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
2260:
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
1830:
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
234:
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.
2380:
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
1751:
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.
2261:
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
617:
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
781:
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.
3123:
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.
1912:
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.
2872:
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.
1886:
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.
586:
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.
3172:
2202:
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.
1393:
1346:
1280:
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.
1936:
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.
227:
This common structure may be represented in an underlying abstract polytope, a purely algebraic partially ordered set which captures the pattern of connections (or
1710:
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.
1551:
This need to identify each element of the polytope with a unique symbol applies to many other abstract polytopes and is therefore common practice.
359:
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
1724:
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
1809:
All polytopes of rank ≤ 2 are regular. The most famous regular polyhedra are the five
Platonic solids. The hemicube (shown) is also regular.
1296:
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,
267:
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,
3362:
in his 1980 PhD dissertation. In it he defined "regular incidence complexes" and "regular incidence polytopes". Subsequently, he and
3731:
3358:
having laid the groundwork, the basic theory of the combinatorial structures now known as abstract polytopes was first described by
285:
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
247:
of the associated abstract polytope. A realization is a mapping or injection of the abstract object into a real space, typically
179:
structures than traditional definitions of a polytope, thus allowing new objects that have no counterpart in traditional theory.
100:
72:
3467:
808:
For a given polytope, all flags contain the same number of faces. Other posets do not, in general, satisfy this requirement.
3117:
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:
3384:
1419:
is a polygon with just 2 edges. Unlike any other polygon, both edges have the same two vertices. For this reason, it is
1675:
may be derived from a cube by identifying opposite vertices, edges, and faces. It has 4 vertices, 6 edges, and 3 faces.
53:
17:
3723:
119:
3327:" — that is, they are the shape one gets if one considers opposite faces of the icosahedra to be actually the
2349:
are three dimensional, even though they can be regarded as tessellations of the two-dimensional surface of a ball.
86:
1720:
300:, although it differs from traditional geometry and some other areas of mathematics. For example, in the square
759:† Traditionally "face" has meant a rank 2 face or 2-face. In abstract theory the term "face" denotes a face of
57:
68:
2164:
Given this fact, the search for polytopes with particular facets and vertex figures usually goes as follows:
2405:. From the definition of an abstract polytope, it can be proven that there is a unique flag differing from
619:
599:
3776:
3771:
1823:
This condition alone is sufficient to ensure that any regular abstract polytope has isomorphic regular (
3715:
3381:
1775:
884:
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.
3472:
1839:
46:
3135:
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.
1288:
1117:
if P has rank ≤ 1, or, given any two proper faces F and G, there is a sequence of proper faces
93:
1947:
of infinite dimension. The realization cone of the abstract polytope has uncountably infinite
1263:
2480:. (The action of this group on the flags of the polytope is an example of what is called the
2373:
1704:
1612:
1378:
1331:
1194:
332:
149:
1866:
in a
Euclidean space equipped with a surjection from the vertex set of an abstract polytope
1854:
that apply to polytopes that have some, but not all of their faces equivalent in each rank.
282:
according to their associated real dimension: vertices have rank 0, edges rank 1 and so on.
3688:
2888:
Further information is gained by counting each occurrence. This numerative usage enables a
2113:
2032:
1835:
1700:
1680:
1672:
1644:
8:
3736:
3378:
2477:
2222:
1948:
1656:
1559:
557:
168:
3132:
least whenever no holes or stars etc. are considered), as for any such incidence matrix
2021:
with square faces, joined three per vertex (that is, there are polytopes of type {4,3}).
271:
is used to refer to any such element e.g. a vertex (0-face), edge (1-face) or a general
3781:
3708:
3692:
3499:
3481:
3297:. He developed a theory of polystromata, showing examples of new objects including the
3289:
issued a call to the geometric community to consider generalizations of the concept of
1956:
1772:
832:
Any subset P' of a poset P is a poset (with the same relation <, restricted to P').
201:
3347:
3286:
2242:
3719:
3696:
2916:. Thus the corresponding incidence matrix of this abstract polytope may be shown as:
1851:
1688:
561:
373:
There is also a single face of which all the others are subfaces. This is called the
213:
3754:
3676:
3491:
3406:
3351:
3324:
3316:
3290:
2862:
2584:
2238:
1843:
1692:
3744:
3507:
3503:
1834:
There are several other weaker concepts, some not yet fully standardized, such as
296:
if either F = G or F < G or G < F. This usage of "incidence" also occurs in
3684:
1847:
1684:
1424:
1258:
773:
594:
297:
248:
172:
1932:
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:
3703:
3664:
3396:
3370:
Since then, research in the theory of abstract polytopes has focused mostly on
3363:
3336:
3128:
2897:
2346:
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
1696:
820:
The graph (left) and Hasse
Diagram of a triangular prism, showing a 1-section (
347:
Just as the number zero is necessary in mathematics, so also every set has the
217:
3495:
2571:
The exchange maps and the flag action in particular can be used to prove that
3765:
3309:
2281:
to be particular polytopes, they are allowed to be any polytope with a given
1803:
1048:
611:
603:
312:
are not abstractly incident (although they are both incident with vertex B).
176:
2880:
2253:
polytope with hemi-icosahedral facets and hemi-dodecahedral vertex figures.
610:
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
132:
3401:
3375:
3359:
3355:
2334:
2317:
and corresponds itself to a tessellation of some manifold. For example, if
2286:
1940:
1601:
2567:
is isomorphic to the automorphism group, otherwise, it is strictly larger.
1407:
1038:/∅ is not usually significant and the two are often treated as identical.
3320:
2869:; it would suffice to show only a 1 when the row face ≤ the column face.
1944:
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.
156:
without specifying purely geometric properties such as points and lines.
141:
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.
3312:
2356:
if its facets and vertex figures are, topologically, either spheres or
2194:} is the cube (also written {4,3}). The hemicube is the quotient {4,3}/
1952:
1909:-space. The characterization of this effect is an outstanding problem.
1745:
1652:
1640:
1635:
1597:
1593:
The concept of an abstract polytope is more general and also includes:
221:
192:
2372:
polytopes, since their facets and vertex figures are tessellations of
2269:
The amalgamation problem has, historically, been pursued according to
1234:
differ by 2, then there are exactly 2 faces that lie strictly between
3486:
3331:
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
1396:
816:
348:
209:
1348:, we have (the abstract equivalent of) the traditional polygon with
394:. It is sometimes realized as the interior of the geometric figure.
35:
2889:
2290:
2282:
1962:
1875:
681:
For some ranks, their face-types are named in the following table.
160:
153:
3271:{\displaystyle I_{ii}\cdot I_{ij}=I_{ji}\cdot I_{jj}\ \ (i<j).}
2031:, with six square faces, twelve edges and eight vertices, and the
1901:. In general, only a restricted set of abstract polytopes of rank
251:, to construct a traditional polytope as a real geometric figure.
3594:
3592:
3562:
3340:
3332:
3305:
3298:
2365:
2361:
2325:
are both squares (and so are topologically the same as circles),
2257:
2234:
1967:
An important question in the theory of abstract polytopes is the
1771:
Formally, an abstract polytope is defined to be "regular" if its
1623:
1619:
259:
realizations. A conventional polytope is a faithful realization.
1554:
A polytope can only be fully described using vertex notation if
2306:
1655:
and higher dimensional hosotopes, which can all be realized as
516:
The relation < comprises a set of pairs, which here include
205:
3604:
3589:
2249:
projective polytope. It is self-dual and universal: it is the
1430:
Faces are sometimes described using "vertex notation" - e.g. {
1151:
The above condition ensures that a pair of disjoint triangles
3538:
2857:
The table shows a 1 wherever a face is a subface of another,
2377:
2330:
2218:
1778:
transitively on the set of its flags. In particular, any two
1608:
1416:
1368:
1679:
Four examples of non-traditional abstract polyhedra are the
1548:
where E' and E" are the two edges, and G the greatest face.
3640:
3616:
2028:
1667:
583:
respectively, but such notation is not always appropriate.
3579:
3577:
2575:
abstract polytope is a quotient of some regular polytope.
2452:
always. Some other properties of the exchange maps :
674:
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
1170:
if every section of P (including P itself) is connected.
667:
of any face. It is always the rank of the greatest face F
204:
can nonetheless share a common structure. For example, a
187:
3628:
3574:
2558:
If the polytope is regular, the group generated by the
1827:−1)-faces and isomorphic regular vertex figures.
3550:
3526:
2884:
A square pyramid and the associated abstract polytope.
2590:
The following incidence matrix is that of a triangle:
2413:
element, and the same otherwise. If we call this flag
136:
A square pyramid and the associated abstract polytope.
3182:
3138:
2583:
A polytope can also be represented by tabulating its
2059:
polytope with these facets and vertex figures, which
1816:, this means that there is no way to distinguish any
1381:
1334:
3757:
and O'Rourke, J., 2nd Ed., Chapman & Hall, 2004.
3749:
Schulte, E.; "Symmetry of polytopes and polyhedra",
3468:"On the Complexity of Polytope Isomorphism Problems"
2209:
is, instead, also a square, the universal polytope {
2175:
These two problems are, in general, very difficult.
2079:
with these facets and vertex figures can be written
2013:
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
1292:
The graph (left) and Hasse
Diagram of a line segment
397:
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
3763:
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
3702:
3667:(1994), "Realizations of regular apeirotopes",
3646:
3610:
3598:
3568:
3544:
3520:
3514:
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
3465:
2228:
1943:of realizations of an abstract polytope is a
1766:
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
3466:Kaibel, Volker; Schwartz, Alexander (2003).
2124:, with the partial order induced by that of
1148:, i < k, is incident with its successor.
262:
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
1283:
342:
212:both comprise an alternating chain of four
1630:
1565:
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:
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 {
1270:
606:of a quadrilateral, showing ranks (right)
120:Learn how and when to remove this message
3663:
3634:
3622:
3583:
3556:
3532:
2879:
1971:. This is a series of questions such as
1905:may be realized faithfully in any given
1666:
1634:
1406:
1308:have rank 0, and that the greatest face
1287:
815:
593:
275:-face, and not just a polygonal 2-face.
