{"history":[],"article":{"id":"647e29fe-8ac7-4991-9ae6-3ccb43f76130","title":"Abstract polytope","slug":"abstract-polytope","content":":::summary\n:::\n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), an **abstract [polytope](https://bliptext.com/articles/polytope)** is an algebraic [partially ordered set](https://bliptext.com/articles/partially-ordered-set) which captures the dyadic property of a traditional [polytope](https://bliptext.com/articles/polytope) without specifying purely geometric properties such as points and lines.\nA geometric [polytope](https://bliptext.com/articles/polytope) is said to be a realization of an abstract [polytope](https://bliptext.com/articles/polytope) in some real [N-dimensional space](https://bliptext.com/articles/n-dimensional-space), typically [Euclidean](https://bliptext.com/articles/euclidean-space).\nThis abstract definition allows more general [combinatorial](https://bliptext.com/articles/combinatorics) structures than traditional definitions of a polytope, thus allowing new objects that have no counterpart in traditional theory.\n# [A square pyramid and the associated abstract polytope. | upload.wikimedia.org/wikipedia/commons/e/ed/Pyramid///abstract///polytope.svg]\n\n# Introductory concepts\n\n## Traditional versus abstract polytopes\nIn Euclidean geometry, two shapes that are not [similar](https://bliptext.com/articles/similar-geometry) can nonetheless share a common structure.\nFor example, a [square](https://bliptext.com/articles/square) and a [trapezoid](https://bliptext.com/articles/trapezoid) both comprise an alternating chain of four [vertices](https://bliptext.com/articles/vertex-geometry) and four sides, which makes them [quadrilaterals](https://bliptext.com/articles/quadrilaterals).\nThey are said to be [isomorphic](https://bliptext.com/articles/isomorphic) or “structure preserving”.\nThis 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.\nThe measurable properties of traditional polytopes such as angles, edge-lengths, skewness, straightness and convexity have no meaning for an abstract polytope.\nWhat is true for traditional polytopes (also called classical or geometric polytopes) may not be so for abstract ones, and vice versa.\nFor 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.\n# [Isomorphic quadrilaterals. | upload.wikimedia.org/wikipedia/commons/f/f4/Isomorphic///Tetragons.svg]\n\n### Realizations\nA traditional polytope is said to be a realization of the associated abstract polytope.\nA realization is a mapping or injection of the abstract object into a real space, typically [Euclidean](https://bliptext.com/articles/euclidean-space), to construct a traditional polytope as a real geometric figure.\nThe six quadrilaterals shown are all distinct realizations of the abstract quadrilateral, each with different geometric properties.\nSome of them do not conform to traditional definitions of a quadrilateral and are said to be unfaithful realizations.\nA conventional polytope is a faithful realization.\n\n## Faces, ranks and ordering\nIn an abstract polytope, each structural element (vertex, edge, cell, etc.) is associated with a corresponding member of the set.\nThe 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.\nThe faces are ranked according to their associated real dimension: vertices have rank 0, edges rank 1 and so on.\nIncident 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.\nF, 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](https://bliptext.com/articles/finite-geometry), although it differs from traditional geometry and some other areas of mathematics.\nFor example, in the square ABCD, edges AB and BC are not abstractly incident (although they are both incident with vertex B).\nA polytope is then defined as a set of faces **P** with an order relation **<**.\nFormally, **P** (with **<**) will be a (strict) [partially ordered set](https://bliptext.com/articles/partially-ordered-set), or poset.\n\n## Least and greatest faces\nJust as the number zero is necessary in mathematics, so also every set has the [empty set](https://bliptext.com/articles/empty-set) ∅ as a subset.\nIn an abstract polytope ∅ is by convention identified as the least or null face and is a subface of all the others.\nSince the least face is one level below the vertices or 0-faces, its rank is −1 and it may be denoted as F−1.\nThus F−1 ≡ ∅ and the abstract polytope also contains the empty set as an element.\nIt is not usually realized.\nThere is also a single face of which all the others are subfaces.\nThis is called the greatest face.\nIn an n-dimensional polytope, the greatest face has rank = n and may be denoted as Fn. It is sometimes realized as the interior of the geometric figure.\nThese least and greatest faces are sometimes called improper faces, with all others being proper faces.\n\n## A simple example\nThe faces of the abstract quadrilateral or square are shown in the table below:\nThe relation < comprises a set of pairs, which here include\nOrder relations are [transitive](https://bliptext.com/articles/transitive-relation), 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](https://bliptext.com/articles/covering-relation) of the other, i.e. where F < H and no G satisfies F < G < H.\nThe edges W, X, Y and Z are sometimes written as **ab**, **ad**, **bc**, and **cd** respectively, but such notation is not always appropriate.\nAll four edges are structurally similar and the same is true of the vertices.\nThe figure therefore has the symmetries of a square and is usually referred to as the square.\n\n## The Hasse diagram\nSmaller posets, and polytopes in particular, are often best visualized in a [Hasse diagram](https://bliptext.com/articles/hasse-diagram), as shown.\nBy convention, faces of equal rank are placed on the same vertical level.\nEach \"line\" between faces, say F, G, indicates an ordering relation < such that F < G where F is below G in the diagram.\nThe Hasse diagram defines the unique poset and therefore fully captures the structure of the polytope.\nIsomorphic polytopes give rise to isomorphic Hasse diagrams, and vice versa.\nThe same is not generally true for the [graph](https://bliptext.com/articles/graph-discrete-mathematics) representation of polytopes.\n# [The graph (left) and Hasse diagram of a quadrilateral, showing ranks (right) | upload.wikimedia.org/wikipedia/commons/4/49/A///Square///and///its///Hasse///Diagram.PNG]\n\n## Rank\nThe rank of a face F is defined as (m − 2), where m is the maximum number of faces in any [chain](https://bliptext.com/articles/total-order) (F', F\", ..., F) satisfying F' < F\" < ... < F. F' is always the least face, F−1.\nThe ra**n**k of a**n** abstract polytope **P** is the maximum ra**n**k **n** of a**n**y face.\nIt is always the rank of the greatest face Fn.\nThe rank of a face or polytope usually corresponds to the dimension of its counterpart in traditional theory.\nFor some ranks, their face-types are named in the following table.\n† Traditionally \"face\" has meant a rank 2 face or 2-face.\nIn abstract theory the term \"face\" denotes a face of any rank.\n\n## Flags\nIn geometry, a **[flag](https://bliptext.com/articles/flag-geometry)** is a maximal [chain](https://bliptext.com/articles/total-order) 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](https://bliptext.com/articles/total-order).\nGiven any two distinct faces F, G in a flag, either F < G or F > G.\nFor ex**a**mple, {**ø**, **a**, **a**b, **a**bc} is **a** fl**a**g in the tri**a**ngle **a**bc.\nFor a given polytope, all flags contain the same number of faces.\nOther posets do not, in general, satisfy this requirement.\n\n## Sections\nAny subset P' of a poset P is a poset (with the same relation <, restricted to P').\nIn an abstract polytope, given any two faces F, H of P with F ≤ H, the set {G | F ≤ G ≤ H} is called a **section** of P, and denoted H/F.\n(In order theory, a section is called a [closed interval](https://bliptext.com/articles/partially-ordered-set) of the poset and denoted [F, H].\nFor example, in the prism **abc**xyz**** (see diagram) the section **xyz**/**ø** (highlighted green) is the triangle\nA k-section is a section of rank k.\nP is thus a section of itself.\nThis concept of section does not have the same meaning as in traditional geometry.\n# [The graph (left) and Hasse Diagram of a triangular prism, showing a 1-section ( red ), and a 2-section ( green ). | upload.wikimedia.org/wikipedia/commons/8/86/Triangular///3-Prism.PNG]\n\n### Facets\nThe **facet** for a given j-face F is the (j−1)-section F/∅, where Fj is the greatest face.\nFor example, in the triangle ******ab****c**, the facet at ****ab**** is ****ab****/**∅** = {**∅**, a, b, ****ab****}, which is a line segment.\nThe distinction between F and F/∅ is not usually significant and the two are often treated as identical.\n\n### Vertex figures\nThe **[vertex figure](https://bliptext.com/articles/vertex-figure)** at a given vertex V is the (n−1)-section Fn/V, where Fn is the greatest face.\nFor example, in the triangle **a****b****c**, the vertex figure at ****b**** is **a****b****c**/****b**** = {****b****, a****b****, ****b****c, **a****b****c**}, which is a line segment.\nThe vertex figures of a cube are triangles.\n\n### Connectedness\nA 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\nsuch that F = H1, G = Hk, and each Hi, i < k, is incident with its successor.\nThe above condition ensures that a pair of disjoint triangles **abc** and **xyz** is not a (single) polytope.\nA poset P is **strongly connected** if every section of P (including P itself) is connected.\nWith this additional requirement, two pyramids that share just a vertex are also excluded.\nHowever, two square pyramids, for example, can, be \"glued\" at their square faces - giving an octahedron.\nThe \"common face\" is not then a face of the octahedron.\n\n# Formal definition\nAn **abstract polytope** is a [partially ordered set](https://bliptext.com/articles/partially-ordered-set), whose elements we call faces, satisfying the 4 axioms:\nAn n-polytope is a polytope of rank n.\nThe abstract polytope associated with a real [convex polytope](https://bliptext.com/articles/convex-polytope) is also referred to as its **[face lattice](https://bliptext.com/articles/convex-polytope)**.\n\n# The simplest polytopes\n\n## Rank < 1\nThere is just one poset for each rank −1 and 0.\nThese are, respectively, the null face and the point.\nThese are not always considered to be valid abstract polytopes.\n\n## Rank 1: the line segment\nThere is only one polytope of rank 1, which is the line segment.\nIt has a least face, just two 0-faces and a greatest face, for example {ø, **a, b, ab**}.\nIt follows th**a**t the vertices **a** **a**nd **b** h**a**ve r**a**nk 0, **a**nd th**a**t the gre**a**test f**a**ce **a****b**, **a**nd therefore the poset, **b**oth h**a**ve r**a**nk 1.\n# [The graph (left) and Hasse Diagram of a line segment | upload.wikimedia.org/wikipedia/commons/d/dd/An///Edge///%28Line///Segment%29///and///its///Hasse///Diagram.PNG]\n\n## Rank 2: polygons\nFor each p, 3 ≤ p < \\infty, we have (the abstract equivalent of) the traditional polygon with p vertices and p edges, or a p-gon.\nFor p = 3, 4, 5, ... we have the triangle, square, pentagon, ....\nFor p = 2, we have the [digon](https://bliptext.com/articles/digon), and p = \\infty we get the [apeirogon](https://bliptext.com/articles/apeirogon).\n\n### The digon\nA [digon](https://bliptext.com/articles/digon) is a polygon with just 2 edges.\nUnlike any other polygon, both edges have the same two vertices.\nFor this reason, it is degenerate in the [Euclidean plane](https://bliptext.com/articles/euclidean-plane).\nF**a****c**es **a**re sometimes des**c**ri**b**ed using \"vertex not**a**tion\" - e.g. {**ø**, **a**, **b**, **c**, **a****b**, **a****c**, **b****c**, **a****b****c**} for the tri**a**ngle **a****b****c**.\nThis method has the advantage of implying the **<** relation.\nWith the digon this vertex notation cannot be used.\nIt is necessary to give the faces individual symbols and specify the subface pairs F < G.\nThus, **a** digon is defined **a**s **a** set {**ø**, **a**, **b**, E', E\", G} with the rel**a**tion **<** given **b**y\nwhere E' and E\" are the two edges, and G the greatest face.\nThis need to identify each element of the polytope with a unique symbol applies to many other abstract polytopes and is therefore common practice.\nA polytope can only be fully described using vertex notation if every face is incident with a unique set of vertices.\nA polytope having this property is said to be [atomistic](https://bliptext.com/articles/atom-order-theory).\n# [The graph (left) and Hasse Diagram of a digon | upload.wikimedia.org/wikipedia/commons/8/85/Digon///and///Hasse///Diagram.PNG]\n\n# Examples of higher rank\nThe set of j-faces (−1 ≤ j ≤ n) of a traditional n-polytope form an abstract n-polytope.\nThe concept of an abstract polytope is more general and also includes:\n\n## Hosohedra and hosotopes\nThe digon is generalized by the [hosohedron](https://bliptext.com/articles/hosohedron) and higher dimensional hosotopes, which can all be realized as [spherical polyhedra](https://bliptext.com/articles/spherical-polyhedra) – they tessellate the sphere.\n# [A hexagonal hosohedron, realized as a spherical polyhedron. | upload.wikimedia.org/wikipedia/commons/f/f9/Hexagonal///Hosohedron.svg]\n\n## Projective polytopes\nFour examples of non-traditional abstract polyhedra are the [Hemicube](https://bliptext.com/articles/hemi-cube-geometry) (shown), [Hemi-octahedron](https://bliptext.com/articles/hemi-octahedron), [Hemi-dodecahedron](https://bliptext.com/articles/hemi-dodecahedron), and the [Hemi-icosahedron](https://bliptext.com/articles/hemi-icosahedron).\nThese are the projective counterparts of the [Platonic solids](https://bliptext.com/articles/platonic-solid), and can be realized as (globally) [projective polyhedra](https://bliptext.com/articles/projective-polyhedra) – they tessellate the [real projective plane](https://bliptext.com/articles/real-projective-plane).\nThe 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.\n# [The Hemicube may be derived from a cube by identifying opposite vertices, edges, and faces. It has 4 vertices, 6 edges, and 3 faces. | upload.wikimedia.org/wikipedia/commons/8/8d/Hemicube.svg]\n\n# Duality\nEvery geometric polytope has a [dual](https://bliptext.com/articles/dual-polyhedron) twin.\nAbstractly, the dual is the same polytope but with the ranking reversed in order: the Hasse diagram differs only in its annotations.\nIn an n-polytope,\neach of the original k-faces maps to an (n − k − 1)-face in the dual.\nThus, for example, the n-face maps to the (−1)-face.\nThe dual of a dual is ([isomorphic](https://bliptext.com/articles/isomorphic) to) the original.\nA polytope is self-dual if it is the same as, i.e. isomorphic to, its dual.\nHence, the Hasse diagram of a self-dual polytope must be symmetrical about the horizontal axis half-way between the top and bottom.\nThe square pyramid in the example above is self-dual.\nThe vertex figure at a vertex V is the dual of the facet to which V maps in the dual polytope.\n\n# Abstract regular polytopes\nFormally, an abstract polytope is defined to be \"regular\" if its [automorphism group](https://bliptext.com/articles/automorphism-group) [acts](https://bliptext.com/articles/group-action-mathematics) transitively on the set of its flags.\nIn 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.\nWhen an abstract polytope is regular, its automorphism group is isomorphic to a quotient of a [Coxeter group](https://bliptext.com/articles/coxeter-group).\nAll polytopes of rank ≤ 2 are regular.\nThe most famous regular polyhedra are the five Platonic solids.\nThe hemicube (shown) is also regular.\nInformally, 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.\nFor 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.\nThis condition alone is sufficient to ensure that any regular abstract polytope has isomorphic regular (n−1)-faces and isomorphic regular vertex figures.\nThis is a weaker condition than regularity for traditional polytopes, in that it refers to the (combinatorial) automorphism group, not the (geometric) symmetry group.\nFor example, any abstract polygon is regular, since angles, edge-lengths, edge curvature, skewness etc. do not exist for abstract polytopes.\nThere are several other weaker concepts, some not yet fully standardized, such as [semi-regular](https://bliptext.com/articles/semiregular-polytope), [quasi-regular](https://bliptext.com/articles/quasiregular-polyhedron), [uniform](https://bliptext.com/articles/uniform-polytope), [chiral](https://bliptext.com/articles/chiral-polytope), and [Archimedean](https://bliptext.com/articles/archimedean-solid) that apply to polytopes that have some, but not all of their faces equivalent in each rank.\n\n# Realization\nA 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](https://bliptext.com/articles/isometry) permutations of V is called a realization of an abstract polytope.\nTwo realizations are called congruent if the natural bijection between their sets of vertices is induced by an isometry of their ambient Euclidean spaces.\nIf 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.\nIn general, only a restricted set of abstract polytopes of rank n may be realized faithfully in any given n-space.\nThe characterization of this effect is an outstanding problem.\nFor 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.\n\n## Moduli space\nThe 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.\nThe product of the two reflections can be decomposed as a product of a non-zero translation, finitely many rotations, and possibly trivial reflection.\nGenerally, the [moduli space](https://bliptext.com/articles/moduli-space) of realizations of an abstract polytope is a [convex cone](https://bliptext.com/articles/convex-cone) of infinite dimension.\nThe realization cone of the abstract polytope has uncountably infinite [algebraic dimension](https://bliptext.com/articles/algebraic-dimension) and cannot be [closed](https://bliptext.com/articles/closed-set) in the [Euclidean topology](https://bliptext.com/articles/euclidean-topology).\n\n# The amalgamation problem and universal polytopes\nAn important question in the theory of abstract polytopes is the amalgamation problem.\nThis is a series of questions such as\nFor example, if K is the square, and L is the triangle, the answers to these questions are\nIt 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.\nThat is, suppose P is the universal polytope with facets K and vertex figures L.\nThen any other polytope Q with these facets and vertex figures can be written Q=P/N, where\nQ=P/N is called a **quotient** of P, and we say P **covers** Q.\nGiven this fact, the search for polytopes with particular facets and vertex figures usually goes as follows:\nThese two problems are, in general, very difficult.\nReturning 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}).\nThe 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.\nIf L is, instead, also a square, the universal polytope {K,L} (that is, {4,4}) is the tessellation of the Euclidean plane by squares.\nThis tessellation has infinitely many quotients with square faces, four per vertex, some regular and some not.\nExcept for the universal polytope itself, they all correspond to various ways to tessellate either a [torus](https://bliptext.com/articles/torus) or an infinitely long [cylinder](https://bliptext.com/articles/cylinder-geometry) with squares.\n\n## The 11-cell and the 57-cell\nThe [11-cell](https://bliptext.com/articles/11-cell), discovered independently by [H. S. M. Coxeter](https://bliptext.com/articles/h-s-m-coxeter) and [Branko Grünbaum](https://bliptext.com/articles/branko-gr-nbaum), is an abstract 4-polytope.\nIts facets are hemi-icosahedra.\nSince its facets are, topologically, projective planes instead of spheres, the 11-cell is not a tessellation of any manifold in the usual sense.\nInstead, the 11-cell is a locally projective polytope.\nIt is self-dual and universal: it is the only polytope with hemi-icosahedral facets and hemi-dodecahedral vertex figures.\nThe [57-cell](https://bliptext.com/articles/57-cell) is also self-dual, with hemi-dodecahedral facets.\nIt was discovered by H. S. M. Coxeter shortly after the discovery of the 11-cell.\nLike the 11-cell, it is also universal, being the only polytope with hemi-dodecahedral facets and hemi-icosahedral vertex figures.\nOn the other hand, there are many other polytopes with hemi-dodecahedral facets and Schläfli type {5,3,5}.\nThe 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.\n\n## Local topology\nThe amalgamation problem has, historically, been pursued according to local topology.\nThat is, rather than restricting K and L to be particular polytopes, they are allowed to be any polytope with a given [topology](https://bliptext.com/articles/topology), that is, any polytope [tessellating](https://bliptext.com/articles/tessellation) a given [manifold](https://bliptext.com/articles/manifold).\nIf K and L are spherical (that is, tessellations of a topological [sphere](https://bliptext.com/articles/sphere)), then P is called locally spherical and corresponds itself to a tessellation of some manifold.\nFor 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](https://bliptext.com/articles/torus) or [Klein bottle](https://bliptext.com/articles/klein-bottle) by squares.\nA tessellation of an n-dimensional manifold is actually a rank n + 1 polytope.\nThis is in keeping with the common intuition that the [Platonic solids](https://bliptext.com/articles/platonic-solid) are three dimensional, even though they can be regarded as tessellations of the two-dimensional surface of a ball.\nIn general, an abstract polytope is called locally X if its facets and vertex figures are, topologically, either spheres or X, but not both spheres.\nThe [11-cell](https://bliptext.com/articles/11-cell) and [57-cell](https://bliptext.com/articles/57-cell) are examples of rank 4 (that is, four-dimensional) locally projective polytopes, since their facets and vertex figures are tessellations of [real projective planes](https://bliptext.com/articles/real-projective-plane).\nThere is a weakness in this terminology however.\nIt does not allow an easy way to describe a polytope whose facets are [tori](https://bliptext.com/articles/torus) and whose vertex figures are projective planes, for example.\nWorse still if different facets have different topologies, or no well-defined topology at all.\nHowever, much progress has been made on the complete classification of the locally toroidal regular polytopes\n\n# Exchange maps\nLet Ψ be a flag of an abstract n-polytope, and let −1 < i < n.\nFrom 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.\nIf 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.\nSome other properties of the exchange maps :\nThe exchange maps and the flag action in particular can be used to prove that any abstract polytope is a quotient of some regular polytope.\n\n# Incidence matrices\nA polytope can also be represented by tabulating its [incidences](https://bliptext.com/articles/incidence-matrix).\nThe following incidence matrix is that of a triangle:\nThe table shows a 1 wherever a face is a subface of another, or vice versa (so the table is [symmetric](https://bliptext.com/articles/symmetric-matrix) 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.\nSince 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.\n\n## Square pyramid\nFurther information is gained by counting each occurrence.\nThis numerative usage enables a [symmetry](https://bliptext.com/articles/symmetry) grouping, as in the [Hasse Diagram](https://bliptext.com/articles/abstract-polytope) of the [square pyramid](https://bliptext.com/articles/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**.\nThus the corresponding incidence matrix of this abstract polytope may be shown as:\nIn this accumulated incidence matrix representation the diagonal entries represent the total counts of either element type.\nElements 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.\nThe 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.\nAlready this simple [square pyramid](https://bliptext.com/articles/square-pyramid) shows that the symmetry-accumulated incidence matrices are no longer symmetrical.\nBut 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_{ij}) holds:\n# [A square pyramid and the associated abstract polytope. | upload.wikimedia.org/wikipedia/commons/e/ed/Pyramid///abstract///polytope.svg]\n\n# History\nIn the 1960s [Branko Grünbaum](https://bliptext.com/articles/branko-gr-nbaum) issued a call to the geometric community to consider generalizations of the concept of [regular polytopes](https://bliptext.com/articles/regular-polytope) that he called polystromata.\nHe developed a theory of polystromata, showing examples of new objects including the [11-cell](https://bliptext.com/articles/11-cell).\nThe [11-cell](https://bliptext.com/articles/11-cell) is a [self-dual](https://bliptext.com/articles/dual-polytope) [4-polytope](https://bliptext.com/articles/4-polytope) whose [facets](https://bliptext.com/articles/facet-geometry) are not [icosahedra](https://bliptext.com/articles/icosahedron), but are \"hemi-[icosahedra](https://bliptext.com/articles/icosahedron)\" — that is, they are the shape one gets if one considers opposite faces of the [icosahedra](https://bliptext.com/articles/icosahedron) to be actually the same face (Grünbaum, 1977).\nA few years after Grünbaum's discovery of the [11-cell](https://bliptext.com/articles/11-cell), [H.S.M. Coxeter](https://bliptext.com/articles/h-s-m-coxeter) discovered a similar polytope, the [57-cell](https://bliptext.com/articles/57-cell) (Coxeter 1982, 1984), and then independently rediscovered the [11-cell](https://bliptext.com/articles/11-cell).\nWith the earlier work by [Branko Grünbaum](https://bliptext.com/articles/branko-gr-nbaum), [H. S. M. Coxeter](https://bliptext.com/articles/h-s-m-coxeter) and [Jacques Tits](https://bliptext.com/articles/jacques-tits) having laid the groundwork, the basic theory of the combinatorial structures now known as abstract polytopes was first described by [Egon Schulte](https://bliptext.com/articles/egon-schulte) in his 1980 PhD dissertation.\nIn it he defined \"regular incidence complexes\" and \"regular incidence polytopes\".\nSubsequently, he and [Peter McMullen](https://bliptext.com/articles/peter-mcmullen) developed the basics of the theory in a series of research articles that were later collected into a book.\nNumerous other researchers have since made their own contributions, and the early pioneers (including Grünbaum) have also accepted Schulte's definition as the \"correct\" one.\nSince then, research in the theory of abstr[act](https://bliptext.com/articles/group-action-mathematics) polytopes has focused mostly on regular polytopes, that is, those whose [automorphism](https://bliptext.com/articles/automorphism) [groups](https://bliptext.com/articles/group-mathematics) [act](https://bliptext.com/articles/group-action-mathematics) [transitively](https://bliptext.com/articles/group-action-mathematics) on the set of flags of the polytope.","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T21:13:46.777Z","updated_at":"2025-01-25T21:13:46.777Z","search_vector":"'abstract':1 'polytop':2","search_content":" \n \n\n# \nIn [mathematics](https://bliptext.com/articles/mathematics), an **abstract [polytope](https://bliptext.com/articles/polytope)** is an algebraic [partially ordered set](https://bliptext.com/articles/partially-ordered-set) which captures the dyadic property of a traditional [polytope](https://bliptext.com/articles/polytope) without specifying purely geometric properties such as points and lines.\nA geometric [polytope](https://bliptext.com/articles/polytope) is said to be a realization of an abstract [polytope](https://bliptext.com/articles/polytope) in some real [N-dimensional space](https://bliptext.com/articles/n-dimensional-space), typically [Euclidean](https://bliptext.com/articles/euclidean-space).\nThis abstract definition allows more general [combinatorial](https://bliptext.com/articles/combinatorics) structures than traditional definitions of a polytope, thus allowing new objects that have no counterpart in traditional theory.\n# [A square pyramid and the associated abstract polytope. | upload.wikimedia.org/wikipedia/commons/e/ed/Pyramid///abstract///polytope.svg]\n\n# Introductory concepts\n\n## Traditional versus abstract polytopes\nIn Euclidean geometry, two shapes that are not [similar](https://bliptext.com/articles/similar-geometry) can nonetheless share a common structure.\nFor example, a [square](https://bliptext.com/articles/square) and a [trapezoid](https://bliptext.com/articles/trapezoid) both comprise an alternating chain of four [vertices](https://bliptext.com/articles/vertex-geometry) and four sides, which makes them [quadrilaterals](https://bliptext.com/articles/quadrilaterals).\nThey are said to be [isomorphic](https://bliptext.com/articles/isomorphic) or “structure preserving”.\nThis 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.\nThe measurable properties of traditional polytopes such as angles, edge-lengths, skewness, straightness and convexity have no meaning for an abstract polytope.\nWhat is true for traditional polytopes (also called classical or geometric polytopes) may not be so for abstract ones, and vice versa.\nFor 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.\n# [Isomorphic quadrilaterals. | upload.wikimedia.org/wikipedia/commons/f/f4/Isomorphic///Tetragons.svg]\n\n### Realizations\nA traditional polytope is said to be a realization of the associated abstract polytope.\nA realization is a mapping or injection of the abstract object into a real space, typically [Euclidean](https://bliptext.com/articles/euclidean-space), to construct a traditional polytope as a real geometric figure.\nThe six quadrilaterals shown are all distinct realizations of the abstract quadrilateral, each with different geometric properties.\nSome of them do not conform to traditional definitions of a quadrilateral and are said to be unfaithful realizations.\nA conventional polytope is a faithful realization.\n\n## Faces, ranks and ordering\nIn an abstract polytope, each structural element (vertex, edge, cell, etc.) is associated with a corresponding member of the set.\nThe 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.\nThe faces are ranked according to their associated real dimension: vertices have rank 0, edges rank 1 and so on.\nIncident faces of different ranks, for example, a vertex F of an edge G, are ordered by the relation F G.\nFor ex**a**mple, {**ø**, **a**, **a**b, **a**bc} is **a** fl**a**g in the tri**a**ngle **a**bc.\nFor a given polytope, all flags contain the same number of faces.\nOther posets do not, in general, satisfy this requirement.\n\n## Sections\nAny subset P' of a poset P is a poset (with the same relation <, restricted to P').\nIn an abstract polytope, given any two faces F, H of P with F ≤ H, the set {G | F ≤ G ≤ H} is called a **section** of P, and denoted H/F.\n(In order theory, a section is called a [closed interval](https://bliptext.com/articles/partially-ordered-set) of the poset and denoted [F, H].\nFor example, in the prism **abc**xyz**** (see diagram) the section **xyz**/**ø** (highlighted green) is the triangle\nA k-section is a section of rank k.\nP is thus a section of itself.\nThis concept of section does not have the same meaning as in traditional geometry.\n# [The graph (left) and Hasse Diagram of a triangular prism, showing a 1-section ( red ), and a 2-section ( green ). | upload.wikimedia.org/wikipedia/commons/8/86/Triangular///3-Prism.PNG]\n\n### Facets\nThe **facet** for a given j-face F is the (j−1)-section F/∅, where Fj is the greatest face.\nFor example, in the triangle ******ab****c**, the facet at ****ab**** is ****ab****/**∅** = {**∅**, a, b, ****ab****}, which is a line segment.\nThe distinction between F and F/∅ is not usually significant and the two are often treated as identical.\n\n### Vertex figures\nThe **[vertex figure](https://bliptext.com/articles/vertex-figure)** at a given vertex V is the (n−1)-section Fn/V, where Fn is the greatest face.\nFor example, in the triangle **a****b****c**, the vertex figure at ****b**** is **a****b****c**/****b**** = {****b****, a****b****, ****b****c, **a****b****c**}, which is a line segment.\nThe vertex figures of a cube are triangles.\n\n### Connectedness\nA 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\nsuch that F = H1, G = Hk, and each Hi, i < k, is incident with its successor.\nThe above condition ensures that a pair of disjoint triangles **abc** and **xyz** is not a (single) polytope.\nA poset P is **strongly connected** if every section of P (including P itself) is connected.\nWith this additional requirement, two pyramids that share just a vertex are also excluded.\nHowever, two square pyramids, for example, can, be \"glued\" at their square faces - giving an octahedron.\nThe \"common face\" is not then a face of the octahedron.\n\n# Formal definition\nAn **abstract polytope** is a [partially ordered set](https://bliptext.com/articles/partially-ordered-set), whose elements we call faces, satisfying the 4 axioms:\nAn n-polytope is a polytope of rank n.\nThe abstract polytope associated with a real [convex polytope](https://bliptext.com/articles/convex-polytope) is also referred to as its **[face lattice](https://bliptext.com/articles/convex-polytope)**.\n\n# The simplest polytopes\n\n## Rank < 1\nThere is just one poset for each rank −1 and 0.\nThese are, respectively, the null face and the point.\nThese are not always considered to be valid abstract polytopes.\n\n## Rank 1: the line segment\nThere is only one polytope of rank 1, which is the line segment.\nIt has a least face, just two 0-faces and a greatest face, for example {ø, **a, b, ab**}.\nIt follows th**a**t the vertices **a** **a**nd **b** h**a**ve r**a**nk 0, **a**nd th**a**t the gre**a**test f**a**ce **a****b**, **a**nd therefore the poset, **b**oth h**a**ve r**a**nk 1.\n# [The graph (left) and Hasse Diagram of a line segment | upload.wikimedia.org/wikipedia/commons/d/dd/An///Edge///%28Line///Segment%29///and///its///Hasse///Diagram.PNG]\n\n## Rank 2: polygons\nFor each p, 3 ≤ p < \\infty, we have (the abstract equivalent of) the traditional polygon with p vertices and p edges, or a p-gon.\nFor p = 3, 4, 5, ... we have the triangle, square, pentagon, ....\nFor p = 2, we have the [digon](https://bliptext.com/articles/digon), and p = \\infty we get the [apeirogon](https://bliptext.com/articles/apeirogon).\n\n### The digon\nA [digon](https://bliptext.com/articles/digon) is a polygon with just 2 edges.\nUnlike any other polygon, both edges have the same two vertices.\nFor this reason, it is degenerate in the [Euclidean plane](https://bliptext.com/articles/euclidean-plane).\nF**a****c**es **a**re sometimes des**c**ri**b**ed using \"vertex not**a**tion\" - e.g. {**ø**, **a**, **b**, **c**, **a****b**, **a****c**, **b****c**, **a****b****c**} for the tri**a**ngle **a****b****c**.\nThis method has the advantage of implying the **<** relation.\nWith the digon this vertex notation cannot be used.\nIt is necessary to give the faces individual symbols and specify the subface pairs F < G.\nThus, **a** digon is defined **a**s **a** set {**ø**, **a**, **b**, E', E\", G} with the rel**a**tion **<** given **b**y\nwhere E' and E\" are the two edges, and G the greatest face.\nThis need to identify each element of the polytope with a unique symbol applies to many other abstract polytopes and is therefore common practice.\nA polytope can only be fully described using vertex notation if every face is incident with a unique set of vertices.\nA polytope having this property is said to be [atomistic](https://bliptext.com/articles/atom-order-theory).\n# [The graph (left) and Hasse Diagram of a digon | upload.wikimedia.org/wikipedia/commons/8/85/Digon///and///Hasse///Diagram.PNG]\n\n# Examples of higher rank\nThe set of j-faces (−1 ≤ j ≤ n) of a traditional n-polytope form an abstract n-polytope.\nThe concept of an abstract polytope is more general and also includes:\n\n## Hosohedra and hosotopes\nThe digon is generalized by the [hosohedron](https://bliptext.com/articles/hosohedron) and higher dimensional hosotopes, which can all be realized as [spherical polyhedra](https://bliptext.com/articles/spherical-polyhedra) – they tessellate the sphere.\n# [A hexagonal hosohedron, realized as a spherical polyhedron. | upload.wikimedia.org/wikipedia/commons/f/f9/Hexagonal///Hosohedron.svg]\n\n## Projective polytopes\nFour examples of non-traditional abstract polyhedra are the [Hemicube](https://bliptext.com/articles/hemi-cube-geometry) (shown), [Hemi-octahedron](https://bliptext.com/articles/hemi-octahedron), [Hemi-dodecahedron](https://bliptext.com/articles/hemi-dodecahedron), and the [Hemi-icosahedron](https://bliptext.com/articles/hemi-icosahedron).\nThese are the projective counterparts of the [Platonic solids](https://bliptext.com/articles/platonic-solid), and can be realized as (globally) [projective polyhedra](https://bliptext.com/articles/projective-polyhedra) – they tessellate the [real projective plane](https://bliptext.com/articles/real-projective-plane).\nThe 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.\n# [The Hemicube may be derived from a cube by identifying opposite vertices, edges, and faces. It has 4 vertices, 6 edges, and 3 faces. | upload.wikimedia.org/wikipedia/commons/8/8d/Hemicube.svg]\n\n# Duality\nEvery geometric polytope has a [dual](https://bliptext.com/articles/dual-polyhedron) twin.\nAbstractly, the dual is the same polytope but with the ranking reversed in order: the Hasse diagram differs only in its annotations.\nIn an n-polytope,\neach of the original k-faces maps to an (n − k − 1)-face in the dual.\nThus, for example, the n-face maps to the (−1)-face.\nThe dual of a dual is ([isomorphic](https://bliptext.com/articles/isomorphic) to) the original.\nA polytope is self-dual if it is the same as, i.e. isomorphic to, its dual.\nHence, the Hasse diagram of a self-dual polytope must be symmetrical about the horizontal axis half-way between the top and bottom.\nThe square pyramid in the example above is self-dual.\nThe vertex figure at a vertex V is the dual of the facet to which V maps in the dual polytope.\n\n# Abstract regular polytopes\nFormally, an abstract polytope is defined to be \"regular\" if its [automorphism group](https://bliptext.com/articles/automorphism-group) [acts](https://bliptext.com/articles/group-action-mathematics) transitively on the set of its flags.\nIn 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.\nWhen an abstract polytope is regular, its automorphism group is isomorphic to a quotient of a [Coxeter group](https://bliptext.com/articles/coxeter-group).\nAll polytopes of rank ≤ 2 are regular.\nThe most famous regular polyhedra are the five Platonic solids.\nThe hemicube (shown) is also regular.\nInformally, 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.\nFor 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.\nThis condition alone is sufficient to ensure that any regular abstract polytope has isomorphic regular (n−1)-faces and isomorphic regular vertex figures.\nThis is a weaker condition than regularity for traditional polytopes, in that it refers to the (combinatorial) automorphism group, not the (geometric) symmetry group.\nFor example, any abstract polygon is regular, since angles, edge-lengths, edge curvature, skewness etc. do not exist for abstract polytopes.\nThere are several other weaker concepts, some not yet fully standardized, such as [semi-regular](https://bliptext.com/articles/semiregular-polytope), [quasi-regular](https://bliptext.com/articles/quasiregular-polyhedron), [uniform](https://bliptext.com/articles/uniform-polytope), [chiral](https://bliptext.com/articles/chiral-polytope), and [Archimedean](https://bliptext.com/articles/archimedean-solid) that apply to polytopes that have some, but not all of their faces equivalent in each rank.\n\n# Realization\nA 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](https://bliptext.com/articles/isometry) permutations of V is called a realization of an abstract polytope.\nTwo realizations are called congruent if the natural bijection between their sets of vertices is induced by an isometry of their ambient Euclidean spaces.\nIf 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.\nIn general, only a restricted set of abstract polytopes of rank n may be realized faithfully in any given n-space.\nThe characterization of this effect is an outstanding problem.\nFor 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.\n\n## Moduli space\nThe 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.\nThe product of the two reflections can be decomposed as a product of a non-zero translation, finitely many rotations, and possibly trivial reflection.\nGenerally, the [moduli space](https://bliptext.com/articles/moduli-space) of realizations of an abstract polytope is a [convex cone](https://bliptext.com/articles/convex-cone) of infinite dimension.\nThe realization cone of the abstract polytope has uncountably infinite [algebraic dimension](https://bliptext.com/articles/algebraic-dimension) and cannot be [closed](https://bliptext.com/articles/closed-set) in the [Euclidean topology](https://bliptext.com/articles/euclidean-topology).\n\n# The amalgamation problem and universal polytopes\nAn important question in the theory of abstract polytopes is the amalgamation problem.\nThis is a series of questions such as\nFor example, if K is the square, and L is the triangle, the answers to these questions are\nIt 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.\nThat is, suppose P is the universal polytope with facets K and vertex figures L.\nThen any other polytope Q with these facets and vertex figures can be written Q=P/N, where\nQ=P/N is called a **quotient** of P, and we say P **covers** Q.\nGiven this fact, the search for polytopes with particular facets and vertex figures usually goes as follows:\nThese two problems are, in general, very difficult.\nReturning 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}).\nThe 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.\nIf L is, instead, also a square, the universal polytope {K,L} (that is, {4,4}) is the tessellation of the Euclidean plane by squares.\nThis tessellation has infinitely many quotients with square faces, four per vertex, some regular and some not.\nExcept for the universal polytope itself, they all correspond to various ways to tessellate either a [torus](https://bliptext.com/articles/torus) or an infinitely long [cylinder](https://bliptext.com/articles/cylinder-geometry) with squares.\n\n## The 11-cell and the 57-cell\nThe [11-cell](https://bliptext.com/articles/11-cell), discovered independently by [H. S. M. Coxeter](https://bliptext.com/articles/h-s-m-coxeter) and [Branko Grünbaum](https://bliptext.com/articles/branko-gr-nbaum), is an abstract 4-polytope.\nIts facets are hemi-icosahedra.\nSince its facets are, topologically, projective planes instead of spheres, the 11-cell is not a tessellation of any manifold in the usual sense.\nInstead, the 11-cell is a locally projective polytope.\nIt is self-dual and universal: it is the only polytope with hemi-icosahedral facets and hemi-dodecahedral vertex figures.\nThe [57-cell](https://bliptext.com/articles/57-cell) is also self-dual, with hemi-dodecahedral facets.\nIt was discovered by H. S. M. Coxeter shortly after the discovery of the 11-cell.\nLike the 11-cell, it is also universal, being the only polytope with hemi-dodecahedral facets and hemi-icosahedral vertex figures.\nOn the other hand, there are many other polytopes with hemi-dodecahedral facets and Schläfli type {5,3,5}.\nThe 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.\n\n## Local topology\nThe amalgamation problem has, historically, been pursued according to local topology.\nThat is, rather than restricting K and L to be particular polytopes, they are allowed to be any polytope with a given [topology](https://bliptext.com/articles/topology), that is, any polytope [tessellating](https://bliptext.com/articles/tessellation) a given [manifold](https://bliptext.com/articles/manifold).\nIf K and L are spherical (that is, tessellations of a topological [sphere](https://bliptext.com/articles/sphere)), then P is called locally spherical and corresponds itself to a tessellation of some manifold.\nFor 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](https://bliptext.com/articles/torus) or [Klein bottle](https://bliptext.com/articles/klein-bottle) by squares.\nA tessellation of an n-dimensional manifold is actually a rank n + 1 polytope.\nThis is in keeping with the common intuition that the [Platonic solids](https://bliptext.com/articles/platonic-solid) are three dimensional, even though they can be regarded as tessellations of the two-dimensional surface of a ball.\nIn general, an abstract polytope is called locally X if its facets and vertex figures are, topologically, either spheres or X, but not both spheres.\nThe [11-cell](https://bliptext.com/articles/11-cell) and [57-cell](https://bliptext.com/articles/57-cell) are examples of rank 4 (that is, four-dimensional) locally projective polytopes, since their facets and vertex figures are tessellations of [real projective planes](https://bliptext.com/articles/real-projective-plane).\nThere is a weakness in this terminology however.\nIt does not allow an easy way to describe a polytope whose facets are [tori](https://bliptext.com/articles/torus) and whose vertex figures are projective planes, for example.\nWorse still if different facets have different topologies, or no well-defined topology at all.\nHowever, much progress has been made on the complete classification of the locally toroidal regular polytopes\n\n# Exchange maps\nLet Ψ be a flag of an abstract n-polytope, and let −1 < i < n.\nFrom 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.\nIf 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.\nSome other properties of the exchange maps :\nThe exchange maps and the flag action in particular can be used to prove that any abstract polytope is a quotient of some regular polytope.\n\n# Incidence matrices\nA polytope can also be represented by tabulating its [incidences](https://bliptext.com/articles/incidence-matrix).\nThe following incidence matrix is that of a triangle:\nThe table shows a 1 wherever a face is a subface of another, or vice versa (so the table is [symmetric](https://bliptext.com/articles/symmetric-matrix) 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.\nSince 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.\n\n## Square pyramid\nFurther information is gained by counting each occurrence.\nThis numerative usage enables a [symmetry](https://bliptext.com/articles/symmetry) grouping, as in the [Hasse Diagram](https://bliptext.com/articles/abstract-polytope) of the [square pyramid](https://bliptext.com/articles/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**.\nThus the corresponding incidence matrix of this abstract polytope may be shown as:\nIn this accumulated incidence matrix representation the diagonal entries represent the total counts of either element type.\nElements 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.\nThe 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.\nAlready this simple [square pyramid](https://bliptext.com/articles/square-pyramid) shows that the symmetry-accumulated incidence matrices are no longer symmetrical.\nBut 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_{ij}) holds:\n# [A square pyramid and the associated abstract polytope. | upload.wikimedia.org/wikipedia/commons/e/ed/Pyramid///abstract///polytope.svg]\n\n# History\nIn the 1960s [Branko Grünbaum](https://bliptext.com/articles/branko-gr-nbaum) issued a call to the geometric community to consider generalizations of the concept of [regular polytopes](https://bliptext.com/articles/regular-polytope) that he called polystromata.\nHe developed a theory of polystromata, showing examples of new objects including the [11-cell](https://bliptext.com/articles/11-cell).\nThe [11-cell](https://bliptext.com/articles/11-cell) is a [self-dual](https://bliptext.com/articles/dual-polytope) [4-polytope](https://bliptext.com/articles/4-polytope) whose [facets](https://bliptext.com/articles/facet-geometry) are not [icosahedra](https://bliptext.com/articles/icosahedron), but are \"hemi-[icosahedra](https://bliptext.com/articles/icosahedron)\" — that is, they are the shape one gets if one considers opposite faces of the [icosahedra](https://bliptext.com/articles/icosahedron) to be actually the same face (Grünbaum, 1977).\nA few years after Grünbaum's discovery of the [11-cell](https://bliptext.com/articles/11-cell), [H.S.M. Coxeter](https://bliptext.com/articles/h-s-m-coxeter) discovered a similar polytope, the [57-cell](https://bliptext.com/articles/57-cell) (Coxeter 1982, 1984), and then independently rediscovered the [11-cell](https://bliptext.com/articles/11-cell).\nWith the earlier work by [Branko Grünbaum](https://bliptext.com/articles/branko-gr-nbaum), [H. S. M. Coxeter](https://bliptext.com/articles/h-s-m-coxeter) and [Jacques Tits](https://bliptext.com/articles/jacques-tits) having laid the groundwork, the basic theory of the combinatorial structures now known as abstract polytopes was first described by [Egon Schulte](https://bliptext.com/articles/egon-schulte) in his 1980 PhD dissertation.\nIn it he defined \"regular incidence complexes\" and \"regular incidence polytopes\".\nSubsequently, he and [Peter McMullen](https://bliptext.com/articles/peter-mcmullen) developed the basics of the theory in a series of research articles that were later collected into a book.\nNumerous other researchers have since made their own contributions, and the early pioneers (including Grünbaum) have also accepted Schulte's definition as the \"correct\" one.\nSince then, research in the theory of abstr[act](https://bliptext.com/articles/group-action-mathematics) polytopes has focused mostly on regular polytopes, that is, those whose [automorphism](https://bliptext.com/articles/automorphism) [groups](https://bliptext.com/articles/group-mathematics) [act](https://bliptext.com/articles/group-action-mathematics) [transitively](https://bliptext.com/articles/group-action-mathematics) on the set of flags of the polytope.","revisionCount":0},"hasMore":false,"nextCursor":null}