Abstract polytope - Knowledge (XXG)

2881: 133: 595: 1636: 193: 817: 36: 1408: 1668: 1289: 1820:-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. 3131:

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

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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

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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

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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.

2217:} (that is, {4,4}) 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 3366:

developed the basics of the theory in a series of research articles that were later collected into a book. Numerous other researchers have since made their own contributions, and the early pioneers (including Grünbaum) have also accepted Schulte's definition as the "correct" one.

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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

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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.

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other polytopes with hemi-dodecahedral facets and Schläfli type {5,3,5}. The universal polytope with hemi-dodecahedral facets and icosahedral (not hemi-icosahedral) vertex figures is finite, but very large, with 10006920 facets and half as many vertices.

2245:, is an abstract 4-polytope. Its facets are hemi-icosahedra. Since its facets are, topologically, projective planes instead of spheres, the 11-cell is not a tessellation of any manifold in the usual sense. Instead, the 11-cell is a 254:

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

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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

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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.

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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.

2900:: 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 614:, 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. 3276: 231:

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.

560:, 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 1897:-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 3120:

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.

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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.

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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.

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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.

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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.

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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.

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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.

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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.

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This common structure may be represented in an underlying abstract polytope, a purely algebraic partially ordered set which captures the pattern of connections (or

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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.

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This need to identify each element of the polytope with a unique symbol applies to many other abstract polytopes and is therefore common practice.

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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

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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

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All polytopes of rank ≤ 2 are regular. The most famous regular polyhedra are the five Platonic solids. The hemicube (shown) is also regular.

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There is only one polytope of rank 1, which is the line segment. It has a least face, just two 0-faces and a greatest face, for example {ø,

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

646:(F', F", ... , F) satisfying F' < F" < ... < F. F' is always the least face, F 3179: 1173:

With this additional requirement, two pyramids that share just a vertex are also excluded. However, two square pyramids, for example,

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in his 1980 PhD dissertation. In it he defined "regular incidence complexes" and "regular incidence polytopes". Subsequently, he and

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having laid the groundwork, the basic theory of the combinatorial structures now known as abstract polytopes was first described by

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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

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of the associated abstract polytope. A realization is a mapping or injection of the abstract object into a real space, typically

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structures than traditional definitions of a polytope, thus allowing new objects that have no counterpart in traditional theory.

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For a given polytope, all flags contain the same number of faces. Other posets do not, in general, satisfy this requirement.

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In this accumulated incidence matrix representation the diagonal entries represent the total counts of either element type.

2376:. There is a weakness in this terminology however. It does not allow an easy way to describe a polytope whose facets are 79: 1419:

is a polygon with just 2 edges. Unlike any other polygon, both edges have the same two vertices. For this reason, it is

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may be derived from a cube by identifying opposite vertices, edges, and faces. It has 4 vertices, 6 edges, and 3 faces.

3776: 3771: 3327:" — that is, they are the shape one gets if one considers opposite faces of the icosahedra to be actually the 1288: 3715: 3384: 3381: 2349:

are three dimensional, even though they can be regarded as tessellations of the two-dimensional surface of a ball.

1775: 86: 46: 300:, although it differs from traditional geometry and some other areas of mathematics. For example, in the square 3472: 759:† Traditionally "face" has meant a rank 2 face or 2-face. In abstract theory the term "face" denotes a face of 68: 2164:

Given this fact, the search for polytopes with particular facets and vertex figures usually goes as follows:

1839: 2405:. From the definition of an abstract polytope, it can be proven that there is a unique flag differing from 2373: 1823:

This condition alone is sufficient to ensure that any regular abstract polytope has isomorphic regular (

1704: 1612: 1194: 884: 332: 149: 2113: 2032: 1835: 1700: 1680: 1672: 1644: 370:≡ ∅ and the abstract polytope also contains the empty set as an element. It is not usually realized. 564:

of the other, i.e. where F < H and no G satisfies F < G < H.

3736: 3378: 2477: 2222: 1948: 1656: 1559: 557: 168: 594: 3781: 3692: 3499: 3481: 1956: 1772: 201: 3347: 3286: 3135: 2242: 1802:. When an abstract polytope is regular, its automorphism group is isomorphic to a quotient of a 1481:. It is necessary to give the faces individual symbols and specify the subface pairs F < G. 1117:

if P has rank ≤ 1, or, given any two proper faces F and G, there is a sequence of proper faces

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of infinite dimension. The realization cone of the abstract polytope has uncountably infinite

1851: 1688: 561: 213: 2480:. (The action of this group on the flags of the polytope is an example of what is called the 1378: 1331: 3754: 3676: 3491: 3406: 3351: 3324: 3316: 3290: 2862: 2584: 2238: 1866:

in a Euclidean space equipped with a surjection from the vertex set of an abstract polytope

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that apply to polytopes that have some, but not all of their faces equivalent in each rank.

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according to their associated real dimension: vertices have rank 0, edges rank 1 and so on.

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Further information is gained by counting each occurrence. This numerative usage enables a

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least whenever no holes or stars etc. are considered), as for any such incidence matrix

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with square faces, joined three per vertex (that is, there are polytopes of type {4,3}).

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is used to refer to any such element e.g. a vertex (0-face), edge (1-face) or a general

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issued a call to the geometric community to consider generalizations of the concept of

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Any subset P' of a poset P is a poset (with the same relation <, restricted to P').

217: 3765: 3696: 3309: 2916:. Thus the corresponding incidence matrix of this abstract polytope may be shown as: 1803: 1048: 611: 603: 373:

There is also a single face of which all the others are subfaces. This is called the

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There are several other weaker concepts, some not yet fully standardized, such as

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if either F = G or F < G or G < F. This usage of "incidence" also occurs in

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is generated by two reflections, the product of which translates each vertex of

1177:, be "glued" at their square faces - giving an octahedron. The "common face" is 778: 643: 413:

The faces of the abstract quadrilateral or square are shown in the table below:

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Since then, research in the theory of abstract polytopes has focused mostly on

2345: + 1 polytope. This is in keeping with the common intuition that the 2039:

It is known that if the answer to the first question is 'Yes' for some regular

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The graph (left) and Hasse Diagram of a triangular prism, showing a 1-section (

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Just as the number zero is necessary in mathematics, so also every set has the

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The exchange maps and the flag action in particular can be used to prove that

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to be particular polytopes, they are allowed to be any polytope with a given

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are not abstractly incident (although they are both incident with vertex B).

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polytope with hemi-icosahedral facets and hemi-dodecahedral vertex figures.

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

351:∅ as a subset. In an abstract polytope ∅ is by convention identified as the 348: 209: 2317:

and corresponds itself to a tessellation of some manifold. For example, if

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is isomorphic to the automorphism group, otherwise, it is strictly larger.