191:
182:
131:
3449:
3447:
3445:
2024:Yes, they are all finite, specifically,
1662:
243:A traditional polytope is said to be a
14:
3764:
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.
2578:
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
3442:
1699:, and can be realized as (globally)
1477:With the digon this vertex notation
1184:
902:(highlighted green) is the triangle
589:
58:adding citations to reliable sources
29:
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
1315:
408:
24:
2171:Attempt to classify its quotients.
1759:is the dual of the facet to which
1382:
1335:
1224:
25:
3793:
3706:; Schulte, Egon (December 2002),
2875:
2337:by squares. A tessellation of an
2264:
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.
1041:
216:and four sides, which makes them
3374:polytopes, that is, those whose
2384:
1651:The digon is generalized by the
1618:Many other objects, such as the
1300:}. It follows that the vertices
1108:
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
1915:
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.
238:
45:needs additional citations for
3459:
3431:
3419:
3262:
3250:
3161:
3145:
2237:, discovered independently by
1857:
1755:The vertex figure at a vertex
1659:– they tessellate the sphere.
1275:
13:
1:
3656:
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
1402:
1027:}, which is a line segment.
801:} is a flag in the triangle
7:
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
3390:
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 .
811:
634:of a face F is defined as (
224:or “structure preserving”.
10:
3798:
3716:Cambridge University Press
3710:Abstract Regular Polytopes
3280:
3167:{\displaystyle I=(I_{ij})}
2075:. Then any other polytope
1983:, are there any polytopes
1812:Informally, for each rank
1767:Abstract regular polytopes
1713:
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
966:
263:Faces, ranks and ordering
3669:Aequationes Mathematicae
3473:Graphs and Combinatorics
3412:
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
766:
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
625:
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:
2918:
2593:
2592:
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:
683:
416:
415:
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
3665:McMullen, Peter
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:
3464:
3460:
3452:
3443:
3436:
3432:
3424:
3420:
3415:
3393:
3348:Branko Grünbaum
3325:hemi-icosahedra
3293:that he called
3287:Branko Grünbaum
3283:
3235:
3231:
3219:
3215:
3203:
3199:
3187:
3183:
3181:
3178:
3177:
3152:
3148:
3137:
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:
1380:
1377:
1376:
1333:
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:
3785:
3784:
3779:
3774:
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:
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:
3648:
3643:
3636:
3635:McMullen 1994
3631:
3624:
3623:McMullen 1994
3619:
3612:
3607:
3600:
3595:
3593:
3585:
3584:McMullen 1994
3580:
3578:
3570:
3565:
3558:
3557:McMullen 1994
3553:
3546:
3541:
3534:
3533:McMullen 1994
3529:
3523:, p. 121
3522:
3517:
3509:
3505:
3501:
3497:
3493:
3488:
3483:
3479:
3475:
3474:
3469:
3462:
3455:
3450:
3448:
3446:
3439:
3434:
3427:
3422:
3418:
3408:
3405:
3403:
3400:
3398:
3395:
3394:
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
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:
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:
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
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:
2262:
2259:
2254:
2252:
2248:
2244:
2240:
2236:
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:
1960:
1958:
1954:
1950:
1946:
1942:
1937:
1935:
1931:
1927:
1923:
1913:
1910:
1908:
1904:
1900:
1896:
1892:
1887:
1885:
1881:
1877:
1873:
1869:
1865:
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:
1764:
1762:
1758:
1753:
1749:
1747:
1743:
1739:
1736: −
1735:
1731:
1727:
1723:
1722:
1711:
1708:
1706:
1702:
1698:
1694:
1690:
1686:
1682:
1674:
1669:
1660:
1658:
1654:
1646:
1642:
1637:
1625:
1621:
1617:
1614:
1610:
1606:
1603:
1602:tessellations
1599:
1596:
1595:
1594:
1591:
1589:
1585:
1581:
1577:
1574:-faces (−1 ≤
1573:
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:
1400:
1398:
1374:
1370:
1366:
1361:
1359:
1355:
1352:vertices and
1351:
1327:
1323:
1313:
1311:
1307:
1303:
1299:
1290:
1281:
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:
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
1106:
1104:
1100:
1096:
1092:
1088:
1083:
1080:
1076:
1072:
1067:
1063:
1059:
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1050:
1049:vertex figure
1039:
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846:
842:
838:
833:
818:
809:
806:
804:
800:
796:
792:
788:
783:
780:
777:is a maximal
776:
775:
764:
762:
752:
749:
746:
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741:
738:
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729:
726:
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710:
685:
682:
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672:
666:
665:
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645:
641:
637:
633:
623:
621:
615:
613:
612:Hasse diagram
605:
604:Hasse diagram
601:
596:
587:
584:
582:
578:
574:
570:
565:
563:
559:
551:
548:<Y, ... ,
547:
544:<G, ... ,
540:
537:<X, ... ,
533:
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194:
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177:combinatorial
174:
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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:
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3054:
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2402:
2398:
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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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