HOME

TheInfoList



OR:

In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, an abstract polytope is an algebraic
partially ordered set In mathematics, especially order theory, a partially ordered set (also poset) 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 ...
which captures the dyadic property of a traditional
polytope In elementary geometry, a polytope is a geometric object with flat sides ('' faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an ...
without specifying purely geometric properties such as points and lines. A geometric
polytope In elementary geometry, a polytope is a geometric object with flat sides ('' faces''). Polytopes are the generalization of three-dimensional polyhedra to any number of dimensions. Polytopes may exist in any general number of dimensions as an ...
is said to be a ''realization'' of an abstract polytope in some real N-dimensional space, typically Euclidean. This abstract definition allows more general
combinatorial Combinatorics is an area of mathematics primarily concerned with counting, both as a means and an end in obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ap ...
structures than traditional definitions of a polytope, thus allowing new objects that have no counterpart in traditional theory.


Introductory concepts


Traditional versus abstract polytopes

In Euclidean geometry, two shapes that are not similar can nonetheless share a common structure. For example a
square In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...
and a
trapezoid A quadrilateral with at least one pair of parallel sides is called a trapezoid () in American and Canadian English. In British and other forms of English, it is called a trapezium (). A trapezoid is necessarily a convex quadrilateral in Eu ...
both comprise an alternating chain of four vertices and four sides, which makes them quadrilaterals. They are said to be
isomorphic In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
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.


Realizations

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.


Faces, ranks and ordering

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 In mathematics, especially order theory, a partially ordered set (also poset) 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 ...
, or ''poset''.


Least and greatest faces

Just as the number zero is necessary in mathematics, so also every set has the
empty set In mathematics, the empty set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in othe ...
∅ as a subset. In an abstract polytope ∅ is by convention identified as the ''least'' or ''null'' face and is a subface of all the others. Since the least face is one level below the vertices or 0-faces, its rank is −1 and it may be denoted as ''F''−1. Thus F−1 ≡ ∅ and the abstract polytope also contains the empty set as an element. It is not usually realized. There is also a single face of which all the others are subfaces. This is called the ''greatest'' face. In an ''n''-dimensional polytope, the greatest face has rank = ''n'' and may be denoted as ''F''''n''. It is sometimes realized as the interior of the geometric figure. These least and greatest faces are sometimes called ''improper'' faces, with all others being ''proper'' faces.


A simple example

The faces of the abstract quadrilateral or square are shown in the table below: The relation < comprises a set of pairs, which here include : ''F''−1−1−1transitive, 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 Successor may refer to: * An entity that comes after another (see Succession (disambiguation)) Film and TV * ''The Successor'' (film), a 1996 film including Laura Girling * ''The Successor'' (TV program), a 2007 Israeli television program Musi ...
of the other, i.e. where F < H and no G satisfies F < G < H. The edges W, X, Y and Z are sometimes written as ab, ad, bc, and cd respectively, but such notation is not always appropriate. All four edges are structurally similar and the same is true of the vertices. The figure therefore has the symmetries of a square and is usually referred to as the square.


The Hasse diagram

Smaller posets, and polytopes in particular, are often best visualized in a
Hasse diagram In order theory, 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. Concretely, for a partially ordered set ''(S, ≤)'' one represents ...
, 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 Graph may refer to: Mathematics *Graph (discrete mathematics), a structure made of vertices and edges **Graph theory, the study of such graphs and their properties *Graph (topology), a topological space resembling a graph in the sense of discre ...
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 A chain is a serial assembly of connected pieces, called links, typically made of metal, with an overall character similar to that of a rope in that it is flexible and curved in compression but linear, rigid, and load-bearing in tension. ...
(F', F", ... , F) satisfying F' < F" < ... < F. F' is always the least face, F−1. The ''rank'' of an abstract polytope P is the maximum rank ''n'' of any face. It is always the rank of the greatest face Fn. The rank of a face or polytope usually corresponds to the ''dimension'' of its counterpart in traditional theory. For some ranks, their face-types are named in the following table. † Traditionally "face" has meant a rank 2 face or 2-face. In abstract theory the term "face" denotes a face of ''any'' rank.


Flags

In geometry, a
flag A flag is a piece of fabric (most often rectangular or quadrilateral) with a distinctive design and colours. It is used as a symbol, a signalling device, or for decoration. The term ''flag'' is also used to refer to the graphic design empl ...
is a maximal
chain A chain is a serial assembly of connected pieces, called links, typically made of metal, with an overall character similar to that of a rope in that it is flexible and curved in compression but linear, rigid, and load-bearing in tension. ...
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, is a flag in the triangle abc. For a given polytope, all flags contain the same number of faces. Other posets do not, in general, satisfy this requirement.


Sections

Any subset P' of a poset P is a poset (with the same relation <, restricted to P'). In an abstract polytope, given any two faces ''F'', ''H'' of P with ''F'' ≤ ''H'', the set is called a section of ''P'', and denoted ''H''/''F''. (In order theory, a section is called a
closed interval In mathematics, a (real) interval is a set of real numbers that contains all real numbers lying between any two numbers of the set. For example, the set of numbers satisfying is an interval which contains , , and all numbers in between. Other ...
of the poset and denoted 'F'', ''H'' For example, in the prism abcxyz (see diagram) the section xyz/ø (highlighted green) is the triangle :. A ''k''-section is a section of rank ''k''. P is thus a section of itself. This concept of section ''does not'' have the same meaning as in traditional geometry.


Facets

The facet for a given ''j''-face ''F'' is the (''j''−''1'')-section ''F''/∅, where ''F''''j'' is the greatest face. For example, in the triangle abc, the facet at ab is ab/b = , which is a line segment. The distinction between ''F'' and ''F''/∅ is not usually significant and the two are often treated as identical.


Vertex figures

The
vertex figure In geometry, a vertex figure, broadly speaking, is the figure exposed when a corner of a polyhedron or polytope is sliced off. Definitions Take some corner or vertex of a polyhedron. Mark a point somewhere along each connected edge. Draw line ...
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 = , 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 set In mathematics, especially order theory, a partially ordered set (also poset) 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 ...
, whose elements we call ''faces'', satisfying the 4 axioms: # It has just one least face and one greatest face. # All
flags A flag is a piece of textile, fabric (most often rectangular or quadrilateral) with a distinctive design and colours. It is used as a symbol, a signalling device, or for decoration. The term ''flag'' is also used to refer to the graphic desi ...
contain the same number of faces. # It is
strongly connected In the mathematical theory of directed graphs, a graph is said to be strongly connected if every vertex is reachable from every other vertex. The strongly connected components of an arbitrary directed graph form a partition into subgraphs that ...
. # If the ranks of two faces ''a > b'' differ by 2, then there are exactly 2 faces that lie strictly between ''a'' and ''b''. An ''n''-polytope is a polytope of rank ''n''. The abstract polytope associated with a real
convex polytope A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the w ...
is also referred to as its
face lattice A convex polytope is a special case of a polytope, having the additional property that it is also a convex set contained in the n-dimensional Euclidean space \mathbb^n. Most texts. use the term "polytope" for a bounded convex polytope, and the wo ...
.


The simplest polytopes


Rank < 1

There is just one poset for each rank −1 and 0. These are, respectively, the null face and the point. These are not always considered to be valid abstract polytopes.


Rank 1: the line segment

There is only one polytope of rank 1, which is the line segment. It has a least face, just two 0-faces and a greatest face, for example . 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'' < \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 In geometry, a digon is a polygon with two sides (edges) and two vertices. Its construction is degenerate in a Euclidean plane because either the two sides would coincide or one or both would have to be curved; however, it can be easily visu ...
, and ''p'' = \infty we get the
apeirogon In geometry, an apeirogon () or infinite polygon is a generalized polygon with a countably infinite number of sides. Apeirogons are the two-dimensional case of infinite polytopes. In some literature, the term "apeirogon" may refer only to t ...
.


The digon

A
digon In geometry, a digon is a polygon with two sides (edges) and two vertices. Its construction is degenerate in a Euclidean plane because either the two sides would coincide or one or both would have to be curved; however, it can be easily visu ...
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 In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions ...
. Faces are sometimes described using "vertex notation" - e.g. 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 with the relation < given by ::: 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.


Examples of higher rank

The set of ''j''-faces (−1 ≤ ''j'' ≤ ''n'') of a traditional ''n''-polytope form an abstract ''n''-polytope. The concept of an abstract polytope is more general and also includes: * Apeirotopes or infinite polytopes, which include
tessellation A tessellation or tiling is the covering of a surface, often a plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety of ...
s (tilings) * Proper decompositions of unbounded manifolds such as the
torus In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle. If the axis of revolution does n ...
or
real projective plane In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold; in other words, a one-sided surface. It cannot be embedded in standard three-dimensional space without intersecting itself. It has b ...
. * Many other objects, such as the
11-cell In mathematics, the 11-cell (or hendecachoron) is a self-dual abstract regular 4-polytope ( four-dimensional polytope). Its 11 cells are hemi-icosahedral. It has 11 vertices, 55 edges and 55 faces. It has Schläfli symbol , with 3 hemi-icosahedra ...
and the 57-cell, that cannot be faithfully realized in Euclidean spaces.