1038:/∅ is not usually significant and the two are often treated as identical. 2889: 2869:; it would suffice to show only a 1 when the row face ≤ the column face. 2290: 2282: 1875: 1794:-polytope are "the same", i.e. that there is an automorphism which maps 1105:}, which is a line segment. The vertex figures of a cube are triangles. 160: 156:

without specifying purely geometric properties such as points and lines.

153: 1360:-gon. For p = 3, 4, 5, ... we have the triangle, square, pentagon, .... 3680: 3343:(Coxeter 1982, 1984), and then independently rediscovered the 11-cell. 3340: 3332: 3305: 3298: 2365: 2361: 2356:

if its facets and vertex figures are, topologically, either spheres or

2257: 2234: 2194:} is the cube (also written {4,3}). The hemicube is the quotient {4,3}/ 1909:-space. The characterization of this effect is an outstanding problem. 1623: 1619: 1593:

The concept of an abstract polytope is more general and also includes:

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polytopes, since their facets and vertex figures are tessellations of

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The amalgamation problem has, historically, been pursued according to

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differ by 2, then there are exactly 2 faces that lie strictly between

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face (Grünbaum, 1977). A few years after Grünbaum's discovery of the

2417:, then this defines a collection of maps on the polytopes flags, say 2306: 205: 1667: 1348:, we have (the abstract equivalent of) the traditional polygon with 394:. It is sometimes realized as the interior of the geometric figure. 681:

For some ranks, their face-types are named in the following table.

3271:{\displaystyle I_{ii}\cdot I_{ij}=I_{ji}\cdot I_{jj}\ \ (i<j).} 2377: 2330: 2218: 2031:, with six square faces, twelve edges and eight vertices, and the 1901:. In general, only a restricted set of abstract polytopes of rank 1634: 1608: 1416: 1368: 251:, to construct a traditional polytope as a real geometric figure. 3594: 3592: 2325:

are both squares (and so are topologically the same as circles),

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An important question in the theory of abstract polytopes is the

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Formally, an abstract polytope is defined to be "regular" if its

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realizations. A conventional polytope is a faithful realization.

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A polytope can only be fully described using vertex notation if

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and higher dimensional hosotopes, which can all be realized as

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The relation < comprises a set of pairs, which here include

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projective polytope. It is self-dual and universal: it is the

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Faces are sometimes described using "vertex notation" - e.g. {

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The above condition ensures that a pair of disjoint triangles

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The table shows a 1 wherever a face is a subface of another,

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transitively on the set of its flags. In particular, any two

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Four examples of non-traditional abstract polyhedra are the

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where E' and E" are the two edges, and G the greatest face.

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respectively, but such notation is not always appropriate.

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abstract polytope is a quotient of some regular polytope.

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always. Some other properties of the exchange maps :

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The rank of a face or polytope usually corresponds to the

1740: − 1)-face in the dual. Thus, for example, the 1626:, that cannot be faithfully realized in Euclidean spaces. 1607:

Proper decompositions of unbounded manifolds such as the

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if every section of P (including P itself) is connected.

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of any face. It is always the rank of the greatest face F

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can nonetheless share a common structure. For example, a

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If the polytope is regular, the group generated by the

1827:−1)-faces and isomorphic regular vertex figures. 2884:

A square pyramid and the associated abstract polytope.

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The following incidence matrix is that of a triangle:

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element, and the same otherwise. If we call this flag

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A square pyramid and the associated abstract polytope.

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A polytope can also be represented by tabulating its

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polytope with these facets and vertex figures, which

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and O'Rourke, J., 2nd Ed., Chapman & Hall, 2004.

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Schulte, E.; "Symmetry of polytopes and polyhedra",

3468:"On the Complexity of Polytope Isomorphism Problems" 2209:

is, instead, also a square, the universal polytope {

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These two problems are, in general, very difficult.

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with these facets and vertex figures can be written

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is the triangle, the answers to these questions are

1744:-face maps to the (−1)-face. The dual of a dual is ( 1556:

every face is incident with a unique set of vertices

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The graph (left) and Hasse Diagram of a line segment

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These least and greatest faces are sometimes called

381:-dimensional polytope, the greatest face has rank = 152:

which captures the dyadic property of a traditional

2368:are examples of rank 4 (that is, four-dimensional) 2047:, then there is a unique polytope whose facets are 60:. Unsourced material may be challenged and removed. 27:

