mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, an abstract polytope is an algebraic
partially ordered set
In mathematics, especially order theory, a partial order on a Set (mathematics), set is an arrangement such that, for certain pairs of elements, one precedes the other. The word ''partial'' is used to indicate that not every pair of elements need ...
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
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coo ...
, typically Euclidean. This abstract definition allows more general
combinatorial
Combinatorics is an area of mathematics primarily concerned with counting, both as a means and as an end to obtaining results, and certain properties of finite structures. It is closely related to many other areas of mathematics and has many ...
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 geometry, a square is a regular polygon, regular quadrilateral. It has four straight sides of equal length and four equal angles. Squares are special cases of rectangles, which have four equal angles, and of rhombuses, which have four equal si ...
and a
trapezoid
In geometry, a trapezoid () in North American English, or trapezium () in British English, is a quadrilateral that has at least one pair of parallel sides.
The parallel sides are called the ''bases'' of the trapezoid. The other two sides are ...
both comprise an alternating chain of four vertices and four sides, which makes them
quadrilaterals
In geometry a quadrilateral is a four-sided polygon, having four edges (sides) and four corners (vertices). The word is derived from the Latin words ''quadri'', a variant of four, and ''latus'', meaning "side". It is also called a tetragon, ...
. They are said to be
isomorphic
In mathematics, an isomorphism is a structure-preserving mapping or morphism 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 the ...
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
A finite geometry is any geometry, geometric system that has only a finite set, finite number of point (geometry), points.
The familiar Euclidean geometry is not finite, because a Euclidean line contains infinitely many points. A geometry based ...
, 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 partial order on a Set (mathematics), set is an arrangement such that, for certain pairs of elements, one precedes the other. The word ''partial'' is used to indicate that not every pair of elements need ...
, 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 or void set is the unique Set (mathematics), set having no Element (mathematics), elements; its size or cardinality (count of elements in a set) is 0, zero. Some axiomatic set theories ensure that the empty set exi ...
∅ 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 usually not realized, though the lack of its realization could be interpreted as it being realized as the set containing no points, the empty set.
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'' (1996 film), a film including Laura Girling
* The Successor (2023 film), a French drama film
* ''The Successor'' ( ...
of the other, i.e. where F < H and no G satisfies F < G < H.
The edges W, X, Y and Z are sometimes written as ab, ad, bc, and cd respectively, but such notation is not always appropriate.
All four edges are structurally similar and the same is true of the vertices. The figure therefore has the symmetries of a square and is usually referred to as the square.
The Hasse diagram
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,\le) one represents each ...
, 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 discret ...
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. A ...
(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 textile, fabric (most often rectangular) with distinctive colours and design. It is used as a symbol, a signalling device, or for decoration. The term ''flag'' is also used to refer to the graphic design employed, and fla ...
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. A ...
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 the set of all real numbers lying between two fixed endpoints with no "gaps". Each endpoint is either a real number or positive or negative infinity, indicating the interval extends without a bound. A real in ...
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/∅ = , 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 general -polytope is sliced off.
Definitions
Take some corner or Vertex (geometry), vertex of a polyhedron. Mark a point somewhere along each connected ed ...
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 partial order on a Set (mathematics), set is an arrangement such that, for certain pairs of elements, one precedes the other. The word ''partial'' is used to indicate that not every pair of elements need ...
, 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 fabric (most often rectangular) with distinctive colours and design. It is used as a symbol, a signalling device, or for decoration. The term ''flag'' is also used to refer to the graphic design employed, and flags have ...
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 a directed graph form a partition into subgraphs that are thems ...
.
# 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 wo ...
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'' < , 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 bigon, digon, or a ''2''-gon, is a polygon with two sides (edge (geometry), edges) and two Vertex (geometry), vertices. Its construction is Degeneracy (mathematics), degenerate in a Euclidean plane because either the two sides wou ...
, and ''p'' = we get the
apeirogon
In geometry, an apeirogon () or infinite polygon is a polygon with an infinite number of sides. Apeirogons are the rank 2 case of infinite polytopes. In some literature, the term "apeirogon" may refer only to the regular apeirogon, with an in ...
.
The digon
A
digon
In geometry, a bigon, digon, or a ''2''-gon, is a polygon with two sides (edge (geometry), edges) and two Vertex (geometry), vertices. Its construction is Degeneracy (mathematics), degenerate in a Euclidean plane because either the two sides wou ...
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, a Euclidean plane is a Euclidean space of Two-dimensional space, dimension two, denoted \textbf^2 or \mathbb^2. It is a geometric space in which two real numbers are required to determine the position (geometry), position of eac ...
.
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:
*
Apeirotope
In geometry, an apeirotope or infinite polytope is a generalized polytope which has infinitely many Facet (geometry), facets.
Definition Abstract apeirotope
An Abstract polytope, abstract ''n''-polytope is a partially ordered set ''P'' (whose ele ...
s 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 ...
s (tilings)
* Proper decompositions of unbounded manifolds such as the
torus
In geometry, a torus (: tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanarity, coplanar with the circle. The main types of toruses inclu ...
or
real projective plane
In mathematics, the real projective plane, denoted or , is a two-dimensional projective space, similar to the familiar Euclidean plane in many respects but without the concepts of distance, circles, angle measure, or parallelism. It is the sett ...
.
* Many other objects, such as the 11-cell 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''. A polyhedron whose vertices are equidistant from ...
– they tessellate the sphere.