Hosohedra and hosotopes

The digon is generalized by the
hosohedron In spherical geometry, an -gonal hosohedron is a tessellation of lunes on a spherical surface, such that each lune shares the same two polar opposite vertices. A regular -gonal hosohedron has Schläfli symbol with each spherical lune ha ...
and higher dimensional hosotopes, which can all be realized as
spherical polyhedra In geometry, a spherical polyhedron or spherical tiling is a tiling of the sphere in which the surface is divided or partitioned by great arcs into bounded regions called spherical polygons. Much of the theory of symmetrical polyhedra is most co ...
– they tessellate the sphere.


Projective polytopes

Four examples of non-traditional abstract polyhedra are the Hemicube (shown), Hemi-octahedron,
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 (a tessellation of the real projective plane by 6 pentagons), which can be visualized by const ...
, and the
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 (a tessellation of the real projective plane by 10 triangles), which can be visualized by const ...
. These are the projective counterparts of the
Platonic solid In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all e ...
s, and can be realized as (globally) projective polyhedra – they tessellate the
real projective plane In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold; in other words, a one-sided surface. It cannot be embedded in standard three-dimensional space without intersecting itself. It has b ...
. 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.


Duality

Every geometric polytope has a '' dual'' twin. Abstractly, the dual is the same polytope but with the ranking reversed in order: the Hasse diagram differs only in its annotations. 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 In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
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.


Abstract regular polytopes

Formally, an abstract polytope is defined to be "regular" if its
automorphism group In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the g ...
acts The Acts of the Apostles ( grc-koi, Πράξεις Ἀποστόλων, ''Práxeis Apostólōn''; la, Actūs Apostolōrum) is the fifth book of the New Testament; it tells of the founding of the Christian Church and the spread of its message ...
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 In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean refle ...
. 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 standardized, such as semi-regular, quasi-regular,
uniform A uniform is a variety of clothing worn by members of an organization while participating in that organization's activity. Modern uniforms are most often worn by armed forces and paramilitary organizations such as police, emergency services, ...
,
chiral Chirality is a property of asymmetry important in several branches of science. The word ''chirality'' is derived from the Greek (''kheir''), "hand", a familiar chiral object. An object or a system is ''chiral'' if it is distinguishable from i ...
, and Archimedean that apply to polytopes that have some, but not all of their faces equivalent in each rank.


Realization

A set of points ''V'' in a Euclidean space equipped with a surjection from the vertex set of an abstract polytope ''P'' such that automorphisms of ''P'' induce isometric permutations of ''V'' is called a ''realization'' of an abstract polytope. Two realizations are called congruent if the natural bijection between their sets of vertices is induced by an isometry of their ambient Euclidean spaces. 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 characterization of this effect is an outstanding problem. For a regular abstract polytope, if the combinatorial automorphisms of the abstract polytope are realized by geometric symmetries then the geometric figure will be a regular polytope.


Moduli space

The group ''G'' of symmetries of a realization ''V'' of an abstract polytope ''P'' is generated by two reflections, the product of which translates each vertex of ''P'' to the next. The product of the two reflections can be decomposed as a product of a non-zero translation, finitely many rotations, and possibly trivial reflection. Generally, the
moduli space In mathematics, in particular algebraic geometry, a moduli space is a geometric space (usually a scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of such objects. Such sp ...
of realizations of an abstract polytope is a
convex cone In linear algebra, a ''cone''—sometimes called a linear cone for distinguishing it from other sorts of cones—is a subset of a vector space that is closed under scalar multiplication; that is, is a cone if x\in C implies sx\in C for every . W ...
of infinite dimension. The realization cone of the abstract polytope has uncountably infinite algebraic dimension and cannot be
closed Closed may refer to: Mathematics * Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set * Closed set, a set which contains all its limit points * Closed interval, ...
in the
Euclidean topology In mathematics, and especially general topology, the Euclidean topology is the natural topology induced on n-dimensional Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, ...
.