Poset representing certain properties of a polytope

3707: 3270: 3166: 1387: 1340: 963:have the same meaning as in traditional geometry. 2168:Attempt to find the applicable universal polytope 567:The edges W, X, Y and Z are sometimes written as 2035:, with three faces, six edges and four vertices. 1963:The amalgamation problem and universal polytopes 1558:. A polytope having this property is said to be 3751:Handbook of discrete and computational geometry 3667:(1994), "Realizations of regular apeirotopes", 3646: 3610: 3598: 3568: 3544: 3520: 3453: 3437: 3425: 2865:about the diagonal)- so in fact, the table has 1695:. These are the projective counterparts of the 1257:. The abstract polytope associated with a real 200:In Euclidean geometry, two shapes that are not 175:. This abstract definition allows more general 1943:of realizations of an abstract polytope is a 1411:The graph (left) and Hasse Diagram of a digon 1312:, and therefore the poset, both have rank 1. 1210: 1206: 835:In an abstract polytope, given any two faces 315:A polytope is then defined as a set of faces 8: 3466:Kaibel, Volker; Schwartz, Alexander (2003). 2124:, with the partial order induced by that of 1148:, i < k, is incident with its successor. 2352:In general, an abstract polytope is called 2098:is a subgroup of the automorphism group of 2063:all other such polytopes. That is, suppose 212:both comprise an alternating chain of four 883:. (In order theory, a section is called a 678:of its counterpart in traditional theory. 3732:Jaron's World: Shapes in Other Dimensions 3485: 3235: 3219: 3203: 3187: 3181: 3152: 3137: 2535:is an automorphism of the polytope, then 2430:, since they swap pairs of flags : ( 2341:-dimensional manifold is actually a rank 2305:(that is, tessellations of a topological 2186:is the triangle, the universal polytope { 1380: 1333: 606:of a quadrilateral, showing ranks (right) 120:Learn how and when to remove this message 3634: 3622: 3583: 3556: 3532: 2918: 2879: 2592: 1971:. This is a series of questions such as 1905:may be realized faithfully in any given 1666: 1406: 1308:have rank 0, and that the greatest face 1287: 815: 683: 593: 415: 275:-face, and not just a polygonal 2-face. 191: 131: 3449: 3447: 3445: 3418: 2024:Yes, they are all finite, specifically, 243:A traditional polytope is said to be a 2893: 2067:is the universal polytope with facets 642:is the maximum number of faces in any 3387:on the set of flags of the polytope. 2397:-polytope, and let −1 < 2329:will be a tessellation of the plane, 1600:or infinite polytopes, which include 188:Traditional versus abstract polytopes 167:of an abstract polytope in some real 7: 1699:, and can be realized as (globally) 1477:With the digon this vertex notation 902:(highlighted green) is the triangle 58:adding citations to reliable sources 3339:discovered a similar polytope, the 2273:. That is, rather than restricting 2178:Returning to the example above, if 1484:Thus, a digon is defined as a set { 1466:. This method has the advantage of 2171:Attempt to classify its quotients. 1759:is the dual of the facet to which 1382: 1335: 1224: 25: 3706:; Schulte, Egon (December 2002), 2337:by squares. A tessellation of an 2001:What finite ones are there ? 1998:If so, are they all finite ? 1728:-polytope, each of the original 1220:contain the same number of faces. 216:and four sides, which makes them 3374:polytopes, that is, those whose 1651:The digon is generalized by the 1618:Many other objects, such as the 1300:}. It follows that the vertices 34: 3110: 3107: 3085: 3082: 3042: 3039: 3019: 3016: 2990: 2987: 2967: 2964: 2850: 2818: 2815: 2812: 2789: 2786: 2783: 2760: 2757: 2754: 2722: 2719: 2716: 2693: 2690: 2687: 2664: 2661: 2658: 2626: 1924:of symmetries of a realization 1718:Every geometric polytope has a 1496:, E', E", G} with the relation 1181:then a face of the octahedron. 956:P is thus a section of itself. 45:needs additional citations for 3262: 3250: 3161: 3145: 2237:, discovered independently by 1755:The vertex figure at a vertex 1659:– they tessellate the sphere. 1: 2484:of the group on the polytope) 2051:and whose vertex figures are 1991:and whose vertex figures are 1975:For given abstract polytopes 1085:For example, in the triangle 1007:For example, in the triangle 622:representation of polytopes. 401:faces, with all others being 2360:, but not both spheres. The 1027:}, which is a line segment. 801:} is a flag in the triangle 3647:McMullen & Schulte 2002 3611:McMullen & Schulte 2002 3599:McMullen & Schulte 2002 3569:McMullen & Schulte 2002 3545:McMullen & Schulte 2002 3521:McMullen & Schulte 2002 3454:McMullen & Schulte 2002 3438:McMullen & Schulte 2002 3426:McMullen & Schulte 2002 2229:The 11-cell and the 57-cell 1870:such that automorphisms of 1763:maps in the dual polytope. 1586:-polytope form an abstract 1261:is also referred to as its 1201:, satisfying the 4 axioms: 890:For example, in the prism 887:of the poset and denoted . 634:of a face F is defined as ( 224:or “structure preserving”. 3798: 3716:Cambridge University Press 3710:Abstract Regular Polytopes 3167:{\displaystyle I=(I_{ij})} 2075:. Then any other polytope 1983:, are there any polytopes 1812:Informally, for each rank 1767:Abstract regular polytopes 1230:If the ranks of two faces 1217: 894:(see diagram) the section 196:Isomorphic quadrilaterals. 3496:10.1007/s00373-002-0503-y 3346:With the earlier work by 2393:be a flag of an abstract 2017:Yes, there are polytopes 1893:-polytope is realized in 1540:<E", E'<G, E"<G} 1197:, whose elements we call 263:Faces, ranks and ordering 3669:Aequationes Mathematicae 3473:Graphs and Combinatorics 2426:. These maps are called 2285:, that is, any polytope 1928:of an abstract polytope 1284:Rank 1: the line segment 1030:The distinction between 959:This concept of section 719: 713: 707: 704: 701: 698: 695: 692: 689: 686: 657:of an abstract polytope 343:Least and greatest faces 1631:Hosohedra and hosotopes 1566:Examples of higher rank 1388:{\displaystyle \infty } 1341:{\displaystyle \infty } 1089:, the vertex figure at 638: − 2), where 319:with an order relation 3743:Dr. Richard Klitzing, 3272: 3168: 2885: 2374:real projective planes 2221:or an infinitely long 1703:– they tessellate the 1676: 1648: 1412: 1389: 1342: 1293: 1271:The simplest polytopes 1253:is a polytope of rank 1082:is the greatest face. 1004:is the greatest face. 829: 607: 385:and may be denoted as 220:. They are said to be 197: 137: 3273: 3169: 2883: 2867:redundant information 2112:is the collection of 1705:real projective plane 1670: 1638: 1613:real projective plane 1410: 1390: 1343: 1291: 1195:partially ordered set 1163:a (single) polytope. 949:is a section of rank 819: 597: 333:partially ordered set 331:) will be a (strict) 195: 183:Introductory concepts 150:partially ordered set 135: 3180: 3136: 3127:Already this simple 2892:grouping, as in the 2120:under the action of 1969:amalgamation problem 1701:projective polyhedra 1663:Projective polytopes 1645:spherical polyhedron 1379: 1332: 824:), and a 2-section ( 661:is the maximum rank 556:Order relations are 552:<G, ... , Z<G. 292:F, G are said to be 54:improve this article 3625:, pp. 229–230. 3571:, pp. 140–141. 2464:is the identity map 2182:is the square, and 2071:and vertex figures 2009:is the square, and 1949:algebraic dimension 1732:-faces maps to an ( 1657:spherical polyhedra 1582:) of a traditional 1462:} for the triangle 169:N-dimensional space 69:"Abstract polytope" 3777:Incidence geometry 3772:Algebraic topology 3745:Incidence Matrices 3681:10.1007/BF01832961 3268: 3164: 2886: 2579:Incidence matrices 2370:locally projective 1957:Euclidean topology 1773:automorphism group 1748:to) the original. 1677: 1649: 1413: 1385: 1338: 1294: 1225:strongly connected 1168:strongly connected 1060:−1)-section 1052:at a given vertex 830: 608: 198: 138: 18:Abstract polyhedra 3291:regular polytopes 3249: 3246: 3115: 3114: 2861:(so the table is 2855: 2854: 2315:locally spherical 1987:whose facets are 1689:Hemi-dodecahedron 1367:= 2, we have the 1191:abstract polytope 1185:Formal definition 757: 756: 747:Subfacet or ridge 590:The Hasse diagram 514: 513: 146:abstract polytope 130: 129: 122: 104: 16:(Redirected from 3789: 3728: 3714:(1st ed.), 3713: 3699: 3675:(2–3): 223–239, 3650: 3644: 3638: 3632: 3626: 3620: 3614: 3608: 3602: 3596: 3587: 3581: 3572: 3566: 3560: 3554: 3548: 3542: 3536: 3530: 3524: 3518: 3512: 3511: 3506:. Archived from 3489: 3463: 3457: 3451: 3440: 3435: 3429: 3423: 3407:Regular polytope 3352:H. S. M. Coxeter 3277: 3275: 3274: 3269: 3247: 3244: 3243: 3242: 3227: 3226: 3211: 3210: 3195: 3194: 3173: 3171: 3170: 3165: 3160: 3159: 2919: 2593: 2401: < 2239:H. S. M. Coxeter 2005:For example, if 1878:permutations of 1862:A set of points 1693:Hemi-icosahedron 1643:, realized as a 1394: 1392: 1391: 1386: 1347: 1345: 1344: 1339: 1316:Rank 2: polygons 1205:It has just one 827: 823: 684: 416: 409:A simple example 163:is said to be a 148:is an algebraic 125: 118: 114: 111: 105: 103: 62: 38: 30: 21: 3797: 3796: 3792: 3791: 3790: 3788: 3787: 3786: 3762: 3761: 3760: 3726: 3704:McMullen, Peter 3702: 3665:McMullen, Peter 3663: 3659: 3654: 3653: 3645: 3641: 3633: 3629: 3621: 3617: 3609: 3605: 3597: 3590: 3582: 3575: 3567: 3563: 3555: 3551: 3543: 3539: 3531: 3527: 3519: 3515: 3465: 3464: 3460: 3452: 3443: 3436: 3432: 3424: 3420: 3415: 3393: 3348:Branko Grünbaum 3325:hemi-icosahedra 3293:that he called 3287:Branko Grünbaum 3283: 3231: 3215: 3199: 3183: 3178: 3177: 3148: 3134: 3133: 2924: A 2878: 2581: 2566: 2552: 2543: 2528: 2520: 2511: 2503: 2475: 2463: 2447: 2438: 2425: 2387: 2347:Platonic solids 2267: 2243:Branko Grünbaum 2231: 2116:of elements of 1965: 1939:Generally, the 1918: 1889:If an abstract 1860: 1769: 1716: 1697:Platonic solids 1685:Hemi-octahedron 1665: 1633: 1568: 1425:Euclidean plane 1405: 1377: 1376: 1330: 1329: 1318: 1286: 1278: 1273: 1259:convex polytope 1187: 1147: 1143: 1139: 1136:such that F = H 1132: 1128: 1124: 1111: 1081: 1068: 1044: 1011:, the facet at 1003: 969: 885:closed interval 825: 821: 814: 771:In geometry, a 769: 670: 649: 628: 592: 543: 536: 525: 454: 411: 393: 369: 365: 345: 298:finite geometry 265: 241: 190: 185: 126: 115: 109: 106: 63: 61: 51: 39: 28: 23: 22: 15: 12: 11: 5: 3795: 3793: 3785: 3784: 3779: 3774: 3764: 3763: 3759: 3758: 3755:Goodman, J. E. 3747: 3741: 3729: 3724: 3700: 3660: 3658: 3655: 3652: 3651: 3639: 3637:, p. 232. 