59:
45:
52:
38:
66:
31:
2091:
84:
3092:
1967:
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
2352:
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
91:
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
1955:
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
1220:
1085:
223:
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
2462:
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.
848:. 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
1742:
274:, 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
470:
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
2102:
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.
927:
787:
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 (
2380:
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.
1258:
1889:
2224:
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
1422:
2012:
1499:
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
1388:
1364:
1317:
1131:
988:
1136:
2072:
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}.
1107:
1013:
1835:
2996:
1018:
692:
670:
627:
605:
1769:
1621:
1291:
1789:
1594:
1338:
963:
243:
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.
2816:, 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."
1910:
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.
3023:
2484:
3891:
2256:-dimensional polytopes was made acceptable. Schläfli's polytopes were rediscovered many times in the following decades, even during his lifetime.
285:
and other unusual constructions led to the idea of a polyhedron as a bounding surface, ignoring its interior. In this light convex polytopes in
2894:
2657:
2186:
1629:
2045:-polytope has a dual structure, obtained by interchanging its vertices for facets, edges for ridges, and so on generally interchanging its (
2372:, convex and nonconvex, in four or more dimensions remains an outstanding problem. The convex uniform 4-polytopes were fully enumerated by
2982:
3326:
2648:
2440:
2604:
3126:
340:
The idea of constructing a higher polytope from those of lower dimension is also sometimes extended downwards in dimension, with an (
3076:
2930:
2706:
3913:
1796:
864:
if it contains at least one vertex. Every bounded nonempty polytope is pointed. An example of a non-pointed polytope is the set
867:
791: − 3)-dimensional boundaries of the original polytope, and so on. These bounding sub-polytopes may be referred to as
933:
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
3016:
2876:
2300:
2075:
In the case of a geometric polytope, some geometric rule for dualising is necessary, see for example the rules described for
137:
is a 2-polytope and a three-dimensional polyhedron is a 3-polytope. In this context, "flat sides" means that the sides of a
2349:
in complex space, where each real dimension has an imaginary one associated with it. Coxeter developed the theory further.
1812:
1441:
have the highest degree of symmetry of all polytopes. The symmetry group of a regular polytope acts transitively on its
3111:
2794:, Egon Schulte. p. 12: "However, there are many more uniform polytopes but a complete list is known only for d = 4 ."
1538:
2698:
2416:
2388:
2221:
852:. This definition allows a polytope to be neither bounded nor finite. Polytopes are defined in this way, e.g., in
3009:
2489:
2473:
2392:
2239:
2161:
1225:
3046:
2881:
2812:
2383:
In modern times, polytopes and related concepts have found many important applications in fields as diverse as
2305:
2205:
2030:
2026:
1527:
1843:
1537:
include one additional convex solid with fourfold symmetry and two with fivefold symmetry. There are ten star
3319:
3254:
3249:
3229:
2494:
2342:
1930:
2353:
or connection of the various elements with one another. These developments led eventually to the theory of
2296:
in higher dimensions. Polytopes also began to be studied in non-Euclidean spaces such as hyperbolic space.
2238:
was the first to consider analogues of polygons and polyhedra in these higher spaces. He described the six
1808:
3239:
3234:
3214:
1109:
denotes a vector of all ones, and the inequality is component-wise. It follows from this definition that
359:
In certain fields of mathematics, the terms "polytope" and "polyhedron" are used in a different sense: a
3863:
3856:
3849:
3244:
3224:
3219:
2807:
2157:
1964:
1396:
849:
380:
58:
2082:
If the dual is reversed, then the original polytope is recovered. Thus, polytopes exist in dual pairs.