Projective polytopes
Four examples of non-traditional abstract polyhedra are the
Hemicube Hemicube can mean:
* Hemicube (computer graphics), a concept in 3D computer graphics rendering
*Hemicube (geometry)
In abstract geometry, a hemicube is an abstract, regular polyhedron, produced by cutting a cube in half with a plane that passes ...
(shown),
Hemi-octahedron
In geometry, a hemi-octahedron is an abstract polytope, abstract regular polyhedron, containing half the faces of a regular octahedron.
It has 4 triangular faces, 6 edges, and 3 vertices. Its dual polyhedron is the Hemicube (geometry), hemicube ...
,
Hemi-dodecahedron
In geometry, a hemi-dodecahedron is an abstract polytope, abstract, regular polyhedron, containing half the Face (geometry), faces of a regular dodecahedron. It can be realized as a projective polyhedron (a tessellation of the real projective pla ...
, and the
Hemi-icosahedron
In geometry, a hemi-icosahedron is an abstract polytope, 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), ...
. These are the projective counterparts of the
Platonic solid
In geometry, a Platonic solid is a Convex polytope, convex, regular polyhedron in three-dimensional space, three-dimensional Euclidean space. Being a regular polyhedron means that the face (geometry), faces are congruence (geometry), congruent (id ...
s, and can be realized as (globally)
projective polyhedra
In geometry, a (globally) projective polyhedron is a tessellation of the real projective plane. These are projective analogs of spherical polyhedra – tessellations of the sphere – and toroidal polyhedra – tessellations of the toroids.
Pr ...
– they tessellate the
real projective plane
In mathematics, the real projective plane, denoted or , is a two-dimensional projective space, similar to the familiar Euclidean plane in many respects but without the concepts of distance, circles, angle measure, or parallelism. It is the sett ...
.
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 or morphism 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 the ...
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 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 ref ...
.
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. do not 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 costume worn by members of an organization while usually participating in that organization's activity. Modern uniforms are most often worn by armed forces and paramilitary organizations such as police, emergency serv ...
,
chiral
Chirality () is a property of asymmetry important in several branches of science. The word ''chirality'' is derived from the Greek language, Greek (''kheir''), "hand", a familiar chiral object.
An object or a system is ''chiral'' if it is dist ...
, 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 (mathematics), scheme or an algebraic stack) whose points represent algebro-geometric objects of some fixed kind, or isomorphism classes of suc ...
of realizations of an abstract polytope is a
convex cone
In linear algebra, a cone—sometimes called a linear cone to distinguish it from other sorts of cones—is a subset of a real vector space that is closed under positive scalar multiplication; that is, C is a cone if x\in C implies sx\in C for e ...
of infinite dimension. The realization cone of the abstract polytope has uncountably infinite algebraic dimension and cannot be closed in the
Euclidean topology
In mathematics, and especially general topology, the Euclidean topology is the natural topology induced on n-dimensional Euclidean space \R^n by the Euclidean metric.
Definition
The Euclidean norm on \R^n is the non-negative function \, \cdot ...
.
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
A cube or regular hexahedron is a three-dimensional space, three-dimensional solid object in geometry, which is bounded by six congruent square (geometry), square faces, a type of polyhedron. It has twelve congruent edges and eight vertices. It i ...
, 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
In celestial mechanics, an orbit (also known as orbital revolution) is the curved trajectory of an physical body, object such as the trajectory of a planet around a star, or of a natural satellite around a planet, or of an satellite, artificia ...
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 (: tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanarity, coplanar with the circle. The main types of toruses inclu ...
or an infinitely long
cylinder
A cylinder () 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 infinite ...
Branko Grünbaum
Branko Grünbaum (; 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
Topology (from the Greek language, Greek words , and ) is the branch of mathematics concerned with the properties of a Mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deformat ...
, that is, any polytope
tessellating
A tessellation or tiling is the covering of a surface, often a plane (mathematics), plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to high-dimensiona ...
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 (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
), 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 (: tori or toruses) is a surface of revolution generated by revolving a circle in three-dimensional space one full revolution about an axis that is coplanarity, coplanar with the circle. The main types of toruses inclu ...
or
Klein bottle
In mathematics, the Klein bottle () is an example of a Orientability, non-orientable Surface (topology), surface; that is, informally, a one-sided surface which, if traveled upon, could be followed back to the point of origin while flipping the ...
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 polytope, convex, regular polyhedron in three-dimensional space, three-dimensional Euclidean space. Being a regular polyhedron means that the face (geometry), faces are congruence (geometry), congruent (id ...
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
In mathematics, the real projective plane, denoted or , is a two-dimensional projective space, similar to the familiar Euclidean plane in many respects but without the concepts of distance, circles, angle measure, or parallelism. It is the sett ...
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 iden ...
. (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 () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is invariant under some transformations ...
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 () in everyday life refers to a sense of harmonious and beautiful proportion and balance. In mathematics, the term has a more precise definition and is usually used to refer to an object that is Invariant (mathematics), invariant und ...
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,\le) one represents each ...
of the
square pyramid
In geometry, a square pyramid is a Pyramid (geometry), pyramid with a square base and four triangles, having a total of five faces. If the Apex (geometry), apex of the pyramid is directly above the center of the square, it is a ''right square p ...
: 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 (geometry), pyramid with a square base and four triangles, having a total of five faces. If the Apex (geometry), apex of the pyramid is directly above the center of the square, it is a ''right square p ...
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: