70:
56:
63:
49:
77:
42:
2102:
95:
3103:
1978:
of elements or members, which obeys certain rules. It is a purely algebraic structure, and the theory was developed in order to avoid some of the issues which make it difficult to reconcile the various geometric classes within a consistent mathematical framework. A geometric polytope is said to be a
2363:
The conceptual issues raised by complex polytopes, non-convexity, duality and other phenomena led Grünbaum and others to the more general study of abstract combinatorial properties relating vertices, edges, faces and so on. A related idea was that of incidence complexes, which studied the incidence
102:
is a 2-dimensional polytope. Polygons can be characterised according to various criteria. Some examples are: open (excluding its boundary), bounding circuit only (ignoring its interior), closed (including both its boundary and its interior), and self-intersecting with varying densities of different
1966:
attempts to detach polytopes from the space containing them, considering their purely combinatorial properties. This allows the definition of the term to be extended to include objects for which it is difficult to define an intuitive underlying space, such as the
1231:
1096:
234:
is a broad term that covers a wide class of objects, and various definitions appear in the mathematical literature. Many of these definitions are not equivalent to each other, resulting in different overlapping sets of objects being called
2473:
is used in to calculate the scattering amplitudes of subatomic particles when they collide. The construct is purely theoretical with no known physical manifestation, but is said to greatly simplify certain calculations.
859:. The convex polytopes are the simplest kind of polytopes, and form the basis for several different generalizations of the concept of polytopes. A convex polytope is sometimes defined as the intersection of a set of
1753:
285:, with the additional property that, for any two simplices that have a nonempty intersection, their intersection is a vertex, edge, or higher dimensional face of the two. However this definition does not allow
481:
A polytope comprises elements of different dimensionality such as vertices, edges, faces, cells and so on. Terminology for these is not fully consistent across different authors. For example, some authors use
2113:
If a polytope has the same number of vertices as facets, of edges as ridges, and so forth, and the same connectivities, then the dual figure will be similar to the original and the polytope is self-dual.
938:
798:
of the original polytope. Every ridge arises as the intersection of two facets (but the intersection of two facets need not be a ridge). Ridges are once again polytopes whose facets give rise to (
2391:
using a computer in 1965; in higher dimensions this problem was still open as of 1997. The full enumeration for nonconvex uniform polytopes is not known in dimensions four and higher as of 2008.
1269:
1900:
2235:
discovered that two mirror-image solids can be superimposed by rotating one of them through a fourth mathematical dimension. By the 1850s, a handful of other mathematicians such as
1433:
2023:
1510:
Dimensions two, three and four include regular figures which have fivefold symmetries and some of which are non-convex stars, and in two dimensions there are infinitely many
1399:
1375:
1328:
1142:
999:
1147:
2083:
for regular polytopes, where the symbol for the dual polytope is simply the reverse of the original. For example, {4, 3, 3} is dual to {3, 3, 4}.
1118:
1024:
1846:
3007:
1029:
703:
681:
638:
616:
1780:
1632:
1302:
1800:
1605:
1349:
974:
254:
and others begins with the extension by analogy into four or more dimensions, of the idea of a polygon and polyhedron respectively in two and three dimensions.
2827:, p. 408. "There are also starry analogs of the Archimedean polyhedra...So far as we know, nobody has yet enumerated the analogs in four or higher dimensions."
1921:
Not all manifolds are finite. Where a polytope is understood as a tiling or decomposition of a manifold, this idea may be extended to infinite manifolds.
3034:
2495:
3902:
2267:-dimensional polytopes was made acceptable. Schläfli's polytopes were rediscovered many times in the following decades, even during his lifetime.
296:
and other unusual constructions led to the idea of a polyhedron as a bounding surface, ignoring its interior. In this light convex polytopes in
2905:
2668:
2197:
1640:
2056:-polytope has a dual structure, obtained by interchanging its vertices for facets, edges for ridges, and so on generally interchanging its (
2383:, convex and nonconvex, in four or more dimensions remains an outstanding problem. The convex uniform 4-polytopes were fully enumerated by
2993:
3337:
2659:
2451:
2615:
3137:
351:
The idea of constructing a higher polytope from those of lower dimension is also sometimes extended downwards in dimension, with an (
3087:
2941:
2717:
3924:
1807:
875:
if it contains at least one vertex. Every bounded nonempty polytope is pointed. An example of a non-pointed polytope is the set
878:
802: − 3)-dimensional boundaries of the original polytope, and so on. These bounding sub-polytopes may be referred to as
944:
if it is defined in terms of a finite number of objects, e.g., as an intersection of a finite number of half-planes. It is an
3027:
2887:
2311:
2086:
In the case of a geometric polytope, some geometric rule for dualising is necessary, see for example the rules described for
148:
is a 2-polytope and a three-dimensional polyhedron is a 3-polytope. In this context, "flat sides" means that the sides of a
2360:
in complex space, where each real dimension has an imaginary one associated with it. Coxeter developed the theory further.
1823:
1452:
have the highest degree of symmetry of all polytopes. The symmetry group of a regular polytope acts transitively on its
3122:
2805:, Egon Schulte. p. 12: "However, there are many more uniform polytopes but a complete list is known only for d = 4 ."
1549:
2709:
2427:
2399:
2232:
863:. This definition allows a polytope to be neither bounded nor finite. Polytopes are defined in this way, e.g., in
3020:
2500:
2484:
2403:
2250:
2172:
1236:
3057:
2892:
2823:
2394:
In modern times, polytopes and related concepts have found many important applications in fields as diverse as
2316:
2216:
2041:
2037:
1538:
1854:
1548:
include one additional convex solid with fourfold symmetry and two with fivefold symmetry. There are ten star
3330:
3265:
3260:
3240:
2505:
2353:
1941:
2364:
or connection of the various elements with one another. These developments led eventually to the theory of
2307:
in higher dimensions. Polytopes also began to be studied in non-Euclidean spaces such as hyperbolic space.
2249:
was the first to consider analogues of polygons and polyhedra in these higher spaces. He described the six
1819:
3250:
3245:
3225:
1120:
denotes a vector of all ones, and the inequality is component-wise. It follows from this definition that
370:
In certain fields of mathematics, the terms "polytope" and "polyhedron" are used in a different sense: a
3874:
3867:
3860:
3255:
3235:
3230:
2818:
2168:
1975:
1407:
860:
391:
69:
2093:
If the dual is reversed, then the original polytope is recovered. Thus, polytopes exist in dual pairs.
