, all its edges are congruent, as are its vertices and...
s, and can be realized as (globally) projective polyhedra – they tessellate the real projective plane
Real projective plane
In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold, that is, a one-sided surface. It cannot be embedded in our usual three-dimensional space without intersecting itself...
.
The hemicube is another example of where vertex notation can't be used to define a polytope - all the 2-faces and the 3-face have the same vertex set.
Duality
Every polytope has a dual, a 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 in the example above is self-dual.
The vertex figure at a vertex V is the dual of the facet to which V maps in the dual polytope.
Regular abstract polytopes
Formally, an abstract polytope is defined to be regularRegular polytope
In mathematics, a regular polytope is a polytope whose symmetry is transitive on its flags, thus giving it the highest degree of symmetry. All its elements or j-faces — cells, faces and so on — are also transitive on the symmetries of the polytope, and are regular polytopes of...
if its automorphism group acts
Group action
In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...
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
Coxeter group
In mathematics, a Coxeter group, named after H.S.M. Coxeter, is an abstract group that admits a formal description in terms of mirror symmetries. Indeed, the finite Coxeter groups are precisely the finite Euclidean reflection groups; the symmetry groups of regular polyhedra are an example...
.
All polytopes of rank ≤ 2 are regular. The most famous regular polyhedra are the five Platonic solids. The hemicube (shown) is also regular.
Informally, for each rank k, this means that there is no way to distinguish any k-face from any other - the faces must be identical, and must have identical neighbors, and so forth. For example, a cube is regular because all the faces are squares, each square's vertices are attached to three squares, and each of these squares is attached to identical arrangements of other faces, edges and vertices, and so on.
This condition alone is sufficient to ensure that any regular abstract polytope has isomorphic regular (n−1)-faces and isomorphic regular vertex figures.
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
Quasiregular polyhedron
In geometry, a quasiregular polyhedron is a semiregular polyhedron that has exactly two kinds of regular faces, which alternate around each vertex. They are edge-transitive and hence step closer to regularity than the semiregular which are merely vertex-transitive.There are only two convex...
, uniform
Uniform polytope
A uniform polytope is a vertex-transitive polytope made from uniform polytope facets of a lower dimension. Uniform polytopes of 2 dimensions are the regular polygons....
, chiral
Chiral polytope
In mathematics, a polytope P is chiral if it hastwo orbits of flags under its group of symmetries, withadjacent flags in different orbits.- References :* Schulte, E. Chiral polytopes in ordinary space, I. Discrete Comput. Geom. 32 , 55–99....
, and Archimedean
Archimedean solid
In geometry an Archimedean solid is a highly symmetric, semi-regular convex polyhedron composed of two or more types of regular polygons meeting in identical vertices...
that apply to polytopes that have some, but not all of their faces equivalent in each rank.
An irregular example
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
Square pyramid
In geometry, a square pyramid is a pyramid having a square base. If the apex is perpendicularly above the center of the square, it will have C4v symmetry.- Johnson solid :...
, though this still has many symmetries.
An example of a polyhedron with no symmetries is shown - no pair of vertices, edges, or 2-faces are "the same", as defined above. This is possibly the simplest such polytope.
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 tessellationTessellation
A tessellation or tiling of the plane is a pattern of plane figures that fills the plane with no overlaps and no gaps. One may also speak of tessellations of parts of the plane or of other surfaces. Generalizations to higher dimensions are also possible. Tessellations frequently appeared in the art...
s or tilings of the plane, or other piecewise linear manifolds in two and higher dimensions. The latter include, for example, the projective
Projective space
In mathematics a projective space is a set of elements similar to the set P of lines through the origin of a vector space V. The cases when V=R2 or V=R3 are the projective line and the projective plane, respectively....
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
Hemi-cube (geometry)
In abstract geometry, a hemi-cube is an abstract regular polyhedron, containing half the faces of a cube.It can be realized as a projective polyhedron , which can be visualized by constructing the projective plane as a hemisphere where opposite points along the boundary are connected and dividing...
and the hemi-dodecahedron
Hemi-dodecahedron
A hemi-dodecahedron is an abstract regular polyhedron, containing half the faces of a regular dodecahedron. It can be realized as a projective polyhedron , which can be visualized by constructing the projective plane as a hemisphere where opposite points along the boundary are connected and...