3627: 3615: 3613:, p. 127. 3603: 3601:, p. 141. 3588: 3586:, p. 231. 3573: 3561: 3559:, p. 229. 3549: 3547:, p. 126. 3537: 3535:, p. 225. 3525: 3513: 3510:on 2015-07-21. 3480:(2): 215–230. 3458: 3441: 3430: 3417: 3416: 3414: 3411: 3410: 3409: 3404: 3399: 3397:Eulerian poset 3392: 3389: 3364:Peter McMullen 3337:H.S.M. Coxeter 3282: 3279: 3267: 3264: 3261: 3258: 3255: 3252: 3241: 3238: 3234: 3230: 3225: 3222: 3218: 3214: 3209: 3206: 3202: 3198: 3193: 3190: 3186: 3163: 3158: 3155: 3151: 3147: 3144: 3141: 3129:square pyramid 3113: 3112: 3109: 3106: 3103: 3100: 3097: 3094: 3088: 3087: 3084: 3081: 3078: 3075: 3072: 3069: 3051: 3050: 3047: 3044: 3041: 3038: 3035: 3032: 3028: 3027: 3024: 3021: 3018: 3015: 3012: 3009: 3005: 3004: 3001: 2998: 2995: 2992: 2989: 2986: 2982: 2981: 2978: 2975: 2972: 2969: 2966: 2963: 2959: 2958: 2951: 2934: 2931: 2928: 2925: 2922: 2898:square pyramid 2877: 2876:Square pyramid 2874: 2853: 2852: 2849: 2846: 2843: 2840: 2837: 2834: 2831: 2828: 2824: 2823: 2820: 2817: 2814: 2811: 2808: 2805: 2802: 2799: 2795: 2794: 2791: 2788: 2785: 2782: 2779: 2776: 2773: 2770: 2766: 2765: 2762: 2759: 2756: 2753: 2750: 2747: 2744: 2741: 2737: 2736: 2733: 2730: 2727: 2724: 2721: 2718: 2715: 2712: 2708: 2707: 2704: 2701: 2698: 2695: 2692: 2689: 2686: 2683: 2679: 2678: 2675: 2672: 2669: 2666: 2663: 2660: 2657: 2654: 2650: 2649: 2646: 2643: 2640: 2637: 2634: 2631: 2628: 2625: 2621: 2620: 2617: 2614: 2611: 2608: 2605: 2602: 2599: 2596: 2580: 2577: 2569: 2568: 2562: 2556: 2548: 2539: 2529: 2524: 2516: 2507: 2499: 2485: 2471: 2465: 2459: 2443: 2434: 2421: 2386: 2383: 2271:local topology 2266: 2265:Local topology 2263: 2230: 2227: 2225:with squares. 2173: 2172: 2169: 2130: 2129: 2103: 2037: 2036: 2025: 2022: 2003: 2002: 1999: 1996: 1964: 1961: 1951:and cannot be 1917: 1914: 1859: 1856: 1768: 1765: 1715: 1712: 1664: 1661: 1632: 1629: 1628: 1627: 1616: 1605: 1567: 1564: 1546: 1545: 1544: 1543: 1542: 1541: 1479:cannot be used 1404: 1401: 1384: 1337: 1317: 1314: 1285: 1282: 1277: 1274: 1272: 1269: 1244: 1243: 1228: 1221: 1214: 1186: 1183: 1145: 1141: 1137: 1134: 1133: 1130: 1126: 1122: 1110: 1107: 1103:b, ab, bc, abc 1077: 1064: 1043: 1042:Vertex figures 1040: 999: 968: 965: 940: 939: 875:, and denoted 867:} is called a 813: 810: 785:For example, { 768: 765: 755: 754: 751: 748: 745: 743: 740: 737: 734: 731: 728: 724: 723: 718: 712: 706: 703: 700: 697: 694: 691: 688: 668: 647: 627: 624: 591: 588: 554: 553: 541: 534: 523: 512: 511: 508: 505: 502: 498: 497: 494: 491: 488: 484: 483: 466: 463: 460: 456: 455: 452: 447: 444: 441: 437: 436: 430: 427: 420: 410: 407: 389: 367: 363: 344: 341: 278:The faces are 264: 261: 240: 237: 218:quadrilaterals 189: 186: 184: 181: 128: 127: 42: 40: 33: 26: 24: 14: 13: 10: 9: 6: 4: 3: 2: 3794: 3783: 3780: 3778: 3775: 3773: 3770: 3769: 3767: 3756: 3752: 3748: 3746: 3742: 3739: 3738: 3737:Discover mag. 3733: 3730: 3727: 3725:0-521-81496-0 3721: 3717: 3712: 3711: 3705: 3701: 3698: 3694: 3690: 3686: 3682: 3678: 3674: 3670: 3666: 3662: 3661: 3656: 3648: 3643: 3640: 3636: 3635:McMullen 1994 3631: 3628: 3624: 3623:McMullen 1994 3619: 3616: 3612: 3607: 3604: 3600: 3595: 3593: 3589: 3585: 3584:McMullen 1994 3580: 3578: 3574: 3570: 3565: 3562: 3558: 3557:McMullen 1994 3553: 3550: 3546: 3541: 3538: 3534: 3533:McMullen 1994 3529: 3526: 3523:, p. 121 3522: 3517: 3514: 3509: 3505: 3501: 3497: 3493: 3488: 3483: 3479: 3475: 3474: 3469: 3462: 3459: 3455: 3450: 3448: 3446: 3442: 3439: 3434: 3431: 3427: 3422: 3419: 3412: 3408: 3405: 3403: 3400: 3398: 3395: 3394: 3390: 3388: 3386: 3383: 3380: 3377: 3373: 3368: 3365: 3361: 3357: 3353: 3349: 3344: 3342: 3338: 3334: 3330: 3326: 3322: 3318: 3314: 3311: 3307: 3302: 3300: 3296: 3292: 3288: 3285:In the 1960s 3280: 3278: 3265: 3259: 3256: 3253: 3239: 3236: 3232: 3228: 3223: 3220: 3216: 3212: 3207: 3204: 3200: 3196: 3191: 3188: 3184: 3175: 3156: 3153: 3149: 3142: 3139: 3130: 3125: 3121: 3118: 3104: 3101: 3098: 3095: 3093: 3090: 3089: 3079: 3076: 3073: 3070: 3068: 3064: 3060: 3056: 3053: 3052: 3048: 3045: 3036: 3033: 3030: 3029: 3025: 3022: 3013: 3010: 3007: 3006: 3002: 2999: 2996: 2993: 2984: 2983: 2979: 2976: 2973: 2970: 2961: 2960: 2956: 2952: 2950: 2946: 2942: 2938: 2935: 2932: 2929: 2926: 2923: 2921: 2920: 2917: 2915: 2911: 2907: 2903: 2899: 2895: 2894:Hasse Diagram 2891: 2882: 2875: 2873: 2870: 2868: 2864: 2860: 2859:or vice versa 2847: 2844: 2841: 2838: 2835: 2832: 2829: 2826: 2825: 2821: 2809: 2806: 2803: 2800: 2797: 2796: 2792: 2780: 2777: 2774: 2771: 