1999:
1226:{\displaystyle (t+1){\mathcal {P}}^{\circ }\cap \mathbb {Z} ^{d}=t{\mathcal {P}}\cap \mathbb {Z} ^{d}}
55:
3531:
3478:
2951:
2923:
2856:
2510:
2300:
2259:
2149:
2134:
2079:
For an abstract polytope, this simply reverses the ordering of the set. This reversal is seen in the
1926:
1612:
1471:
258:
1380:
1356:
1309:
1123:
980:
62:
48:
3886:
3785:
3535:
3132:
3127:
2466:
2443:
2336:
2138:
1475:
784:
751:
325:
184:
76:
1101:
1091:{\displaystyle {\mathcal {P}}=\{\mathbf {x} \in \mathbb {R} ^{d}:\mathbf {Ax} \leq \mathbf {1} \}}
1007:
41:
3919:
3755:
3705:
3655:
3612:
3582:
3542:
3505:
3323:
3306:
3147:
3102:
2897:
2846:
2814:
2431:
2281:
1608:
1545:
1503:
864:
828:
278:
219:
2915:
2771:
2340:
2246:
2208:
2130:
2080:
247:
195:
2996:– application of polytopes to a database of articles used to support custom news feeds via the
2276:
1828:
3894:
3142:
2976:
2937:
2901:
2757:
2713:
2664:
2611:
2435:
2411:
2395:
2365:
2324:
2240:
2193:
2189:
1963:
1957:
1930:
945:
562:
551:
364:
360:
188:
688:
666:
623:
601:
3898:
3463:
3452:
3441:
3430:
3421:
3412:
3399:
3377:
3365:
3351:
3347:
3072:
2927:
2864:
2783:
2380:
2357:
2345:
2254:
2033:
1988:
1449:
1444:
794:
733:
511:
301:
2101:
2090:. Depending on circumstance, the dual figure may or may not be another geometric polytope.
1758:
1617:
1275:
3488:
3473:
3117:
3062:
2959:
2933:
2489:
2418:
was discovered as a simplifying construct in certain calculations of theoretical physics.
2179:
2161:
1511:
1463:
There are three main classes of regular polytope which occur in any number of dimensions:
1453:
1402:
850:
803:
715:
592:
582:
572:
352:
345:
333:
321:
293:
240:
203:
121:
116:
2860:
3838:
3199:
3184:
2702:
2697:
2573:
2569:
2462:
2455:
2369:
2296:
2271:
2253:
in 1852 but his work was not published until 1901, six years after his death. By 1854,
2212:
2186:
2087:
1785:
1590:
1526:
1334:
1002:
959:
390:. With this terminology, a convex polyhedron is the intersection of a finite number of
386:
polyhedron. This terminology is typically confined to polytopes and polyhedra that are
309:
251:
211:
2299:
not only rediscovered Schläfli's regular polytopes but also investigated the ideas of
3913:
3855:
3743:
3736:
3729:
3693:
3686:
3679:
3643:
3636:
3360:
3189:
2470:
2415:
2236:
2157:
1994:
1567:
1561:
1457:
286:
129:
to any number of dimensions. Polytopes may exist in any general number of dimensions
2802:
1566:
A non-convex polytope may be self-intersecting; this class of polytopes include the
3795:
3209:
3174:
3067:
2304:
2285:
1922:
1530:
270:
176:
2263:
had firmly established the geometry of higher dimensions, and thus the concept of
277:. An example of this approach defines a polytope as a set of points that admits a
2076: − 1), while retaining the connectivity or incidence between elements.
3804:
3765:
3715:
3665:
3622:
3592:
3524:
3510:
3294:
3077:
2919:
2868:
2388:
2384:
1534:
395:
387:
383:
289:
with interior structures, and so is restricted to certain areas of mathematics.
94:
2280:
to refer to this more general concept of polygons and polyhedra. In due course
871:
if there is a ball of finite radius that contains it. A polytope is said to be
171:
Some theories further generalize the idea to include such objects as unbounded
3790:
3774:
3724:
3674:
3631:
3601:
3515:
3289:
3169:
2979:
2332:
1934:
1916:
833:
468:
458:
356:
341:
329:
269:
as analogous to a polytope. In this approach, a polytope may be regarded as a
266:
172:
126:
85:
2647:
The part of the polytope that lies in one of the hyperplanes is called a cell
2167:
Numerous compact, paracompact and noncompact hyperbolic tilings, such as the
3846:
3760:
3710:
3660:
3617:
3587:
3556:
3270:
3179:
3092:
3043:
2984:
2450:-dimensional polytope. In linear programming, polytopes occur in the use of
2407:
2153:
1945:
1495:
1481:
430:
420:
282:
31:
2788:
2660:
Computing the
Continuous Discretely: Integer-point enumeration in polyhedra
1522:≥ 5) star. But in higher dimensions there are no other regular polytopes.
17:
3575:
3571:
3498:
3194:
3082:
2997:
2683:
M. A. Perles and G. C. Shephard. 1967. "Angle sums of convex polytopes".
2328:
1552:, all with fivefold symmetry, giving in all sixteen regular 4-polytopes.
337:
274:
262:
180:
108:
3829:
3799:
3566:
3561:
3552:
3493:
3204:
2515:
2204:
2126:
1968:
1748:{\displaystyle \chi =n_{0}-n_{1}+n_{2}-\cdots \pm n_{d-1}=1+(-1)^{d-1}}
1467:
823:
448:
261:
of polyhedra to higher-dimensional polytopes led to the development of
145:
99:
3008:
Regular and semi-regular convex polytopes a short historical overview:
2610:
Nemhauser and Wolsey, "Integer and
Combinatorial Optimization," 1999,
1541:
with fivefold symmetry, bringing the total to nine regular polyhedra.
3769:
3719:
3669:
3626:
3596:
3547:
3483:
2439:
2142:
2106:
1499:
1485:
822:, and consists of a line segment. A 2-dimensional face consists of a
313:
1979:
realization in some real space of the associated abstract polytope.
818:, and consists of a single point. A 1-dimensional face is called an
363:
as a 0-polytope. This approach is used for example in the theory of
3161:
2958:, Graduate Texts in Mathematics, vol. 152, Berlin, New York:
2851:
93:
2837:
2663:, Undergraduate Texts in Mathematics, New York: Springer-Verlag,
2109:(4-simplex) is self-dual with 5 vertices and 5 tetrahedral cells.
194:
Polytopes of more than three dimensions were first discovered by
3519:
1489:
525:
The terms adopted in this article are given in the table below:
3016:
2320:, summarizing work to date and adding new findings of his own.
281:. In this definition, a polytope is the union of finitely many
1944:
and the infinite series of tilings represented by the regular
401:
Polytopes in lower numbers of dimensions have standard names:
2575:
Euler's Gem: The
Polyhedron Formula and the Birth of Topology
1377:
only by lattice points gained on the boundary. Equivalently,
398:
of a finite number of points and is defined by its vertices.