1988:
1215:{\displaystyle (t+1){\mathcal {P}}^{\circ }\cap \mathbb {Z} ^{d}=t{\mathcal {P}}\cap \mathbb {Z} ^{d}}
44:
3520:
3467:
2940:
2912:
2845:
2499:
2289:
2248:
2138:
2123:
2068:
For an abstract polytope, this simply reverses the ordering of the set. This reversal is seen in the
1915:
1601:
1460:
247:
1369:
1345:
1298:
1112:
969:
51:
37:
3875:
3774:
3524:
3121:
3116:
2455:
2432:
2325:
2127:
1464:
773:
740:
314:
173:
65:
1090:
1080:{\displaystyle {\mathcal {P}}=\{\mathbf {x} \in \mathbb {R} ^{d}:\mathbf {Ax} \leq \mathbf {1} \}}
996:
30:
3908:
3744:
3694:
3644:
3601:
3571:
3531:
3494:
3312:
3295:
3136:
3091:
2886:
2835:
2803:
2420:
2270:
1597:
1534:
1492:
853:
817:
267:
208:
2904:
2760:
2329:
2235:
2197:
2119:
2069:
236:
184:
2985:– application of polytopes to a database of articles used to support custom news feeds via the
2265:
1817:
3883:
3131:
2965:
2926:
2890:
2746:
2702:
2653:
2600:
2424:
2400:
2384:
2354:
2313:
2229:
2182:
2178:
1952:
1946:
1919:
934:
551:
540:
353:
349:
177:
677:
655:
612:
590:
3887:
3452:
3441:
3430:
3419:
3410:
3401:
3388:
3366:
3354:
3340:
3336:
3061:
2916:
2853:
2772:
2369:
2346:
2334:
2243:
2022:
1977:
1438:
1433:
783:
722:
500:
290:
2090:
2079:. Depending on circumstance, the dual figure may or may not be another geometric polytope.
1747:
1606:
1264:
3477:
3462:
3106:
3051:
2948:
2922:
2478:
2407:
was discovered as a simplifying construct in certain calculations of theoretical physics.
2168:
2150:
1500:
1452:
There are three main classes of regular polytope which occur in any number of dimensions:
1442:
1391:
839:
792:
704:
581:
571:
561:
341:
334:
322:
310:
282:
229:
192:
110:
105:
2849:
3827:
3188:
3173:
2691:
2686:
2562:
2558:
2451:
2444:
2358:
2285:
2260:
2242:
in 1852 but his work was not published until 1901, six years after his death. By 1854,
2201:
2175:
2076:
1774:
1579:
1515:
1323:
991:
948:
379:. With this terminology, a convex polyhedron is the intersection of a finite number of
375:
polyhedron. This terminology is typically confined to polytopes and polyhedra that are
298:
240:
200:
2288:
not only rediscovered Schläfli's regular polytopes but also investigated the ideas of
3902:
3844:
3732:
3725:
3718:
3682:
3675:
3668:
3632:
3625:
3349:
3178:
2459:
2404:
2225:
2146:
1983:
1556:
1550:
1446:
275:
118:
to any number of dimensions. Polytopes may exist in any general number of dimensions
2791:
1555:
A non-convex polytope may be self-intersecting; this class of polytopes include the
3784:
3198:
3163:
3056:
2293:
2274:
1911:
1519:
259:
165:
2252:
had firmly established the geometry of higher dimensions, and thus the concept of
266:. An example of this approach defines a polytope as a set of points that admits a
2065: − 1), while retaining the connectivity or incidence between elements.
3793:
3754:
3704:
3654:
3611:
3581:
3513:
3499:
3283:
3066:
2908:
2857:
2377:
2373:
1523:
384:
376:
372:
278:
with interior structures, and so is restricted to certain areas of mathematics.