, and their duals, the hemi-octahedron
Hemi-octahedron
A hemi-octahedron is an abstract regular polyhedron, containing half the faces of a regular octahedron.It has 4 triangular faces, 6 edges, and 3 vertices...
and the hemi-icosahedron
Hemi-icosahedron
A hemi-icosahedron is an abstract regular polyhedron, containing half the faces of a regular icosahedron. It can be realized as a projective polyhedron , which can be visualized by constructing the projective plane as a hemisphere where opposite points along the boundary are connected and dividing...
.
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 {(0,0), (0,1), (1,0), (1,1)} 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
Simplex
In geometry, a simplex is a generalization of the notion of a triangle or tetrahedron to arbitrary dimension. Specifically, an n-simplex is an n-dimensional polytope which is the convex hull of its n + 1 vertices. For example, a 2-simplex is a triangle, a 3-simplex is a tetrahedron,...
. 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 and universal polytopes
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'sEgon Schulte
Egon Schulte is a mathematician and a professor of Mathematics at Northeastern University in Boston. He received his Ph.D. in 1980 from University of Dortmund, Germany, his doctoral dissertation was on Regular Incidence Complexes .- External links:* *...
doctoral dissertation, although earlier work by Branko Grünbaum
Branko Grünbaum
Branko Grünbaum is a Croatian-born mathematician and a professor emeritus at the University of Washington in Seattle. He received his Ph.D. in 1957 from Hebrew University of Jerusalem in Israel....
, H. S. M. Coxeter and Jacques Tits
Jacques Tits
Jacques Tits is a Belgian and French mathematician who works on group theory and geometry and who introduced Tits buildings, the Tits alternative, and the Tits group.- Career :Tits received his doctorate in mathematics at the age of 20...
laid the groundwork. Since then, research in the theory of abstract polytopes has focused mostly on regular polytopes, that is, those whose automorphism
Automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphisms of an object forms a group, called the automorphism...
groups
Group (mathematics)
In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
act
Group action
In algebra and geometry, a group action is a way of describing symmetries of objects using groups. The essential elements of the object are described by a set, and the symmetries of the object are described by the symmetry group of this set, which consists of bijective transformations of the set...
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 {4,3}).
- Yes, they are all finite, specifically,
- There is the cubeCubeIn geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. The cube can also be called a regular hexahedron and is one of the five Platonic solids. It is a special kind of square prism, of rectangular parallelepiped and...
, with six square faces, twelve edges and eight vertices, and the hemi-cubeHemi-cube (geometry)In abstract geometry, a hemi-cube is an abstract regular polyhedron, containing half the faces of a cube.It can be realized as a projective polyhedron , which can be visualized by constructing the projective plane as a hemisphere where opposite points along the boundary are connected and dividing...
, 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.
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 tesselation of the Euclidean plane by squares. This tesselation 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 tesselate either a torus
Torus
In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...
or an infinitely long cylinder
Cylinder (geometry)
A cylinder is one of the most basic curvilinear geometric shapes, the surface formed by the points at a fixed distance from a given line segment, the axis of the cylinder. The solid enclosed by this surface and by two planes perpendicular to the axis is also called a cylinder...
with squares.
The 11-cell and the 57-cell
The 11-cell11-cell
In mathematics, the 11-cell is a self-dual abstract regular 4-polytope . Its 11 cells are hemi-icosahedral. It has 11 vertices, 55 edges and 55 faces. Its symmetry group is the projective special linear group L2, so it has660 symmetries...
, discovered independently by H. S. M. Coxeter and Branko Grünbaum
Branko Grünbaum
Branko Grünbaum is a Croatian-born mathematician and a professor emeritus at the University of Washington in Seattle. He received his Ph.D. in 1957 from Hebrew University of Jerusalem in Israel....
, 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 (defined below): 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 topologyTopology
Topology is a major area of mathematics concerned with properties that are preserved under continuous deformations of objects, such as deformations that involve stretching, but no tearing or gluing...
, that is, any polytope tessellating
Tessellation
A tessellation or tiling of the plane is a pattern of plane figures that fills the plane with no overlaps and no gaps. One may also speak of tessellations of parts of the plane or of other surfaces. Generalizations to higher dimensions are also possible. Tessellations frequently appeared in the art...
a given manifold
Manifold
In mathematics , a manifold is a topological space that on a small enough scale resembles the Euclidean space of a specific dimension, called the dimension of the manifold....
. If K and L are spherical (that is, tessellations of a topological sphere
Sphere
A sphere is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point...
), 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
Torus
In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...
or Klein bottle
Klein bottle
In mathematics, the Klein bottle is a non-orientable surface, informally, a surface in which notions of left and right cannot be consistently defined. Other related non-orientable objects include the Möbius strip and the real projective plane. Whereas a Möbius strip is a surface with boundary, a...
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 solid
Platonic solid
In geometry, a Platonic solid is a convex polyhedron that is regular, in the sense of a regular polygon. Specifically, the faces of a Platonic solid are congruent regular polygons, with the same number of faces meeting at each vertex; thus, all its edges are congruent, as are its vertices and...
s 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
11-cell
In mathematics, the 11-cell is a self-dual abstract regular 4-polytope . Its 11 cells are hemi-icosahedral. It has 11 vertices, 55 edges and 55 faces. Its symmetry group is the projective special linear group L2, so it has660 symmetries...
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 plane
Real projective plane
In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold, that is, a one-sided surface. It cannot be embedded in our usual three-dimensional space without intersecting itself...
s. There is a weakness in this terminology however. It does not allow an easy way to describe a polytope whose facets are tori
Torus
In geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...
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 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 φi. These maps are called exchange maps, since they swap pairs of flags : (Ψφi)φi = Ψ always. Some other properties of the exchange maps :- φi2 is the identity map
- The φi generate a groupGroup (mathematics)In mathematics, a group is an algebraic structure consisting of a set together with an operation that combines any two of its elements to form a third element. To qualify as a group, the set and the operation must satisfy a few conditions called group axioms, namely closure, associativity, identity...
. (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, φiφj = φjφi
- If α is an automorphism of the polytope, then αφi = φiα
- If the polytope is regular, the group generated by the φi 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.
Incidence matrices
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 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 dot 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.
Further information is gained by counting each occurrence of an incidence as 1 (and hence non-incidence as 0). This numerative usage enables a symmetry grouping, as in the Hasse Diagram of the square pyramid
Square pyramid
In geometry, a square pyramid is a pyramid having a square base. If the apex is perpendicularly above the center of the square, it will have C4v symmetry.- Johnson solid :...
: 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:
| A | B,C,D,E | f,g,h,j | k,l,m,n | P,Q,R,S | T | |
|---|---|---|---|---|---|---|
| A | 1 | * | 4 | 0 | 4 | 0 |
| B,C,D,E | * | 4 | 1 | 2 | 2 | 1 |
| f,g,h,j | 1 | 1 | 4 | * | 2 | 0 |
| k,l,m,n | 0 | 2 | * | 4 | 1 | 1 |
| P,Q,R,S | 1 | 2 | 2 | 1 | 4 | * |
| T | 0 | 4 | 0 | 4 | * | 1 |
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
Square pyramid
In geometry, a square pyramid is a pyramid having a square base. If the apex is perpendicularly above the center of the square, it will have C4v symmetry.- Johnson solid :...
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:History
An early example of abstract polytopes was the discovery by CoxeterHarold Scott MacDonald Coxeter
Harold Scott MacDonald "Donald" Coxeter, was a British-born Canadian geometer. Coxeter is regarded as one of the great geometers of the 20th century. He was born in London but spent most of his life in Canada....
and Petrie of the three infinite structures {4, 6}, {6, 4} and {6, 6}, which they called regular skew polyhedra
Regular skew polyhedron
In geometry, the regular skew polyhedra are generalizations to the set of regular polyhedron which include the possibility of nonplanar faces or vertex figures....
.
In the 1960s Branko Grünbaum
Branko Grünbaum
Branko Grünbaum is a Croatian-born mathematician and a professor emeritus at the University of Washington in Seattle. He received his Ph.D. in 1957 from Hebrew University of Jerusalem in Israel....
issued a call to the geometric community to consider more abstract types of regular polytope
Regular polytope
In mathematics, a regular polytope is a polytope whose symmetry is transitive on its flags, thus giving it the highest degree of symmetry. All its elements or j-faces — cells, faces and so on — are also transitive on the symmetries of the polytope, and are regular polytopes of...
s that he called polystromata. He developed the theory of polystromata, showing examples of new objects he called regular apeirotopes
Apeirogon
An apeirogon is a degenerate polygon with a countably infinite number of sides. It is the limit of a sequence of polygons with more and more sides.Like any polygon, it is a sequence of line segments and angles...
, that is, regular polytopes with infinitely
Infinity
Infinity is a concept in many fields, most predominantly mathematics and physics, that refers to a quantity without bound or end. People have developed various ideas throughout history about the nature of infinity...
many faces
Face (geometry)
In geometry, a face of a polyhedron is any of the polygons that make up its boundaries. For example, any of the squares that bound a cube is a face of the cube...
.
Grünbaum also discovered the 11-cell
11-cell
In mathematics, the 11-cell is a self-dual abstract regular 4-polytope . Its 11 cells are hemi-icosahedral. It has 11 vertices, 55 edges and 55 faces. Its symmetry group is the projective special linear group L2, so it has660 symmetries...
, a self-dual 4-polytope whose facets are not icosahedra
Icosahedron
In geometry, an icosahedron is a regular polyhedron with 20 identical equilateral triangular faces, 30 edges and 12 vertices. It is one of the five Platonic solids....
, but are "hemi-icosahedra
Hemi-icosahedron
A hemi-icosahedron is an abstract regular polyhedron, containing half the faces of a regular icosahedron. It can be realized as a projective polyhedron , which can be visualized by constructing the projective plane as a hemisphere where opposite points along the boundary are connected and dividing...
" — 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
11-cell
In mathematics, the 11-cell is a self-dual abstract regular 4-polytope . Its 11 cells are hemi-icosahedral. It has 11 vertices, 55 edges and 55 faces. Its symmetry group is the projective special linear group L2, so it has660 symmetries...
, H. S. M. Coxeter independently discovered the same shape. He had earlier discovered a similar polytope, the 57-cell (Coxeter 1982, 1984).
Egon Schulte
Egon Schulte
Egon Schulte is a mathematician and a professor of Mathematics at Northeastern University in Boston. He received his Ph.D. in 1980 from University of Dortmund, Germany, his doctoral dissertation was on Regular Incidence Complexes .- External links:* *...
defined "regular incidence complexes" and "regular incidence polytopes" in his PhD dissertation in the 1980s - the first time the modern definition was introduced. 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) had also accepted Schulte's definition as the "correct" one.
See also
- 11-cell11-cellIn mathematics, the 11-cell is a self-dual abstract regular 4-polytope . Its 11 cells are hemi-icosahedral. It has 11 vertices, 55 edges and 55 faces. Its symmetry group is the projective special linear group L2, so it has660 symmetries...
and 57-cell - two abstract regular 4-polytopes - Regular polytopeRegular polytopeIn mathematics, a regular polytope is a polytope whose symmetry is transitive on its flags, thus giving it the highest degree of symmetry. All its elements or j-faces — cells, faces and so on — are also transitive on the symmetries of the polytope, and are regular polytopes of...
- Graded posetGraded posetIn mathematics, in the branch of combinatorics, a graded poset, sometimes called a ranked poset , is a partially ordered set P equipped with a rank function ρ from P to N compatible with the ordering such that whenever y covers x, then...