3012:
2228:
Polygons and polyhedra have been known since ancient times.
1414:
1386:
1362:
1315:
1203:
1169:
1129:
1035:
986:
788:. These facets are themselves polytopes, whose facets are (
394:
and is defined by its sides while a convex polytope is the
239:. They represent different approaches to generalizing the
933:{\displaystyle \{(x,y)\in \mathbb {R} ^{2}\mid x\geq 0\}}
125:). Polytopes are the generalization of three-dimensional
2368:
as partially ordered sets, or posets, of such elements.
2331:
idea of a polytope as the piecewise decomposition (e.g.
374:
is the generic object in any dimension (referred to as
1933:
are in this sense polytopes, and are sometimes called
1848:
for convex polyhedra to higher-dimensional polytopes:
1831:
36:
2231:
An early hint of higher dimensions came in 1827 when
2002:
1857:
1788:
1761:
1643:
1620:
1593:
1410:
1383:
1359:
1337:
1312:
1278:
1239:
1150:
1126:
1104:
1032:
1010:
983:
962:
881:
691:
669:
626:
604:
490: − 1)-dimensional element while others use
2776:
Biographical
Memoirs of Fellows of the Royal Society
2772:"John Horton Conway. 26 December 1937—11 April 2020"
1634:
of its boundary ∂P is given by the alternating sum:
1330:
differs, in terms of integer lattice points, from a
3282:
3218:
3156:
3110:
3050:
1993:Structures analogous to polytopes exist in complex
1940:Among these, there are regular forms including the
2701:
2017:
1894:
1840:
1794:
1774:
1747:
1626:
1599:
1427:
1393:
1369:
1343:
1322:
1296:
1263:
1225:
1136:
1112:
1090:
1018:
993:
968:
932:
697:
675:
632:
610:
243:to include other objects with similar properties.
214:, and was introduced to English mathematicians as
2152:, in any number of dimensions. These include the
948:if all of its vertices have integer coordinates.
778:-dimensional polytope is bounded by a number of (
494:to denote a 2-face specifically. Authors may use
2442:functions; these maxima and minima occur on the
2310:An important milestone was reached in 1948 with
826:, and a 3-dimensional face, sometimes called a
3331:
3028:
2555:
2553:
2551:
2549:
2543:, pp. 141–144, §7-x. Historical remarks.
1948:, square tiling, cubic honeycomb, and so on.
1822:similarly generalizes the alternating sum of
8:
2628:
2626:
2624:
1085:
1043:
927:
882:
3338:
3324:
3316:
3035:
3021:
3013:
2636:, Cambridge University Press, 2018, p.224.
2060: − 1)-dimensional elements for (
1264:{\displaystyle t\in \mathbb {Z} _{\geq 0}}
246:The original approach broadly followed by
2850:
2787:
2117:Some common self-dual polytopes include:
2009:
2005:
2004:
2001:
1937:because they have infinitely many cells.
1880:
1856:
1830:
1787:
1766:
1760:
1733:
1699:
1680:
1667:
1654:
1642:
1619:
1592:
1419:
1413:
1412:
1409:
1385:
1384:
1382:
1361:
1360:
1358:
1336:
1314:
1313:
1311:
1277:
1252:
1248:
1247:
1238:
1217:
1213:
1212:
1202:
1201:
1189:
1185:
1184:
1174:
1168:
1167:
1149:
1128:
1127:
1125:
1105:
1103:
1080:
1069:
1060:
1056:
1055:
1046:
1034:
1033:
1031:
1011:
1009:
985:
984:
982:
961:
909:
905:
904:
880:
814:-faces. A 0-dimensional face is called a
690:
668:
625:
603:
2687:, Vol 21, No 2. March 1967. pp. 199–218.
2679:
2677:
2496:Intersection of a polyhedron with a line
2100:
1895:{\displaystyle \sum \varphi =(-1)^{d-1}}
951:A certain class of convex polytopes are
527:
403:
359:bounded by a point pair, and a point or
265:and the treatment of a decomposition or
198:before 1853, who called such a figure a
3903:List of regular polytopes and compounds
2932:(2nd ed.), New York & London:
2540:
2533:
2414:and numerous other fields. In 2013 the
2243:had also considered higher dimensions.
1460:of a regular polytope is also regular.
308:, while others may be tilings of other
2758:John Horton Conway: Mathematical Magus
2657:Beck, Matthias; Robins, Sinai (2007),
2372:and Egon Schulte published their book
522: − 1)-dimensional element.
179:, decompositions or tilings of curved
2274:, writing in German, coined the word
2198:tetrahedrally diminished dodecahedron
1518:-fold symmetry, both convex and (for
7:
2323:Meanwhile, the French mathematician
2129:, in any number of dimensions, with
1570:. Some regular polytopes are stars.
1484:or measure polytopes, including the
2803:Symmetry of Polytopes and Polyhedra
2770:Curtis, Robert Turner (June 2022).
2452:generalized barycentric coordinates
2029:real dimensions are accompanied by
2343:published his influential work on
2040:are more appropriately treated as
1498:or cross polytopes, including the
1428:{\displaystyle {\mathcal {P}}^{*}}
25:
2700:; Schulte, Egon (December 2002),
1583:Since a (filled) convex polytope
502:-facet to indicate an element of
332:is understood as a surface whose
144:. For example, a two-dimensional
3101:
2018:{\displaystyle \mathbb {C} ^{n}}
1401:is reflexive if and only if its
1106:
1081:
1073:
1070:
1047:
1012:
344:as a hypersurface whose facets (
210:was coined by the mathematician
75:
68:
61:
54:
47:
40:
27:Geometric object with flat sides
2888:Coxeter, Harold Scott MacDonald
1537:, and there are also four star
1529:include the fivefold-symmetric
1525:In three dimensions the convex
348:) are polyhedra, and so forth.
273:or decomposition of some given
2839:Journal of High Energy Physics
2634:Geometries and Transformations
1877:
1867:
1730:
1720:
1394:{\displaystyle {\mathcal {P}}}
1370:{\displaystyle {\mathcal {P}}}
1323:{\displaystyle {\mathcal {P}}}
1291:
1279:
1163:
1151:
1137:{\displaystyle {\mathcal {P}}}
994:{\displaystyle {\mathcal {P}}}
897:
885:
840:Important classes of polytopes
1:
2578:. Princeton University Press.
1906:Generalisations of a polytope
1808:Euler's formula for polyhedra
2601:, CUP (ppbk 1999) pp 205 ff.
2288:, introduced the anglicised
2068:)-dimensional elements (for
1144:is reflexive if and only if
1113:{\displaystyle \mathbf {1} }
1019:{\displaystyle \mathbf {A} }
792: − 2)-dimensional
782: − 1)-dimensional
762:
744:
726:
708:
663:
643:
598:
588:
578:
568:
558:
546:
464:
454:
444:
436:
426:
416:
88:is a 3-dimensional polytope
2994:"Math will rock your world"
2492:-discrete oriented polytope
2292:into the English language.
510:to refer to a ridge, while
257:Attempts to generalise the
115:is a geometric object with
3941:
3892:
3319:
2710:Cambridge University Press
2704:Abstract Regular Polytopes
2645:Regular polytopes, p. 127
2374:Abstract Regular Polytopes
2251:convex regular 4-polytopes
1986:
1974:An abstract polytope is a
1955:
1914:
1841:{\textstyle \sum \varphi }
1559:
1442:
848:
29:
3303:
3099:
2733:Regular Complex Polytopes
2501:Extension of a polyhedron
2485:List of regular polytopes
2173:order-5 pentagonal tiling
2038:Regular complex polytopes
1550:Schläfli-Hess 4-polytopes
1435:is an integral polytope.
1001:is reflexive if for some
300:-space are equivalent to
160:-polytopes that may have
137:-dimensional polytope or
83:
2918:(2003), Kaibel, Volker;
2824:The Symmetries of Things
2469:, a polytope called the
2356:generalised the idea as
2217:grand stellated 120-cell
1539:Kepler-Poinsot polyhedra
652:-face – element of rank
279:simplicial decomposition
226:Approaches to definition
30:Not to be confused with
3925:Real algebraic geometry
2869:10.1007/JHEP10(2014)030
2354:Geoffrey Colin Shephard
2284:, daughter of logician
2233:August Ferdinand Möbius
2133:{3}. These include the
1544:In four dimensions the
955:polytopes. An integral
698:{\displaystyle \vdots }
676:{\displaystyle \vdots }
656:= −1, 0, 1, 2, 3, ...,
633:{\displaystyle \vdots }
611:{\displaystyle \vdots }
2789:10.1098/rsbm.2021.0034
2110:
2019:
1942:regular skew polyhedra
1896:
1842:
1796:
1776:
1749:
1628:
1601:
1429:
1395:
1371:
1345:
1324:
1298:
1265:
1227:
1138:
1114:
1092:
1020:
995:
970:
934:
810:-dimensional faces or
699:
677:
634:
612:
550:Nullity (necessary in
168:-polytopes in common.
104:
2956:Lectures on Polytopes
2819:Chaim Goodman-Strauss
2817:, Heidi Burgiel, and
2301:semiregular polytopes
2203:In 4 dimensions, the
2185:In 3 dimensions, the
2182:(regular 2-polytopes)
2178:In 2 dimensions, all
2169:icosahedral honeycomb
2104:
2020:
1976:partially ordered set
1897:
1843:
1797:
1777:
1775:{\displaystyle n_{j}}
1750:
1629:
1627:{\displaystyle \chi }
1602:
1430:
1396:
1372:
1346:
1325:
1299:
1297:{\displaystyle (t+1)}
1266:
1228:
1139:
1115:
1093:
1021:
996:
971:
935:
700:
678:
635:
613:
506:dimensions. Some use
378:in this article) and
156:-polytope consist of
97:
3219:Dimensions by number
3002:Business Week Online
2632:Johnson, Norman W.;
2511:Honeycomb (geometry)
2506:Polytope de Montréal
2260:Habilitationsschrift
2150:hypercubic honeycomb
2135:equilateral triangle
2000:
1855:
1829:
1786:
1759:
1641:
1618:
1613:Euler characteristic
1591:
1579:Euler characteristic
1472:equilateral triangle
1408:
1381:
1357:
1335:
1310:
1276:
1271:. In other words, a
1237:
1148:
1124:
1102:
1030:
1008:
981:
960:
879:
768:The polytope itself
689:
667:
624:
602:
259:Euler characteristic
187:, and set-theoretic
3887:pentagonal polytope
3786:Uniform 10-polytope
3346:Fundamental convex
2861:2014JHEP...10..030A
2467:theoretical physics
2139:regular tetrahedron
2097:Self-dual polytopes
1802:-dimensional faces.
1546:regular 4-polytopes
1476:regular tetrahedron
326:toroidal polyhedron
320:−1)-surfaces – see
230:Nowadays, the term
185:spherical polyhedra
3756:Uniform 9-polytope
3706:Uniform 8-polytope
3656:Uniform 7-polytope
3613:Uniform 6-polytope
3583:Uniform 5-polytope
3543:Uniform polychoron
3506:Uniform polyhedron
3354:in dimensions 2–10
3148:Degrees of freedom
3051:Dimensional spaces
2977:Weisstein, Eric W.
2952:Ziegler, Günter M.
2924:Ziegler, Günter M.
2898:Dover Publications
2815:John Horton Conway
2685:Math. Scandinavica
2432:linear programming
2366:abstract polytopes
2327:had developed the
2303:and space-filling
2282:Alicia Boole Stott
2211:{3,4,3}. Also the
2194:elongated pyramids
2190:polygonal pyramids
2111:
2072: = 1 to
2015:
1964:abstract polytopes
1952:Abstract polytopes
1931:hyperbolic tilings
1911:Infinite polytopes
1892:
1838:
1820:Gram–Euler theorem
1792:
1772:
1745:
1624:
1597:
1504:regular octahedron
1425:
1391:
1367:
1341:
1320:
1294:
1261:
1223:
1134:
1110:
1088:
1016:
991:
966:
930:
865:linear programming
855:A polytope may be
806:, or specifically
695:
673:
630:
608:
365:abstract polytopes
220:Alicia Boole Stott
189:abstract polytopes
105:
3908:
3907:
3895:Polytope families
3352:uniform polytopes
3314:
3313:
3123:Lebesgue covering
3088:Algebraic variety
2907:978-0-486-61480-9
2893:Regular Polytopes
2731:Coxeter, H.S.M.;
2669:978-0-387-29139-0
2618:, Definition 2.2.
2436:maxima and minima
2412:quantum mechanics
2396:computer graphics
2381:uniform polytopes
2358:complex polytopes
2317:Regular Polytopes
2241:Hermann Grassmann
1983:Complex polytopes
1958:Abstract polytope
1925:, space-filling (
1806:This generalizes
1795:{\displaystyle j}
1782:is the number of
1600:{\displaystyle d}
1450:Regular polytopes
1439:Regular polytopes
1344:{\displaystyle t}
969:{\displaystyle d}
946:integral polytope
772:
771:
474:
473:
292:The discovery of
92:
91:
16:(Redirected from
3932:
3899:Regular polytope
3460:
3449:
3438:
3397:
3340:
3333:
3326:
3317:
3111:Other dimensions
3105:
3073:Projective space
3037:
3030:
3023:
3014:
2990:
2989:
2962:
2946:
2929:Convex polytopes
2916:Grünbaum, Branko
2910:
2873:
2872:
2854:
2834:
2828:
2812:
2806:
2800:
2794:
2793:
2791:
2767:
2761:
2760:- Richard K. Guy
2755:
2749:
2742:
2736:
2729:
2723:
2722:
2708:(1st ed.),
2707:
2694:
2688:
2681:
2672:
2655:
2649:
2643:
2637:
2630:
2619:
2608:
2602:
2595:
2589:
2586:
2580:
2579:
2566:
2560:
2557:
2544:
2538:
2426:In the field of
2379:Enumerating the
2346:Convex Polytopes
2312:H. S. M. Coxeter
2255:Bernhard Riemann
2180:regular polygons
2081:Schläfli symbols
2024:
2022:
2021:
2016:
2014:
2013:
2008:
1989:Complex polytope
1901:
1899:
1898:
1893:
1891:
1890:
1847:
1845:
1844:
1839:
1801:
1799:
1798:
1793:
1781:
1779:
1778:
1773:
1771:
1770:
1754:
1752:
1751:
1746:
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1710:
1709:
1685:
1684:
1672:
1671:
1659:
1658:
1633:
1631:
1630:
1625:
1611:to a point, the
1606:
1604:
1603:
1598:
1512:regular polygons
1470:, including the
1445:Regular polytope
1434:
1432:
1431:
1426:
1424:
1423:
1418:
1417:
1400:
1398:
1397:
1392:
1390:
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1368:
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1352:
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1295:
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1232:
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1193:
1188:
1179:
1178:
1173:
1172:
1143:
1141:
1140:
1135:
1133:
1132:
1119:
1117:
1116:
1111:
1109:
1097:
1095:
1094:
1089:
1084:
1076:
1065:
1064:
1059:
1050:
1039:
1038:
1025:
1023:
1022:
1017:
1015:
1000:
998:
997:
992:
990:
989:
977:
975:
973:
972:
967:
940:. A polytope is
939:
937:
936:
931:
914:
913:
908:
867:. A polytope is
845:Convex polytopes
832:, consists of a
704:
702:
701:
696:
682:
680:
679:
674:
639:
637:
636:
631:
617:
615:
614:
609:
528:
512:H. S. M. Coxeter
486:to refer to an (
404:
302:tilings of the (
241:convex polytopes
167:
159:
155:
141:
136:
132:
79:
72:
65:
58:
51:
44:
37:
21:
3940:
3939:
3935:
3934:
3933:
3931:
3930:
3929:
3910:
3909:
3878:
3871:
3864:
3747:
3740:
3733:
3697:
3690:
3683:
3647:
3640:
3474:Regular polygon
3467:
3458:
3451:
3447:
3440:
3436:
3427:
3418:
3411:
3407:
3395:
3389:
3385:
3373:
3355:
3344:
3315:
3310:
3299:
3278:
3214:
3152:
3106:
3097:
3063:Euclidean space
3046:
3041:
2975:
2974:
2971:
2966:
2960:Springer-Verlag
2950:
2944:
2934:Springer-Verlag
2914:
2908:
2886:
2882:
2877:
2876:
2836:
2835:
2831:
2813:
2809:
2801:
2797:
2769:
2768:
2764:
2756:
2752:
2744:Wenninger, M.;
2743:
2739:
2730:
2726:
2720:
2698:McMullen, Peter
2696:
2695:
2691:
2682:
2675:
2656:
2652:
2644:
2640:
2631:
2622:
2609:
2605:
2596:
2592:
2588:Grünbaum (2003)
2587:
2583:
2568:
2567:
2563:
2558:
2547:
2539:
2535:
2530:
2525:
2520:
2490:Bounding volume
2480:
2456:slack variables
2424:
2341:Branko Grünbaum
2247:Ludwig Schläfli
2226:
2209:Schläfli symbol
2162:cubic honeycomb
2131:Schläfli symbol
2099:
2050:
2003:
1998:
1997:
1991:
1985:
1960:
1954:
1919:
1913:
1908:
1876:
1853:
1852:
1827:
1826:
1824:internal angles
1816:
1814:Internal angles
1784:
1783:
1762:
1757:
1756:
1729:
1695:
1676:
1663:
1650:
1639:
1638:
1616:
1615:
1589:
1588:
1581:
1576:
1564:
1558:
1527:Platonic solids
1447:
1441:
1411:
1406:
1405:
1379:
1378:
1355:
1354:
1333:
1332:
1331:
1308:
1307:
1274:
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1235:
1234:
1211:
1183:
1166:
1146:
1145:
1122:
1121:
1100:
1099:
1054:
1028:
1027:
1006:
1005:
1003:integral matrix
979:
978:
958:
957:
956:
903:
877:
876:
853:
851:Convex polytope
847:
842:
736:or subfacet – (
687:
686:
665:
664:
622:
621:
600:
599:
537:
532:
479:
408:
322:elliptic tiling
248:Ludwig Schläfli
228:
196:Ludwig Schläfli
161:
157:
149:
139:
134:
130:
35:
28:
23:
22:
15:
12:
11:
5:
3938:
3936:
3928:
3927:
3922:
3912:
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3890:
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3777:
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3763:
3758:
3752:
3751:
3749:
3745:
3738:
3731:
3727:
3722:
3713:
3708:
3702:
3701:
3699:
3695:
3688:
3681:
3677:
3672:
3663:
3658:
3652:
3651:
3649:
3645:
3638:
3634:
3629:
3620:
3615:
3609:
3608:
3606:
3604:
3599:
3590:
3585:
3579:
3578:
3569:
3564:
3559:
3550:
3545:
3539:
3538:
3529:
3527:
3522:
3513:
3508:
3502:
3501:
3496:
3491:
3486:
3481:
3476:
3470:
3469:
3465:
3461:
3456:
3445:
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3416:
3409:
3403:
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3387:
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3328:
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3298:
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3284:
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3279:
3277:
3276:
3268:
3263:
3258:
3253:
3248:
3243:
3238:
3233:
3228:
3222:
3220:
3216:
3215:
3213:
3212:
3207:
3202:
3200:Cross-polytope
3197:
3192:
3187:
3185:Hyperrectangle
3182:
3177:
3172:
3166:
3164:
3154:
3153:
3151:
3150:
3145:
3140:
3135:
3130:
3125:
3120:
3114:
3112:
3108:
3107:
3100:
3098:
3096:
3095:
3090:
3085:
3080:
3075:
3070:
3065:
3060:
3054:
3052:
3048:
3047:
3042:
3040:
3039:
3032:
3025:
3017:
3011:
3010:
3005:
2991:
2970:
2969:External links
2967:
2965:
2964:
2948:
2942:
2912:
2906:
2883:
2881:
2878:
2875:
2874:
2829:
2807:
2795:
2762:
2750:
2737:
2724:
2718:
2689:
2673:
2650:
2638:
2620:
2616:978-0471359432
2603:
2597:Cromwell, P.;
2590:
2581:
2561:
2559:Coxeter (1973)
2545:
2532:
2531:
2529:
2526:
2524:
2521:
2519:
2518:
2513:
2508:
2503:
2498:
2493:
2487:
2481:
2479:
2476:
2465:, a branch of
2463:twistor theory
2423:
2420:
2404:search engines
2370:Peter McMullen
2325:Henri Poincaré
2297:Thorold Gosset
2272:Reinhold Hoppe
2225:
2222:
2221:
2220:
2215:{5,5/2,5} and
2213:great 120-cell
2201:
2183:
2176:
2165:
2146:
2121:Every regular
2098:
2095:
2088:dual polyhedra
2049:
2046:
2042:configurations
2012:
2007:
1995:Hilbert spaces
1987:Main article:
1984:
1981:
1962:The theory of
1956:Main article:
1953:
1950:
1915:Main article:
1912:
1909:
1907:
1904:
1903:
1902:
1889:
1886:
1883:
1879:
1875:
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1683:
1679:
1675:
1670:
1666:
1662:
1657:
1653:
1649:
1646:
1623:
1607:dimensions is
1596:
1580:
1577:
1575:
1572:
1568:star polytopes
1560:Main article:
1557:
1556:Star polytopes
1554:
1508:
1507:
1493:
1479:
1443:Main article:
1440:
1437:
1422:
1416:
1388:
1364:
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1317:
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1290:
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884:
849:Main article:
846:
843:
841:
838:
770:
769:
766:
760:
759:
749:
742:
741:
731:
724:
723:
713:
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694:
683:
672:
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580:
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570:
566:
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556:
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548:
544:
543:
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518:to denote an (
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414:
413:
410:
294:star polyhedra
287:star polytopes
252:Thorold Gosset
227:
224:
212:Reinhold Hoppe
107:In elementary
90:
89:
81:
80:
73:
66:
59:
52:
45:
26:
24:
14:
13:
10:
9:
6:
4:
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3814:
3813:
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3808:
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3662:
3659:
3657:
3654:
3653:
3650:
3648:
3641:
3635:
3633:
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3628:
3624:
3621:
3619:
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3614:
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3607:
3605:
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3600:
3598:
3594:
3591:
3589:
3586:
3584:
3581:
3580:
3577:
3573:
3570:
3568:
3565:
3563:
3562:Demitesseract
3560:
3558:
3554:
3551:
3549:
3546:
3544:
3541:
3540:
3537:
3533:
3530:
3528:
3526:
3523:
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3517:
3514:
3512:
3509:
3507:
3504:
3503:
3500:
3497:
3495:
3492:
3490:
3487:
3485:
3482:
3480:
3477:
3475:
3472:
3471:
3468:
3462:
3459:
3455:
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3341:
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3329:
3327:
3322:
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3318:
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3308:
3302:
3296:
3293:
3291:
3288:
3287:
3285:
3281:
3275:
3273:
3269:
3267:
3264:
3262:
3259:
3257:
3254:
3252:
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3221:
3217:
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3208:
3206:
3203:
3201:
3198:
3196:
3193:
3191:
3190:Demihypercube
3188:
3186:
3183:
3181:
3178:
3176:
3173:
3171:
3168:
3167:
3165:
3163:
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3081:
3079:
3076:
3074:
3071:
3069:
3066:
3064:
3061:
3059:
3056:
3055:
3053:
3049:
3045:
3038:
3033:
3031:
3026:
3024:
3019:
3018:
3015:
3009:
3006:
3003:
2999:
2995:
2992:
2987:
2986:
2981:
2978:
2973:
2972:
2968:
2961:
2957:
2953:
2949:
2945:
2943:0-387-00424-6
2939:
2935:
2931:
2930:
2925:
2921:
2917:
2913:
2909:
2903:
2899:
2895:
2894:
2889:
2885:
2884:
2879:
2870:
2866:
2862:
2858:
2853:
2848:
2844:
2840:
2833:
2830:
2826:
2825:
2820:
2816:
2811:
2808:
2804:
2799:
2796:
2790:
2785:
2781:
2777:
2773:
2766:
2763:
2759:
2754:
2751:
2748:, CUP (1983).
2747:
2741:
2738:
2734:
2728:
2725:
2721:
2719:0-521-81496-0
2715:
2711:
2706:
2705:
2699:
2693:
2690:
2686:
2680:
2678:
2674:
2670:
2666:
2662:
2661:
2654:
2651:
2648:
2642:
2639:
2635:
2629:
2627:
2625:
2621:
2617:
2613:
2607:
2604:
2600:
2594:
2591:
2585:
2582:
2577:
2576:
2571:
2565:
2562:
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2554:
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2550:
2546:
2542:
2537:
2534:
2527:
2522:
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2512:
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2502:
2499:
2497:
2494:
2491:
2488:
2486:
2483:
2482:
2477:
2475:
2472:
2471:amplituhedron
2468:
2464:
2459:
2457:
2453:
2449:
2445:
2441:
2437:
2433:
2429:
2421:
2419:
2417:
2416:amplituhedron
2413:
2409:
2405:
2401:
2397:
2392:
2390:
2386:
2382:
2377:
2375:
2371:
2367:
2361:
2359:
2355:
2350:
2348:
2347:
2342:
2338:
2334:
2330:
2326:
2321:
2319:
2318:
2313:
2308:
2306:
2305:tessellations
2302:
2298:
2293:
2291:
2287:
2283:
2279:
2278:
2273:
2268:
2266:
2262:
2261:
2256:
2252:
2248:
2244:
2242:
2238:
2237:Arthur Cayley
2234:
2229:
2223:
2218:
2214:
2210:
2206:
2202:
2199:
2195:
2191:
2188:
2184:
2181:
2177:
2174:
2171:{3,5,3}, and
2170:
2166:
2163:
2159:
2158:square tiling
2155:
2151:
2147:
2144:
2140:
2136:
2132:
2128:
2124:
2120:
2119:
2118:
2115:
2108:
2103:
2096:
2094:
2091:
2089:
2084:
2082:
2077:
2075:
2071:
2067:
2064: −
2063:
2059:
2055:
2047:
2045:
2043:
2039:
2035:
2032:
2028:
2010:
1996:
1990:
1982:
1980:
1977:
1972:
1970:
1965:
1959:
1951:
1949:
1947:
1943:
1938:
1936:
1932:
1928:
1924:
1923:plane tilings
1918:
1910:
1905:
1887:
1884:
1881:
1873:
1870:
1864:
1861:
1858:
1851:
1850:
1849:
1835:
1832:
1825:
1821:
1813:
1811:
1809:
1789:
1767:
1763:
1740:
1737:
1734:
1726:
1723:
1717:
1714:
1711:
1706:
1703:
1700:
1696:
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1689:
1686:
1681:
1677:
1673:
1668:
1664:
1660:
1655:
1651:
1647:
1644:
1637:
1636:
1635:
1621:
1614:
1610:
1594:
1586:
1578:
1573:
1571:
1569:
1563:
1562:Star polytope
1555:
1553:
1551:
1547:
1542:
1540:
1536:
1532:
1528:
1523:
1521:
1517:
1513:
1505:
1501:
1497:
1494:
1491:
1487:
1483:
1480:
1477:
1473:
1469:
1466:
1465:
1464:
1461:
1459:
1458:dual polytope
1456:; hence, the
1455:
1451:
1446:
1438:
1436:
1420:
1404:
1403:dual polytope
1338:
1288:
1285:
1282:
1256:
1253:
1243:
1240:
1218:
1208:
1198:
1195:
1190:
1180:
1175:
1160:
1157:
1154:
1077:
1066:
1061:
1051:
1040:
1004:
963:
954:
949:
947:
943:
924:
921:
918:
915:
910:
900:
894:
891:
888:
874:
870:
866:
862:
858:
852:
844:
839:
837:
835:
831:
830:
825:
821:
817:
813:
809:
805:
801:
797:
796:
791:
787:
786:
781:
777:
767:
765:
761:
757:
753:
750:
747:
743:
739:
735:
732:
729:
725:
721:
717:
714:
711:
707:
692:
684:
670:
662:
659:
655:
651:
648:
646:
642:
627:
619:
605:
597:
594:
591:
587:
584:
581:
577:
574:
571:
567:
564:
561:
557:
553:
549:
545:
541:
535:
530:
529:
526:
523:
521:
517:
513:
509:
505:
501:
497:
493:
489:
485:
476:
470:
467:
463:
460:
457:
453:
450:
447:
443:
439:
435:
432:
429:
425:
422:
419:
415:
411:
406:
405:
402:
399:
397:
393:
389:
385:
381:
377:
373:
368:
366:
362:
358:
354:
349:
347:
343:
339:
335:
331:
327:
323:
319:
315:
311:
307:
305:
299:
295:
290:
288:
284:
280:
276:
272:
268:
264:
260:
255:
253:
249:
244:
242:
238:
233:
225:
223:
221:
217:
213:
209:
205:
201:
197:
192:
190:
186:
182:
178:
177:tessellations
174:
169:
165:
153:
147:
143:
128:
124:
123:
118:
114:
110:
101:
96:
87:
82:
78:
74:
71:
67:
64:
60:
57:
53:
50:
46:
43:
39:
38:
33:
19:
3882:
3851:
3842:
3834:
3825:
3820:
3816:
3796:10-orthoplex
3532:Dodecahedron
3453:
3442:
3431:
3422:
3413:
3404:
3400:
3390:
3382:
3378:
3370:
3366:
3305:
3271:
3210:Hyperpyramid
3175:Hypersurface
3157:
3068:Affine space
3058:Vector space
3001:
2983:
2955:
2928:
2920:Klee, Victor
2896:, New York:
2891:
2880:Bibliography
2842:
2838:
2832:
2822:
2810:
2798:
2779:
2775:
2765:
2753:
2745:
2740:
2732:
2727:
2703:
2692:
2684:
2671:, MR 2271992
2658:
2653:
2646:
2641:
2633:
2606:
2598:
2593:
2584:
2574:
2570:Richeson, D.
2564:
2541:Coxeter 1973
2536:
2460:
2447:
2434:studies the
2428:optimization
2425:
2422:Applications
2400:optimization
2393:
2378:
2373:
2362:
2351:
2344:
2322:
2315:
2309:
2294:
2289:
2286:George Boole
2275:
2269:
2264:
2258:
2245:
2230:
2227:
2219:{5/2,5,5/2}.
2122:
2116:
2112:
2092:
2085:
2078:
2073:
2069:
2065:
2061:
2057:
2053:
2051:
2030:
2026:
1992:
1973:
1961:
1939:
1920:
1817:
1805:
1609:contractible
1584:
1582:
1565:
1543:
1531:dodecahedron
1524:
1519:
1515:
1509:
1462:
1448:
952:
950:
941:
872:
868:
856:
854:
827:
819:
815:
811:
807:
799:
793:
789:
783:
779:
775:
773:
763:
755:
745:
737:
727:
719:
709:
657:
653:
649:
644:
539:
524:
519:
515:
507:
503:
499:
495:
491:
487:
483:
480:
412:Description
409:of polytope
400:
379:
375:
371:
369:
355:) seen as a
350:
317:
303:
297:
291:
271:tessellation
256:
245:
236:
231:
229:
215:
207:
199:
193:
170:
163:
151:
138:
120:
112:
106:
3805:10-demicube
3766:9-orthoplex
3716:8-orthoplex
3666:7-orthoplex
3623:6-orthoplex
3593:5-orthoplex
3548:Pentachoron
3536:Icosahedron
3511:Tetrahedron
3295:Codimension
3274:-dimensions
3195:Hypersphere
3078:Free module
2782:: 117–138.
2746:Dual Models
2389:Michael Guy
2385:John Conway
2329:topological
2141:{3,3}, and
1935:apeirotopes
1535:icosahedron
1496:Orthoplexes
861:half-spaces
542:-polytope)
533:of element
396:convex hull
173:apeirotopes
3914:Categories
3791:10-simplex
3775:9-demicube
3725:8-demicube
3675:7-demicube
3632:6-demicube
3602:5-demicube
3516:Octahedron
3290:Hyperspace
3170:Hyperplane
2980:"Polytope"
2523:References
2333:CW-complex
2160:{4,4} and
1927:honeycombs
1917:Apeirotope
1574:Properties
1482:Hypercubes
834:polyhedron
758:− 1)-face
740:− 2)-face
722:− 3)-face
469:Polychoron
459:Polyhedron
392:halfspaces
372:polyhedron
357:1-polytope
342:4-polytope
330:polyhedron
312:, flat or
306:−1)-sphere
267:CW-complex
183:including
86:polyhedron
18:1-polytope
3920:Polytopes
3839:orthoplex
3761:9-simplex
3711:8-simplex
3661:7-simplex
3618:6-simplex
3588:5-simplex
3557:Tesseract
3180:Hypercube
3158:Polytopes
3138:Minkowski
3133:Hausdorff
3128:Inductive
3093:Spacetime
3044:Dimension
2985:MathWorld
2852:1312.2007
2599:Polyhedra
2528:Citations
2408:cosmology
2376:in 2002.
2349:in 1967.
2295:In 1895,
2187:canonical
2154:apeirogon
2034:imaginary
1946:apeirogon
1885:−
1871:−
1862:φ
1859:∑
1836:φ
1833:∑
1738:−
1724:−
1704:−
1693:±
1690:⋯
1687:−
1661:−
1645:χ
1622:χ
1468:Simplices
1421:∗
1254:≥
1244:∈
1209:∩
1181:∩
1176:∘
1078:≤
1052:∈
976:-polytope
953:reflexive
922:≥
916:∣
901:∈
693:⋮
671:⋮
628:⋮
606:⋮
531:Dimension
498:-face or
421:Nullitope
407:Dimension
283:simplices
237:polytopes
200:polyschem
181:manifolds
142:-polytope
127:polyhedra
32:Polytrope
3893:Topics:
3856:demicube
3821:polytope
3815:Uniform
3576:600-cell
3572:120-cell
3525:Demicube
3499:Pentagon
3479:Triangle
3307:Category
3283:See also
3083:Manifold
2998:Internet
2954:(1995),
2926:(eds.),
2890:(1973),
2572:(2008).
2478:See also
2444:boundary
2352:In 1952
2337:manifold
2314:'s book
2290:polytope
2270:In 1882
2164:{4,3,4}.
2145:{3,3,3}.
1755:, where
1488:and the
1474:and the
1233:for all
1098:, where
554:theory)
552:abstract
477:Elements
382:means a
380:polytope
376:polytope
338:polygons
314:toroidal
310:elliptic
275:manifold
263:topology
232:polytope
216:polytope
113:polytope
109:geometry
103:regions.
3830:simplex
3800:10-cube
3567:24-cell
3553:16-cell
3494:Hexagon
3348:regular
3205:Simplex
3143:Fractal
2857:Bibcode
2516:Opetope
2335:) of a
2277:polytop
2224:History
2207:, with
2205:24-cell
2127:simplex
2048:Duality
1969:11-cell
1351:-dilate
1304:-dilate
873:pointed
869:bounded
824:polygon
538:(in an
449:Polygon
384:bounded
208:polytop
146:polygon
119:sides (
100:polygon
3770:9-cube
3720:8-cube
3670:7-cube
3627:6-cube
3597:5-cube
3484:Square
3361:Family
3162:shapes
2940:
2904:
2735:, 1974
2716:
2667:
2614:
2446:of an
2440:linear
2196:, and
2175:{5,5}.
2148:Every
2143:5-cell
2107:5-cell
2052:Every
2036:ones.
2025:where
1929:) and
1500:square
1486:square
942:finite
857:convex
816:vertex
795:ridges
785:facets
685:
620:
563:Vertex
388:convex
361:vertex
204:German
202:. The
133:as an
3489:p-gon
3266:Eight
3261:Seven
3241:Three
3118:Krull
2847:arXiv
2156:{∞},
2137:{3},
1454:flags
804:faces
752:Facet
734:Ridge
514:uses
440:Dion
431:Monon
346:cells
334:faces
206:term
122:faces
3847:cube
3520:Cube
3350:and
3251:Five
3246:Four
3226:Zero
3160:and
2938:ISBN
2902:ISBN
2843:2014
2714:ISBN
2665:ISBN
2612:ISBN
2461:In
2454:and
2387:and
2239:and
2192:and
2105:The
1818:The
1533:and
1502:and
1490:cube
829:cell
820:edge
748:− 1
730:− 2
716:Peak
712:− 3
593:Cell
583:Face
573:Edge
536:Term
516:cell
508:edge
492:face
484:face
353:edge
340:, a
336:are
328:. A
324:and
175:and
166:– 1)
154:+ 1)
117:flat
111:, a
3396:(p)
3256:Six
3236:Two
3231:One
3000:– (
2865:doi
2784:doi
2438:of
2257:'s
1587:in
1514:of
1353:of
1306:of
774:An
754:– (
718:– (
547:−1
417:−1
218:by
3916::
3901:•
3897:•
3877:21
3873:•
3870:k1
3866:•
3863:k2
3841:•
3798:•
3768:•
3746:21
3742:•
3739:41
3735:•
3732:42
3718:•
3696:21
3692:•
3689:31
3685:•
3682:32
3668:•
3646:21
3642:•
3639:22
3625:•
3595:•
3574:•
3555:•
3534:•
3518:•
3450:/
3439:/
3429:/
3420:/
3398:/
2982:.
2936:,
2922:;
2900:,
2863:.
2855:.
2845:.
2841:.
2821::
2780:72
2778:.
2774:.
2712:,
2676:^
2623:^
2548:^
2458:.
2430:,
2410:,
2406:,
2402:,
2398:,
2339:.
2044:.
1971:.
1810:.
1026:,
836:.
589:3
579:2
569:1
559:0
465:4
455:3
445:2
437:1
427:0
367:.
250:,
222:.
191:.
98:A
84:A
3885:-
3883:n
3875:k
3868:2
3861:1
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3852:n
3845:-
3843:n
3837:-
3835:n
3828:-
3826:n
3819:-
3817:n
3744:4
3737:2
3730:1
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3637:1
3466:n
3464:H
3457:2
3454:G
3446:4
3443:F
3435:8
3432:E
3426:7
3423:E
3417:6
3414:E
3405:n
3401:D
3394:2
3391:I
3383:n
3379:B
3371:n
3367:A
3339:e
3332:t
3325:v
3272:n
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2988:.
2963:.
2947:.
2911:.
2871:.
2867::
2859::
2849::
2792:.
2786::
2448:n
2265:n
2200:.
2125:-
2123:n
2074:n
2070:j
2066:j
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2054:n
2031:n
2027:n
2011:n
2006:C
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1878:)
1874:1
1868:(
1865:=
1790:j
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1731:)
1727:1
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1697:n
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1648:=
1595:d
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1520:n
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1492:.
1478:.
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1191:d
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1170:P
1164:)
1161:1
1158:+
1155:t
1152:(
1130:P
1107:1
1086:}
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1067::
1062:d
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1044:{
1041:=
1036:P
1013:A
987:P
964:d
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925:0
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911:2
906:R
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892:,
889:x
886:(
883:{
812:j
808:j
800:n
790:n
780:n
776:n
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720:n
710:n
658:n
654:j
650:j
645:j
540:n
520:n
504:j
500:j
496:j
488:n
318:p
316:(
304:p
298:p
164:k
162:(
158:k
152:k
150:(
140:n
135:n
131:n
34:.
20:)
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