The amalgamation problem and universal polytopes

An important question in the theory of abstract polytopes is the ''amalgamation problem''. This is a series of questions such as : For 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 ). : Yes, they are all finite, specifically, : There is the
cube In geometry, a cube is a three-dimensional solid object bounded by six square faces, facets or sides, with three meeting at each vertex. Viewed from a corner it is a hexagon and its net is usually depicted as a cross. The cube is the only ...
, with six square faces, twelve edges and eight vertices, and the hemi-cube, 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 called a quotient of ''P'', and we say ''P'' covers ''Q''. Given this fact, the search for polytopes with particular facets and vertex figures usually goes as follows: # 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 is the cube (also written ). The hemicube is the quotient /''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 (that is, ) is the tessellation of the Euclidean plane by squares. This tessellation has infinitely many quotients with square faces, four per vertex, some regular and some not. Except for the universal polytope itself, they all correspond to various ways to tessellate either a
torus In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle. If the axis of revolution does n ...
or an infinitely long
cylinder A cylinder (from ) has traditionally been a three-dimensional solid, one of the most basic of curvilinear geometric shapes. In elementary geometry, it is considered a prism with a circle as its base. A cylinder may also be defined as an ...
with squares.


The 11-cell and the 57-cell

The
11-cell In mathematics, the 11-cell (or hendecachoron) is a self-dual abstract regular 4-polytope ( four-dimensional polytope). Its 11 cells are hemi-icosahedral. It has 11 vertices, 55 edges and 55 faces. It has Schläfli symbol , with 3 hemi-icosahedra ...
, discovered independently by
H. S. M. Coxeter Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington t ...
and
Branko Grünbaum Branko Grünbaum ( he, ברנקו גרונבאום; 2 October 1929 – 14 September 2018) was a Croatian-born mathematician of Jewish descent57-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 . 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 In mathematics, topology (from the Greek words , and ) is concerned with the properties of a geometric object that are preserved under continuous deformations, such as stretching, twisting, crumpling, and bending; that is, without closing ...
, that is, any polytope tessellating a given
manifold In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a n ...
. If ''K'' and ''L'' are ''spherical'' (that is, tessellations of a topological
sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the c ...
), 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 In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle. If the axis of revolution does n ...
or
Klein bottle In topology, a branch of mathematics, the Klein bottle () is an example of a non-orientable surface; it is a two-dimensional manifold against which a system for determining a normal vector cannot be consistently defined. Informally, it is 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 In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all e ...
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 In mathematics, the 11-cell (or hendecachoron) is a self-dual abstract regular 4-polytope ( four-dimensional polytope). Its 11 cells are hemi-icosahedral. It has 11 vertices, 55 edges and 55 faces. It has Schläfli symbol , with 3 hemi-icosahedra ...
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 In mathematics, the real projective plane is an example of a compact non-orientable two-dimensional manifold; in other words, a one-sided surface. It cannot be embedded in standard three-dimensional space without intersecting itself. It has b ...
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


Exchange maps

Let ''Ψ'' be a flag of an abstract ''n''-polytope, and let −1 < ''i'' < ''n''. From the definition of an abstract polytope, it can be proven that there is a unique flag differing from ''Ψ'' by a rank ''i'' element, and the same otherwise. If we call this flag ''Ψ''(''i''), then this defines a collection of maps on the polytopes flags, say ''φ''''i''. These maps are called exchange maps, since they swap pairs of flags : (''Ψφ''''i'')''φ''''i'' = ''Ψ'' always. Some other properties of the exchange maps : * ''φ''''i''2 is the identity map * The ''φ''''i'' generate a
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
. (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, ''φ''''i''''φ''''j'' = ''φ''''j''''φ''''i'' * If ''α'' is an automorphism of the polytope, then ''αφ''''i'' = ''φ''''i''''α'' * If the polytope is regular, the group generated by the ''φ''''i'' 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: The table shows a 1 wherever a face is a subface of another, ''or vice versa'' (so the table is
symmetric Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
about the diagonal)- so in fact, the table has ''redundant information''; it would suffice to show only a 1 when the row face ≤ the column face. Since both the body and the empty set are incident with all other elements, the first row and column as well as the last row and column are trivial and can conveniently be omitted.


Square pyramid

Further information is gained by counting each occurrence. This numerative usage enables a
symmetry Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...
grouping, as in the
Hasse Diagram In order theory, 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. Concretely, for a partially ordered set ''(S, ≤)'' one represents ...
of the
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 is a right square pyramid, and has symmetry. If all edge lengths are equal, it is an equilateral square pyrami ...
: 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 In geometry, a square pyramid is a pyramid having a square base. If the apex is perpendicularly above the center of the square, it is a right square pyramid, and has symmetry. If all edge lengths are equal, it is an equilateral square pyrami ...
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 I=(I_) holds: I_ \cdot I_ = I_ \cdot I_ \ \ (i