2768: 2767: 2763: 2751: 2748: 2745: 2742: 2739: 2738: 2734: 2731: 2728: 2725: 2713: 2710: 2709: 2705: 2702: 2699: 2696: 2684: 2681: 2680: 2676: 2673: 2670: 2667: 2655: 2652: 2651: 2647: 2644: 2641: 2638: 2635: 2632: 2629: 2623: 2622: 2618: 2615: 2612: 2609: 2606: 2603: 2600: 2597: 2595: 2594: 2591: 2588: 2586: 2578: 2576: 2574: 2565: 2561: 2557: 2555: 2551: 2547: 2542: 2538: 2534: 2530: 2527: 2523: 2519: 2515: 2510: 2506: 2502: 2498: 2494: 2491: − 2490: 2486: 2483: 2479: 2474: 2470: 2466: 2462: 2458: 2455: 2454: 2453: 2451: 2448: = 2446: 2442: 2437: 2433: 2429: 2428:exchange maps 2424: 2420: 2416: 2412: 2408: 2404: 2400: 2396: 2392: 2385:Exchange maps 2384: 2382: 2379: 2375: 2371: 2367: 2363: 2359: 2355: 2350: 2348: 2344: 2340: 2336: 2332: 2328: 2324: 2320: 2316: 2312: 2308: 2304: 2300: 2296: 2292: 2288: 2284: 2280: 2276: 2272: 2264: 2262: 2259: 2254: 2252: 2248: 2244: 2240: 2236: 2228: 2226: 2224: 2220: 2216: 2212: 2208: 2203: 2201: 2197: 2193: 2189: 2185: 2181: 2176: 2170: 2167: 2166: 2165: 2162: 2160: 2157: 2154: 2151:, and we say 2150: 2146: 2142: 2138: 2134: 2127: 2123: 2119: 2115: 2111: 2107: 2104: 2101: 2097: 2094: 2093: 2092: 2090: 2086: 2082: 2078: 2074: 2070: 2066: 2062: 2058: 2055:, called the 2054: 2050: 2046: 2042: 2034: 2030: 2027:There is the 2026: 2023: 2020: 2016: 2015: 2014: 2012: 2008: 2000: 1997: 1994: 1990: 1986: 1982: 1978: 1974: 1973: 1972: 1970: 1962: 1960: 1958: 1954: 1950: 1946: 1942: 1937: 1935: 1931: 1927: 1923: 1915: 1913: 1910: 1908: 1904: 1900: 1896: 1892: 1887: 1885: 1881: 1877: 1873: 1869: 1865: 1857: 1855: 1853: 1849: 1845: 1841: 1840:quasi-regular 1837: 1832: 1828: 1826: 1821: 1819: 1815: 1810: 1807: 1805: 1804:Coxeter group 1801: 1797: 1793: 1789: 1785: 1781: 1777: 1774: 1766: 1764: 1762: 1758: 1753: 1749: 1747: 1743: 1739: 1736: − 1735: 1731: 1727: 1723: 1722: 1713: 1711: 1708: 1706: 1702: 1698: 1694: 1690: 1686: 1682: 1674: 1669: 1662: 1660: 1658: 1654: 1646: 1642: 1637: 1630: 1625: 1621: 1617: 1614: 1610: 1606: 1603: 1602:tessellations 1599: 1596: 1595: 1594: 1591: 1589: 1585: 1581: 1577: 1574:-faces (−1 ≤ 1573: 1565: 1563: 1561: 1557: 1552: 1549: 1539: 1535: 1531: 1527: 1523: 1519: 1515: 1511: 1507: 1506: 1505: 1504: 1503: 1502: 1501: 1499: 1495: 1491: 1487: 1482: 1480: 1475: 1473: 1469: 1465: 1461: 1457: 1453: 1449: 1445: 1441: 1437: 1433: 1428: 1426: 1422: 1418: 1409: 1402: 1400: 1398: 1374: 1370: 1366: 1361: 1359: 1355: 1352:vertices and 1351: 1327: 1323: 1315: 1313: 1311: 1307: 1303: 1299: 1290: 1283: 1281: 1275: 1270: 1268: 1266: 1265: 1260: 1256: 1252: 1250: 1241: 1237: 1233: 1229: 1226: 1222: 1219: 1215: 1212: 1211:greatest face 1208: 1204: 1203: 1202: 1200: 1196: 1192: 1184: 1182: 1180: 1176: 1171: 1169: 1166:A poset P is 1164: 1162: 1158: 1154: 1149: 1120: 1119: 1118: 1116: 1113:A poset P is 1109:Connectedness 1108: 1106: 1104: 1100: 1096: 1092: 1088: 1083: 1080: 1076: 1072: 1067: 1063: 1059: 1055: 1051: 1050: 1049:vertex figure 1041: 1039: 1037: 1033: 1028: 1026: 1022: 1018: 1014: 1010: 1005: 1002: 998: 994: 990: 986: 982: 978: 974: 966: 964: 962: 957: 954: 952: 948: 946: 937: 933: 929: 925: 921: 917: 913: 909: 905: 904: 903: 901: 897: 893: 888: 886: 882: 878: 874: 870: 866: 862: 858: 854: 850: 846: 842: 838: 833: 818: 811: 809: 806: 804: 800: 796: 792: 788: 783: 780: 777:is a maximal 776: 775: 766: 764: 762: 752: 749: 746: 744: 741: 738: 735: 732: 729: 726: 725: 722: 716: 710: 685: 682: 679: 677: 672: 666: 665: 660: 656: 651: 645: 641: 637: 633: 625: 623: 621: 615: 613: 612:Hasse diagram 605: 604:Hasse diagram 601: 596: 589: 587: 584: 582: 578: 574: 570: 565: 563: 559: 551: 548:<Y, ... , 547: 544:<G, ... , 540: 537:<X, ... , 533: 529: 522: 519: 518: 517: 509: 506: 503: 500: 499: 495: 492: 489: 486: 485: 482: 478: 474: 470: 467: 464: 461: 458: 457: 451: 448: 445: 442: 439: 438: 434: 431: 428: 425: 421: 418: 417: 414: 408: 406: 404: 400: 395: 392: 388: 384: 380: 376: 371: 362: 358: 354: 350: 342: 340: 338: 334: 330: 326: 322: 318: 313: 311: 307: 303: 299: 295: 290: 288: 283: 281: 276: 274: 270: 262: 260: 258: 252: 250: 246: 238: 236: 232: 230: 225: 223: 219: 215: 211: 207: 203: 194: 187: 182: 180: 178: 177:combinatorial 174: 170: 166: 162: 157: 155: 151: 147: 143: 134: 124: 121: 113: 102: 99: 95: 92: 88: 85: 81: 78: 74: 71: – 70: 66: 65:Find sources: 59: 55: 49: 48: 43:This article 41: 37: 32: 31: 19: 3753:, edited by 3750: 3735: 3709: 3672: 3668: 3642: 3630: 3618: 3606: 3564: 3552: 3540: 3528: 3516: 3508:the original 3487:math/0106093 3477: 3471: 3461: 3433: 3421: 3402:Graded poset 3385:transitively 3376:automorphism 3371: 3369: 3360:Egon Schulte 3356:Jacques Tits 3345: 3328: 3303: 3295:polystromata 3294: 3284: 3176: 3126: 3122: 3119: 3116: 3091: 3066: 3062: 3058: 3054: 2954: 2948: 2944: 2940: 2936: 2913: 2909: 2905: 2901: 2887: 2871: 2866: 2858: 2856: 2589: 2582: 2572: 2570: 2563: 2559: 2553: 2549: 2545: 2540: 2536: 2532: 2525: 2521: 2517: 2513: 2508: 2504: 2500: 2496: 2492: 2488: 2481: 2472: 2468: 2460: 2456: 2449: 