83:
2269:
to refer to this more general concept of polygons and polyhedra. In due course
860:
if there is a ball of finite radius that contains it. A polytope is said to be
160:
Some theories further generalize the idea to include such objects as unbounded
3779:
3763:
3713:
3663:
3620:
3590:
3504:
3278:
3158:
2968:
2321:
1923:
1905:
822:
457:
447:
345:
330:
318:
258:
as analogous to a polytope. In this approach, a polytope may be regarded as a
255:
161:
115:
74:
2636:
The part of the polytope that lies in one of the hyperplanes is called a cell
2156:
Numerous compact, paracompact and noncompact hyperbolic tilings, such as the
3835:
3749:
3699:
3649:
3606:
3576:
3545:
3259:
3168:
3081:
3032:
2973:
2439:-dimensional polytope. In linear programming, polytopes occur in the use of
2396:
2142:
1934:
1484:
1470:
419:
409:
271:
20:
2777:
2649:
Computing the
Continuous Discretely: Integer-point enumeration in polyhedra
1511:≥ 5) star. But in higher dimensions there are no other regular polytopes.
3564:
3560:
3487:
3183:
3071:
2986:
2672:
M. A. Perles and G. C. Shephard. 1967. "Angle sums of convex polytopes".
2317:
1541:, all with fivefold symmetry, giving in all sixteen regular 4-polytopes.
326:
263:
251:
169:
97:
3818:
3788:
3555:
3550:
3541:
3482:
3193:
2504:
2193:
2115:
1957:
1737:{\displaystyle \chi =n_{0}-n_{1}+n_{2}-\cdots \pm n_{d-1}=1+(-1)^{d-1}}
1456:
812:
437:
250:
of polyhedra to higher-dimensional polytopes led to the development of
134:
88:
2997:
Regular and semi-regular convex polytopes a short historical overview:
2599:
Nemhauser and Wolsey, "Integer and
Combinatorial Optimization," 1999,
1530:
with fivefold symmetry, bringing the total to nine regular polyhedra.
3758:
3708:
3658:
3615:
3585:
3536:
3472:
2428:
2131:
2095:
1488:
1474:
811:, and consists of a line segment. A 2-dimensional face consists of a
302:
1968:
realization in some real space of the associated abstract polytope.
807:, and consists of a single point. A 1-dimensional face is called an
352:
as a 0-polytope. This approach is used for example in the theory of
3150:
2947:, Graduate Texts in Mathematics, vol. 152, Berlin, New York:
2840:
82:
2826:
2652:, Undergraduate Texts in Mathematics, New York: Springer-Verlag,
2098:(4-simplex) is self-dual with 5 vertices and 5 tetrahedral cells.
183:
Polytopes of more than three dimensions were first discovered by
3508:
1478:
514:
The terms adopted in this article are given in the table below:
3005:
2309:, summarizing work to date and adding new findings of his own.
270:. In this definition, a polytope is the union of finitely many
1933:
and the infinite series of tilings represented by the regular
390:
Polytopes in lower numbers of dimensions have standard names:
2564:
Euler's Gem: The
Polyhedron Formula and the Birth of Topology
1366:
only by lattice points gained on the boundary. Equivalently,
387:
of a finite number of points and is defined by its vertices.
3001:
2217:
Polygons and polyhedra have been known since ancient times.
1403:
1375:
1351:
1304:
1192:
1158:
1118:
1024:
975:
777:. These facets are themselves polytopes, whose facets are (
383:
and is defined by its sides while a convex polytope is the
228:. They represent different approaches to generalizing the
922:{\displaystyle \{(x,y)\in \mathbb {R} ^{2}\mid x\geq 0\}}
114:). Polytopes are the generalization of three-dimensional
2357:
as partially ordered sets, or posets, of such elements.
2320:
idea of a polytope as the piecewise decomposition (e.g.
363:
is the generic object in any dimension (referred to as
1922:
are in this sense polytopes, and are sometimes called
1837:
for convex polyhedra to higher-dimensional polytopes:
1820:
25:
2220:
An early hint of higher dimensions came in 1827 when
1991:
1846:
1777:
1750:
1632:
1609:
1582:
1399:
1372:
1348:
1326:
1301:
1267:
1228:
1139:
1115:
1093:
1021:
999:
972:
951:
870:
680:
658:
615:
593:
479: − 1)-dimensional element while others use
2765:
Biographical
Memoirs of Fellows of the Royal Society
2761:"John Horton Conway. 26 December 1937—11 April 2020"
1623:
of its boundary ∂P is given by the alternating sum:
1319:
differs, in terms of integer lattice points, from a
3271:
3207:
3145:
3099:
3039:
1982:Structures analogous to polytopes exist in complex
1929:Among these, there are regular forms including the
2690:
2006:
1883:
1829:
1783:
1763:
1736:
1615:
1588:
1416:
1382:
1358:
1332:
1311:
1285:
1252:
1214:
1125:
1101:
1079:
1007:
982:
957:
921:
686:
664:
621:
599:
232:to include other objects with similar properties.
203:, and was introduced to English mathematicians as
2141:, in any number of dimensions. These include the
937:if all of its vertices have integer coordinates.
767:-dimensional polytope is bounded by a number of (
483:to denote a 2-face specifically. Authors may use
2431:functions; these maxima and minima occur on the
2299:An important milestone was reached in 1948 with
815:, and a 3-dimensional face, sometimes called a
3320:
3017:
2544:
2542:
2540:
2538:
2532:, pp. 141–144, §7-x. Historical remarks.
1937:, square tiling, cubic honeycomb, and so on.
1811:similarly generalizes the alternating sum of
8:
2617:
2615:
2613:
1074:
1032:
916:
871:
3327:
3313:
3305:
3024:
3010:
3002:
2625:, Cambridge University Press, 2018, p.224.
2049: − 1)-dimensional elements for (
1253:{\displaystyle t\in \mathbb {Z} _{\geq 0}}
235:The original approach broadly followed by
2839:
2776:
2106:Some common self-dual polytopes include:
1998:
1994:
1993:
1990:
1926:because they have infinitely many cells.
1869:
1845:
1819:
1776:
1755:
1749:
1722:
1688:
1669:
1656:
1643:
1631:
1608:
1581:
1408:
1402:
1401:
1398:
1374:
1373:
1371:
1350:
1349:
1347:
1325:
1303:
1302:
1300:
1266:
1241:
1237:
1236:
1227:
1206:
1202:
1201:
1191:
1190:
1178:
1174:
1173:
1163:
1157:
1156:
1138:
1117:
1116:
1114:
1094:
1092:
1069:
1058:
1049:
1045:
1044:
1035:
1023:
1022:
1020:
1000:
998:
974:
973:
971:
950:
898:
894:
893:
869:
803:-faces. A 0-dimensional face is called a
679:
657:
614:
592:
2676:, Vol 21, No 2. March 1967. pp. 199–218.
2668:
2666:
2485:Intersection of a polyhedron with a line
2089:
1884:{\displaystyle \sum \varphi =(-1)^{d-1}}
940:A certain class of convex polytopes are
516:
392:
348:bounded by a point pair, and a point or
254:and the treatment of a decomposition or
187:before 1853, who called such a figure a
3892:List of regular polytopes and compounds
2921:(2nd ed.), New York & London:
2529:
2522:
2403:and numerous other fields. In 2013 the
2232:had also considered higher dimensions.
1449:of a regular polytope is also regular.
297:, while others may be tilings of other
2747:John Horton Conway: Mathematical Magus
2646:Beck, Matthias; Robins, Sinai (2007),
2361:and Egon Schulte published their book
511: − 1)-dimensional element.
168:, decompositions or tilings of curved
2263:, writing in German, coined the word
2187:tetrahedrally diminished dodecahedron
1507:-fold symmetry, both convex and (for
7:
2312:Meanwhile, the French mathematician
2118:, in any number of dimensions, with
1559:. Some regular polytopes are stars.
1473:or measure polytopes, including the
2792:Symmetry of Polytopes and Polyhedra
2759:Curtis, Robert Turner (June 2022).
2441:generalized barycentric coordinates
2018:real dimensions are accompanied by
2332:published his influential work on
2029:are more appropriately treated as
1487:or cross polytopes, including the
1417:{\displaystyle {\mathcal {P}}^{*}}
14:
2689:; Schulte, Egon (December 2002),
1572:Since a (filled) convex polytope
491:-facet to indicate an element of
321:is understood as a surface whose
133:. For example, a two-dimensional
3090:
2007:{\displaystyle \mathbb {C} ^{n}}
1390:is reflexive if and only if its
1095:
1070:
1062:
1059:
1036:
1001:
333:as a hypersurface whose facets (
199:was coined by the mathematician
64:
57:
50:
43:
36:
29:
16:Geometric object with flat sides
2877:Coxeter, Harold Scott MacDonald
1526:, and there are also four star
1518:include the fivefold-symmetric
1514:In three dimensions the convex
337:) are polyhedra, and so forth.
262:or decomposition of some given
2828:Journal of High Energy Physics
2623:Geometries and Transformations
1866:
1856:
1719:
1709:
1383:{\displaystyle {\mathcal {P}}}
1359:{\displaystyle {\mathcal {P}}}
1312:{\displaystyle {\mathcal {P}}}
1280:
1268:
1152:
1140:
1126:{\displaystyle {\mathcal {P}}}
983:{\displaystyle {\mathcal {P}}}
886:
874:
829:Important classes of polytopes
1:
2567:. Princeton University Press.
1895:Generalisations of a polytope
1797:Euler's formula for polyhedra
2590:, CUP (ppbk 1999) pp 205 ff.
2277:, introduced the anglicised
2057:)-dimensional elements (for
1133:is reflexive if and only if
1102:{\displaystyle \mathbf {1} }
1008:{\displaystyle \mathbf {A} }
781: − 2)-dimensional
771: − 1)-dimensional
751:
733:
715:
697:
652:
632:
587:
577:
567:
557:
547:
535:
453:
443:
433:
425:
415:
405:
77:is a 3-dimensional polytope
2983:"Math will rock your world"
2481:-discrete oriented polytope
2281:into the English language.
499:to refer to a ridge, while
246:Attempts to generalise the
104:is a geometric object with
3930:
3881:
3308:
2699:Cambridge University Press
2693:Abstract Regular Polytopes
2634:Regular polytopes, p. 127
2363:Abstract Regular Polytopes
2240:convex regular 4-polytopes
1975:
1963:An abstract polytope is a
1944:
1903:
1830:{\textstyle \sum \varphi }
1548:
1431:
837:
18:
3292:
3088:
2722:Regular Complex Polytopes
2490:Extension of a polyhedron
2474:List of regular polytopes
2162:order-5 pentagonal tiling
2027:Regular complex polytopes
1539:Schläfli-Hess 4-polytopes
1424:is an integral polytope.
990:is reflexive if for some
289:-space are equivalent to
149:-polytopes that may have
126:-dimensional polytope or
72:
2907:(2003), Kaibel, Volker;
2813:The Symmetries of Things
2458:, a polytope called the
2345:generalised the idea as
2206:grand stellated 120-cell
1528:Kepler-Poinsot polyhedra
641:-face – element of rank
268:simplicial decomposition
215:Approaches to definition
19:Not to be confused with
3914:Real algebraic geometry
2858:10.1007/JHEP10(2014)030
2343:Geoffrey Colin Shephard
2273:, daughter of logician
2222:August Ferdinand Möbius
2122:{3}. These include the
1533:In four dimensions the
944:polytopes. An integral
687:{\displaystyle \vdots }
665:{\displaystyle \vdots }
645:= −1, 0, 1, 2, 3, ...,
622:{\displaystyle \vdots }
600:{\displaystyle \vdots }
2778:10.1098/rsbm.2021.0034
2099:
2008:
1931:regular skew polyhedra
1885:
1831:
1785:
1765:
1738:
1617:
1590:
1418:
1384:
1360:
1334:
1313:
1287:
1254:
1216:
1127:
1103:
1081:
1009:
984:
959:
923:
799:-dimensional faces or
688:
666:
623:
601:
539:Nullity (necessary in
157:-polytopes in common.
93:
2945:Lectures on Polytopes
2808:Chaim Goodman-Strauss
2806:, Heidi Burgiel, and
2290:semiregular polytopes
2192:In 4 dimensions, the
2174:In 3 dimensions, the
2171:(regular 2-polytopes)
2167:In 2 dimensions, all
2158:icosahedral honeycomb
2093:
2009:
1965:partially ordered set
1886:
1832:
1786:
1766:
1764:{\displaystyle n_{j}}
1739:
1618:
1616:{\displaystyle \chi }
1591:
1419:
1385:
1361:
1335:
1314:
1288:
1286:{\displaystyle (t+1)}
1255:
1217:
1128:
1104:
1082:
1010:
985:
960:
924:
689:
667:
624:
602:
495:dimensions. Some use
367:in this article) and
145:-polytope consist of
86:
3208:Dimensions by number
2991:Business Week Online
2621:Johnson, Norman W.;
2500:Honeycomb (geometry)
2495:Polytope de Montréal
2249:Habilitationsschrift
2139:hypercubic honeycomb
2124:equilateral triangle
1989:
1844:
1818:
1775:
1748:
1630:
1607:
1602:Euler characteristic
1580:
1568:Euler characteristic
1461:equilateral triangle
1397:
1370:
1346:
1324:
1299:
1265:
1260:. In other words, a
1226:
1137:
1113:
1091:
1019:
997:
970:
949:
868:
757:The polytope itself
678:
656:
613:
591:
248:Euler characteristic
176:, and set-theoretic
3876:pentagonal polytope
3775:Uniform 10-polytope
3335:Fundamental convex
2850:2014JHEP...10..030A
2456:theoretical physics
2128:regular tetrahedron
2086:Self-dual polytopes
1791:-dimensional faces.
1535:regular 4-polytopes
1465:regular tetrahedron
315:toroidal polyhedron
309:−1)-surfaces – see
219:Nowadays, the term
174:spherical polyhedra
3745:Uniform 9-polytope
3695:Uniform 8-polytope
3645:Uniform 7-polytope
3602:Uniform 6-polytope
3572:Uniform 5-polytope
3532:Uniform polychoron
3495:Uniform polyhedron
3343:in dimensions 2–10
3137:Degrees of freedom
3040:Dimensional spaces
2966:Weisstein, Eric W.
2941:Ziegler, Günter M.
2913:Ziegler, Günter M.
2887:Dover Publications
2804:John Horton Conway
2674:Math. Scandinavica
2421:linear programming
2355:abstract polytopes
2316:had developed the
2292:and space-filling
2271:Alicia Boole Stott
2200:{3,4,3}. Also the
2183:elongated pyramids
2179:polygonal pyramids
2100:
2061: = 1 to
2004:
1953:abstract polytopes
1941:Abstract polytopes
1920:hyperbolic tilings
1900:Infinite polytopes
1881:
1827:
1809:Gram–Euler theorem
1781:
1761:
1734:
1613:
1586:
1493:regular octahedron
1414:
1380:
1356:
1330:
1309:
1283:
1250:
1212:
1123:
1099:
1077:
1005:
980:
955:
919:
854:linear programming
844:A polytope may be
795:, or specifically
684:
662:
619:
597:
354:abstract polytopes
209:Alicia Boole Stott
178:abstract polytopes
94:
3897:
3896:
3884:Polytope families
3341:uniform polytopes
3303:
3302:
3112:Lebesgue covering
3077:Algebraic variety
2896:978-0-486-61480-9
2882:Regular Polytopes
2720:Coxeter, H.S.M.;
2658:978-0-387-29139-0
2607:, Definition 2.2.
2425:maxima and minima
2401:quantum mechanics
2385:computer graphics
2370:uniform polytopes
2347:complex polytopes
2306:Regular Polytopes
2230:Hermann Grassmann
1972:Complex polytopes
1947:Abstract polytope
1914:, space-filling (
1795:This generalizes
1784:{\displaystyle j}
1771:is the number of
1589:{\displaystyle d}
1439:Regular polytopes
1428:Regular polytopes
1333:{\displaystyle t}
958:{\displaystyle d}
935:integral polytope
761:
760:
463:
462:
281:The discovery of
81:
80:
3921:
3888:Regular polytope
3449:
3438:
3427:
3386:
3329:
3322:
3315:
3306:
3100:Other dimensions
3094:
3062:Projective space
3026:
3019:
3012:
3003:
2979:
2978:
2951:
2935:
2918:Convex polytopes
2905:Grünbaum, Branko
2899:
2862:
2861:
2843:
2823:
2817:
2801:
2795:
2789:
2783:
2782:
2780:
2756:
2750:
2749:- Richard K. Guy
2744:
2738:
2731:
2725:
2718:
2712:
2711:
2697:(1st ed.),
2696:
2683:
2677:
2670:
2661:
2644:
2638:
2632:
2626:
2619:
2608:
2597:
2591:
2584:
2578:
2575:
2569:
2568:
2555:
2549:
2546:
2533:
2527:
2415:In the field of
2368:Enumerating the
2335:Convex Polytopes
2301:H. S. M. Coxeter
2244:Bernhard Riemann
2169:regular polygons
2070:Schläfli symbols
2013:
2011:
2010:
2005:
2003:
2002:
1997:
1978:Complex polytope
1890:
1888:
1887:
1882:
1880:
1879:
1836:
1834:
1833:
1828:
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1698:
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1660:
1648:
1647:
1622:
1620:
1619:
1614:
1600:to a point, the
1595:
1593:
1592:
1587:
1501:regular polygons
1459:, including the
1434:Regular polytope
1423:
1421:
1420:
1415:
1413:
1412:
1407:
1406:
1389:
1387:
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1039:
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1011:
1006:
1004:
989:
987:
986:
981:
979:
978:
966:
964:
962:
961:
956:
929:. A polytope is
928:
926:
925:
920:
903:
902:
897:
856:. A polytope is
834:Convex polytopes
821:, consists of a
693:
691:
690:
685:
671:
669:
668:
663:
628:
626:
625:
620:
606:
604:
603:
598:
517:
501:H. S. M. Coxeter
475:to refer to an (
393:
291:tilings of the (
230:convex polytopes
156:
148:
144:
130:
125:
121:
68:
61:
54:
47:
40:
33:
26:
3929:
3928:
3924:
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3922:
3920:
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3899:
3898:
3867:
3860:
3853:
3736:
3729:
3722:
3686:
3679:
3672:
3636:
3629:
3463:Regular polygon
3456:
3447:
3440:
3436:
3429:
3425:
3416:
3407:
3400:
3396:
3384:
3378:
3374:
3362:
3344:
3333:
3304:
3299:
3288:
3267:
3203:
3141:
3095:
3086:
3052:Euclidean space
3035:
3030:
2964:
2963:
2960:
2955:
2949:Springer-Verlag
2939:
2933:
2923:Springer-Verlag
2903:
2897:
2875:
2871:
2866:
2865:
2825:
2824:
2820:
2802:
2798:
2790:
2786:
2758:
2757:
2753:
2745:
2741:
2733:Wenninger, M.;
2732:
2728:
2719:
2715:
2709:
2687:McMullen, Peter
2685:
2684:
2680:
2671:
2664:
2645:
2641:
2633:
2629:
2620:
2611:
2598:
2594:
2585:
2581:
2577:Grünbaum (2003)
2576:
2572:
2557:
2556:
2552:
2547:
2536:
2528:
2524:
2519:
2514:
2509:
2479:Bounding volume
2469:
2445:slack variables
2413:
2330:Branko Grünbaum
2236:Ludwig Schläfli
2215:
2198:Schläfli symbol
2151:cubic honeycomb
2120:Schläfli symbol
2088:
2039:
1992:
1987:
1986:
1980:
1974:
1949:
1943:
1908:
1902:
1897:
1865:
1842:
1841:
1816:
1815:
1813:internal angles
1805:
1803:Internal angles
1773:
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1639:
1628:
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1605:
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1578:
1577:
1570:
1565:
1553:
1547:
1516:Platonic solids