History

In the 1960s
Branko Grünbaum Branko Grünbaum ( he, ברנקו גרונבאום; 2 October 1929 – 14 September 2018) was a Croatian-born mathematician of Jewish descentregular polytope In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. All its elements or -faces (for all , where is the dimension of the polytope) — cells, ...
s that he called ''polystromata''. He developed a theory of polystromata, showing examples of new objects including the
11-cell In mathematics, the 11-cell (or hendecachoron) is a self-dual abstract regular 4-polytope ( four-dimensional polytope). Its 11 cells are hemi-icosahedral. It has 11 vertices, 55 edges and 55 faces. It has Schläfli symbol , with 3 hemi-icosahedra ...
. The
11-cell In mathematics, the 11-cell (or hendecachoron) is a self-dual abstract regular 4-polytope ( four-dimensional polytope). Its 11 cells are hemi-icosahedral. It has 11 vertices, 55 edges and 55 faces. It has Schläfli symbol , with 3 hemi-icosahedra ...
is a
self-dual In mathematics, a duality translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often (but not always) by means of an involution operation: if the dual of is , then the ...
4-polytope In geometry, a 4-polytope (sometimes also called a polychoron, polycell, or polyhedroid) is a four-dimensional polytope. It is a connected and closed figure, composed of lower-dimensional polytopal elements: vertices, edges, faces (polygons), an ...
whose
facets A facet is a flat surface of a geometric shape, e.g., of a cut gemstone. Facet may also refer to: Arts, entertainment, and media * ''Facets'' (album), an album by Jim Croce * ''Facets'', a 1980 album by jazz pianist Monty Alexander and his tri ...
are not
icosahedra In geometry, an icosahedron ( or ) is a polyhedron with 20 faces. The name comes and . The plural can be either "icosahedra" () or "icosahedrons". There are infinitely many non- similar shapes of icosahedra, some of them being more symmetrica ...
, 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 In mathematics, the 11-cell (or hendecachoron) is a self-dual abstract regular 4-polytope ( four-dimensional polytope). Its 11 cells are hemi-icosahedral. It has 11 vertices, 55 edges and 55 faces. It has Schläfli symbol , with 3 hemi-icosahedra ...
, H.S.M. Coxeter discovered a similar polytope, the 57-cell (Coxeter 1982, 1984), and then independently rediscovered the 11-cell. With the earlier work by
Branko Grünbaum Branko Grünbaum ( he, ברנקו גרונבאום; 2 October 1929 – 14 September 2018) was a Croatian-born mathematician of Jewish descentH. S. M. Coxeter Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington t ...
and
Jacques Tits Jacques Tits () (12 August 1930 – 5 December 2021) was a Belgian-born French mathematician who worked on group theory and incidence geometry. He introduced Tits buildings, the Tits alternative, the Tits group, and the Tits metric. Life an ...
having laid the groundwork, the basic theory of the combinatorial structures now known as abstract polytopes was first described by
Egon Schulte Egon Schulte (born January 7, 1955 in Heggen ( Kreis Olpe), Germany) is a mathematician and a professor of Mathematics at Northeastern University in Boston. He received his Ph.D. in 1980 from the Technical University of Dortmund TU Dortmund ...
in his 1980 PhD dissertation. In it he defined "regular incidence complexes" and "regular incidence polytopes". Subsequently, he and
Peter McMullen Peter McMullen (born 11 May 1942) is a British mathematician, a professor emeritus of mathematics at University College London. Education and career McMullen earned bachelor's and master's degrees from Trinity College, Cambridge, and studied at ...
developed the basics of the theory in a series of research articles that were later collected into a book. Numerous other researchers have since made their own contributions, and the early pioneers (including Grünbaum) have also accepted Schulte's definition as the "correct" one. Since then, research in the theory of abstract polytopes has focused mostly on ''regular'' polytopes, that is, those whose
automorphism 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 automorphis ...
groups A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
act transitively on the set of flags of the polytope.


See also

* Eulerian poset *
Graded poset In mathematics, in the branch of combinatorics, a graded poset is a partially-ordered set (poset) ''P'' equipped with a rank function ''ρ'' from ''P'' to the set N of all natural numbers. ''ρ'' must satisfy the following two properties: * The r ...
*
Regular polytope In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. All its elements or -faces (for all , where is the dimension of the polytope) — cells, ...


Notes


References

* *
Jaron's World: Shapes in Other Dimensions
'' Discover mag.'', Apr 2007 * Dr. Richard Klitzing
Incidence Matrices
*Schulte, E.; "Symmetry of polytopes and polyhedra", ''Handbook of discrete and computational geometry'', edited by Goodman, J. E. and O'Rourke, J., 2nd Ed., Chapman & Hall, 2004. {{refend Algebraic topology Incidence geometry Polytopes