2444: 2440: 2435: 2431: 2427: 2422: 2418: 2414: 2410: 2406: 2402: 2398: 2394: 2390: 2388: 2369: 2357: 2353: 2351: 2342: 2338: 2335:Klein bottle 2326: 2322: 2318: 2314: 2310: 2302: 2298: 2294: 2287:tessellating 2278: 2274: 2270: 2268: 2255: 2250: 2246: 2232: 2214: 2210: 2206: 2204: 2199: 2195: 2191: 2187: 2183: 2179: 2177: 2174: 2163: 2158: 2155: 2152: 2148: 2144: 2143:is called a 2140: 2136: 2132: 2131: 2125: 2121: 2117: 2109: 2105: 2099: 2095: 2088: 2084: 2080: 2076: 2072: 2068: 2064: 2060: 2056: 2052: 2048: 2044: 2040: 2038: 2018: 2010: 2006: 2004: 1992: 1988: 1984: 1980: 1976: 1968: 1966: 1941:moduli space 1938: 1933: 1929: 1925: 1921: 1919: 1916:Moduli space 1911: 1906: 1902: 1898: 1894: 1890: 1888: 1883: 1882:is called a 1879: 1871: 1867: 1863: 1861: 1836:semi-regular 1833: 1829: 1824: 1822: 1817: 1813: 1811: 1808: 1799: 1795: 1791: 1787: 1783: 1779: 1770: 1760: 1756: 1754: 1750: 1741: 1737: 1733: 1729: 1725: 1719: 1717: 1709: 1678: 1650: 1639:A hexagonal 1592: 1587: 1583: 1579: 1575: 1571: 1569: 1555: 1553: 1550: 1547: 1537: 1533: 1529: 1525: 1521: 1517: 1513: 1509: 1497: 1493: 1489: 1485: 1483: 1478: 1476: 1471: 1467: 1463: 1459: 1455: 1451: 1447: 1443: 1439: 1435: 1431: 1429: 1420: 1414: 1372: 1364: 1362: 1357: 1356:edges, or a 1353: 1349: 1325: 1321: 1319: 1309: 1305: 1301: 1297: 1295: 1279: 1264:face lattice 1262: 1254: 1248: 1247: 1245: 1239: 1235: 1231: 1198: 1190: 1188: 1178: 1174: 1172: 1167: 1165: 1160: 1156: 1152: 1150: 1144:, and each H 1135: 1114: 1112: 1102: 1098: 1094: 1090: 1086: 1084: 1078: 1074: 1070: 1065: 1061: 1057: 1053: 1047: 1045: 1035: 1031: 1029: 1024: 1020: 1016: 1012: 1008: 1006: 1000: 996: 992: 988: 984: 980: 976: 975:for a given 972: 970: 960: 958: 955: 950: 944: 943: 941: 935: 931: 927: 923: 919: 915: 911: 907: 899: 895: 891: 889: 880: 876: 872: 868: 864: 860: 856: 852: 848: 844: 840: 836: 834: 831: 807: 802: 798: 794: 790: 786: 784: 772: 770: 760: 758: 720: 714: 708: 680: 675: 673: 663: 662: 658: 654: 652: 639: 635: 631: 629: 616: 609: 585: 580: 576: 572: 568: 566: 555: 549: 545: 538: 531: 527: 520: 515: 480: 476: 472: 468: 449: 432: 423: 412: 402: 398: 396: 390: 386: 382: 378: 377:face. In an 374: 372: 360: 356: 352: 346: 336: 328: 324: 323:. Formally, 320: 316: 314: 309: 305: 301: 293: 291: 286: 284: 279: 277: 272: 268: 266: 256: 253: 244: 242: 239:Realizations 233: 228: 226: 199: 171:, typically 164: 159:A geometric 158: 145: 139: 116: 107: 97: 90: 83: 76: 64: 52:Please help 47:verification 44: 3323:, but are " 2482:flag action 2476:generate a 1945:convex cone 1884:realization 1858:Realization 1852:Archimedean 1831:polytopes. 1598:Apeirotopes 1590:-polytope. 1570:The set of 1395:we get the 1276:Rank < 1 1025:∅, a, b, ab 851:, the set { 602:(left) and 496:W, X, Y, Z 245:realization 229:incidences) 165:realization 142:mathematics 3766:Categories 3740:, Apr 2007 3657:References 3321:icosahedra 3313:4-polytope 2585:incidences 2495:| > 1, 2409:by a rank 2313:is called 1920:The group 1746:isomorphic 1691:, and the 1653:hosohedron 1641:hosohedron 1474:relation. 1421:degenerate 1207:least face 995:/∅, where 991:)-section 843:of P with 727:Face Type 558:transitive 419:Face type 257:unfaithful 222:isomorphic 110:April 2016 80:newspapers 3782:Polytopes 3697:121616949 3310:self-dual 3229:⋅ 3197:⋅ 2863:symmetric 2354:locally X 2303:spherical 2057:universal 2033:hemi-cube 1876:isometric 1683:(shown), 1604:(tilings) 1560:atomistic 1532:<E", 1500:given by 1403:The digon 1397:apeirogon 1383:∞ 1336:∞ 1320:For each 1251:-polytope 1115:connected 753:Greatest 676:dimension 562:successor 349:empty set 249:Euclidean 210:trapezoid 173:Euclidean 3391:See also 3319:are not 3031:k,l,m,n 3008:f,g,h,j 2985:B,C,D,E 2890:symmetry 2309:), then 2291:manifold 2289:a given 2283:topology 2223:cylinder 2198:, where 2145:quotient 2091:, where 1899:faithful 1681:Hemicube 1673:Hemicube 1622:and the 1536:<E', 1528:<E', 1468:implying 1298:a, b, ab 1232:a > b 1209:and one 1129:, ... ,H 1073:, where 1056:is the ( 983:is the ( 961:does not 947:-section 812:Sections 530:, ... , 501:Greatest 399:improper 375:greatest 366:. Thus F 304:, edges 294:incident 214:vertices 161:polytope 154:polytope 3689:1268033 3456:, p. 23 3428:, p. 31 3372:regular 3341:57-cell 3333:11-cell 3306:11-cell 3299:11-cell 3281:History 3174:holds: 2957: 2953: 2933:k,l,m,n 2930:f,g,h,j 2927:B,C,D,E 2896:of the 2366:57-cell 2362:11-cell 2258:57-cell 2247:locally 2235:11-cell 1995: ? 1955:in the 1874:induce 1844:uniform 1782:-faces 1714:Duality 1624:57-cell 1620:11-cell 1423:in the 1140:, G = H 987:− 869:section 435:-faces 405:faces. 287:subface 202:similar 94:scholar 3722: 3695: 3687: 3504:179936 3502: 3379:groups 3317:facets 3315:whose 3248: 3245: 2912:, and 2307:sphere 2156:covers 2114:orbits 2061:covers 1953:closed 1850:, and 1848:chiral 1790:of an 1371:, and 1324:, 3 ≤ 1223:It is 979:-face 967:Facets 892:abcxyz 763:rank. 733:Vertex 579:, and 459:Vertex 429:Count 422:Rank ( 403:proper 327:(with 289:of G. 280:ranked 208:and a 206:square 96: 89: 82: 75: 67: 3693:S2CID 3500:S2CID 3482:arXiv 3413:Notes 3308:is a 2478:group 2331:torus 2293:. If 2219:torus 2102:, and 1609:torus 1417:digon 1369:digon 1328:< 1218:flags 1199:faces 1193:is a 973:facet 826:green 779:chain 767:Flags 750:Facet 730:Least 644:chain 620:graph 600:graph 440:Least 353:least 337:poset 335:, or 144:, an 101:JSTOR 87:books 3720:ISBN 3354:and 3329:same 3304:The 3257:< 2827:abc 2619:abc 2487:If | 2467:The 2389:Let 2378:tori 2364:and 2321:and 2301:are 2297:and 2277:and 2256:The 2251:only 2241:and 2233:The 2043:and 2029:cube 1979:and 1776:acts 1721:dual 1671:The 1520:< 1512:< 1498:< 1472:< 1470:the 1363:For 1304:and 1238:and 1216:All 1155:and 1046:The 1034:and 971:The 774:flag 742:Cell 736:Edge 687:Rank 655:rank 653:The 632:rank 630:The 626:Rank 598:The 526:< 487:Edge 357:null 329:< 321:< 308:and 302:ABCD 269:face 73:news 3677:doi 3492:doi 3382:act 2798:ca 2769:bc 2740:ab 2573:any 2531:If 2333:or 2205:If 2147:of 1798:to 1611:or 1464:abc 1460:abc 1246:An 1189:An 1179:not 1175:can 1161:not 1159:is 1157:xyz 1153:abc 1125:, H 1101:= { 1095:abc 1093:is 1087:abc 1023:= { 1015:is 1009:abc 936:xyz 896:xyz 871:of 822:red 803:abc 799:abc 761:any 717:- 1 711:- 2 705:... 355:or 140:In 56:by 3768:: 3734:, 3718:, 3691:, 3685:MR 3683:, 3673:47 3671:, 3591:^ 3576:^ 3498:. 3490:. 3478:19 3476:. 3470:. 3444:^ 3350:, 3335:, 3301:. 3111:1 3086:* 3049:1 3026:0 3003:1 2980:0 2962:A 2908:, 2904:, 2851:1 2822:1 2793:1 2764:1 2735:1 2711:c 2706:1 2682:b 2677:1 2653:a 2648:1 2624:ø 2616:ca 2613:bc 2610:ab 2587:. 2544:= 2537:αφ 2512:= 2432:Ψφ 2161:. 1959:. 1846:, 1842:, 1838:, 1806:. 1786:, 1707:. 1687:, 1578:≤ 1562:. 1524:, 1516:, 1492:, 1488:, 1458:, 1456:bc 1454:, 1452:ac 1450:, 1448:ab 1446:, 1442:, 1438:, 1434:, 1427:. 1415:A 1399:. 1375:= 1310:ab 1267:. 1017:ab 1013:ab 953:. 942:A 938:}. 934:, 932:yz 930:, 928:xz 926:, 924:xy 922:, 918:, 914:, 910:, 863:≤ 859:≤ 855:| 847:≤ 839:, 828:). 805:. 797:, 795:ab 793:, 789:, 690:-1 671:. 650:. 648:−1 581:cd 577:bc 575:, 573:ad 571:, 569:ab 542:−1 535:−1 524:−1 510:G 479:, 475:, 471:, 453:−1 443:−1 426:) 368:−1 364:−1 339:. 310:BC 306:AB 3679:: 3649:. 3494:: 3484:: 3266:. 3263:) 3260:j 3254:i 3251:( 3240:j 3237:j 3233:I 3224:i 3221:j 3217:I 3213:= 3208:j 3205:i 3201:I 3192:i 3189:i 3185:I 3162:) 3157:j 3154:i 3150:I 3146:( 3143:= 3140:I 3108:* 3105:4 3102:0 3099:4 3096:0 3092:T 3083:4 3080:1 3077:2 3074:2 3071:1 3067:S 3065:, 3063:R 3061:, 3059:Q 3057:, 3055:P 3046:1 3043:4 3040:* 3037:2 3034:0 3023:2 3020:* 3017:4 3014:1 3011:1 3000:2 2997:2 2994:1 2991:4 2988:* 2977:4 2974:0 2971:4 2968:* 2965:1 2955:T 2949:S 2947:, 2945:R 2943:, 2941:Q 2939:, 2937:P 2914:S 2910:R 2906:Q 2902:P 2848:1 2845:1 2842:1 2839:1 2836:1 2833:1 2830:1 2819:1 2816:0 2813:0 2810:1 2807:0 2804:1 2801:1 2790:0 2787:1 2784:0 2781:1 2778:1 2775:0 2772:1 2761:0 2758:0 2755:1 2752:0 2749:1 2746:1 2743:1 2732:1 2729:1 2726:0 2723:1 2720:0 2717:0 2714:1 2703:0 2700:1 2697:1 2694:0 2691:1 2688:0 2685:1 2674:1 2671:0 2668:1 2665:0 2662:0 2659:1 2656:1 2645:1 2642:1 2639:1 2636:1 2633:1 2630:1 2627:1 2607:c 2604:b 2601:a 2598:ø 2564:i 2560:φ 2554:α 2550:i 2546:φ 2541:i 2533:α 2526:i 2522:φ 2518:j 2514:φ 2509:j 2505:φ 2501:i 2497:φ 2493:j 2489:i 2473:i 2469:φ 2461:i 2457:φ 2450:Ψ 2445:i 2441:φ 2439:) 2436:i 2423:i 2419:φ 2415:Ψ 2411:i 2407:Ψ 2403:n 2399:i 2395:n 2391:Ψ 2358:X 2343:n 2339:n 2327:P 2323:L 2319:K 2311:P 2299:L 2295:K 2279:L 2275:K 2215:L 2213:, 2211:K 2207:L 2200:N 2196:N 2192:L 2190:, 2188:K 2184:L 2180:K 2159:Q 2153:P 2149:P 2141:N 2139:/ 2137:P 2135:= 2133:Q 2128:. 2126:P 2122:N 2118:P 2110:N 2108:/ 2106:P 2100:P 2096:N 2089:N 2087:/ 2085:P 2083:= 2081:Q 2077:Q 2073:L 2069:K 2065:P 2053:L 2049:K 2045:L 2041:K 2019:P 2011:L 2007:K 1993:L 1989:K 1985:P 1981:L 1977:K 1934:P 1930:P 1926:V 1922:G 1907:n 1903:n 1895:n 1891:n 1880:V 1872:P 1868:P 1864:V 1825:n 1818:k 1814:k 1800:G 1796:F 1792:n 1788:G 1784:F 1780:k 1761:V 1757:V 1742:n 1738:k 1734:n 1730:k 1726:n 1647:. 1615:. 1588:n 1584:n 1580:n 1576:j 1572:j 1538:b 1534:b 1530:a 1526:a 1522:b 1518:ø 1514:a 1510:ø 1508:{ 1494:b 1490:a 1486:ø 1444:c 1440:b 1436:a 1432:ø 1373:p 1365:p 1358:p 1354:p 1350:p 1326:p 1322:p 1306:b 1302:a 1255:n 1249:n 1242:. 1240:b 1236:a 1227:. 1213:. 1146:i 1142:k 1138:1 1131:k 1127:2 1123:1 1121:H 1099:b 1097:/ 1091:b 1079:n 1075:F 1071:V 1069:/ 1066:n 1062:F 1058:n 1054:V 1036:F 1032:F 1021:∅ 1019:/ 1001:j 997:F 993:F 989:1 985:j 981:F 977:j 951:k 945:k 920:z 916:y 912:x 908:ø 906:{ 900:ø 898:/ 881:F 879:/ 877:H 873:P 865:H 861:G 857:F 853:G 849:H 845:F 841:H 837:F 791:a 787:ø 739:† 721:n 715:n 709:n 702:3 699:2 696:1 693:0 669:n 664:n 659:P 640:m 636:m 550:c 546:b 539:F 532:F 528:a 521:F 507:1 504:2 493:4 490:1 481:d 477:c 473:b 469:a 465:4 462:0 450:F 446:1 433:k 424:k 391:n 387:F 383:n 379:n 361:F 325:P 317:P 273:k 123:) 117:( 112:) 108:( 98:· 91:· 84:· 77:· 50:. 20:)

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