1436:
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1395:
1394:
1368:
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1322:
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1320:
1297:
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1172:
1155:
1135:
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1111:
1110:
1089:
1088:
1043:
1017:
1016:
995:
994:
992:integral matrix
968:
967:
947:
946:
945:
892:
866:
865:
842:
840:Convex polytope
836:
831:
725:or subfacet – (
676:
675:
654:
653:
611:
610:
589:
588:
526:
521:
468:
397:
311:elliptic tiling
237:Ludwig Schläfli
217:
185:Ludwig Schläfli
150:
146:
138:
128:
123:
119:
24:
17:
12:
11:
5:
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3697:
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3640:
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3623:
3618:
3609:
3604:
3598:
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3558:
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3227:
3222:
3217:
3211:
3209:
3205:
3204:
3202:
3201:
3196:
3191:
3189:Cross-polytope
3186:
3181:
3176:
3174:Hyperrectangle
3171:
3166:
3161:
3155:
3153:
3143:
3142:
3140:
3139:
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3124:
3119:
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3021:
3014:
3006:
3000:
2999:
2994:
2980:
2959:
2958:External links
2956:
2954:
2953:
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2901:
2895:
2872:
2870:
2867:
2864:
2863:
2818:
2796:
2784:
2751:
2739:
2726:
2713:
2707:
2678:
2662:
2639:
2627:
2609:
2605:978-0471359432
2592:
2586:Cromwell, P.;
2579:
2570:
2550:
2548:Coxeter (1973)
2534:
2521:
2520:
2518:
2515:
2513:
2510:
2508:
2507:
2502:
2497:
2492:
2487:
2482:
2476:
2470:
2468:
2465:
2454:, a branch of
2452:twistor theory
2412:
2409:
2393:search engines
2359:Peter McMullen
2314:Henri Poincaré
2286:Thorold Gosset
2261:Reinhold Hoppe
2214:
2211:
2210:
2209:
2204:{5,5/2,5} and
2202:great 120-cell
2190:
2172:
2165:
2154:
2135:
2110:Every regular
2087:
2084:
2077:dual polyhedra
2038:
2035:
2031:configurations
2001:
1996:
1984:Hilbert spaces
1976:Main article:
1973:
1970:
1951:The theory of
1945:Main article:
1942:
1939:
1904:Main article:
1901:
1898:
1896:
1893:
1892:
1891:
1878:
1875:
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1651:
1646:
1642:
1638:
1635:
1612:
1596:dimensions is
1585:
1569:
1566:
1564:
1561:
1557:star polytopes
1549:Main article:
1546:
1545:Star polytopes
1543:
1497:
1496:
1482:
1468:
1432:Main article:
1429:
1426:
1411:
1405:
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1353:
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838:Main article:
835:
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507:to denote an (
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283:star polyhedra
276:star polytopes
241:Thorold Gosset
216:
213:
201:Reinhold Hoppe
96:In elementary
79:
78:
70:
69:
62:
55:
48:
41:
34:
15:
13:
10:
9:
6:
4:
3:
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3554:
3552:
3551:Demitesseract
3549:
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3179:Demihypercube
3177:
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3050:
3048:
3045:
3044:
3042:
3038:
3034:
3027:
3022:
3020:
3015:
3013:
3008:
3007:
3004:
2998:
2995:
2992:
2988:
2984:
2981:
2976:
2975:
2970:
2967:
2962:
2961:
2957:
2950:
2946:
2942:
2938:
2934:
2932:0-387-00424-6
2928:
2924:
2920:
2919:
2914:
2910:
2906:
2902:
2898:
2892:
2888:
2884:
2883:
2878:
2874:
2873:
2868:
2859:
2855:
2851:
2847:
2842:
2837:
2833:
2829:
2822:
2819:
2815:
2814:
2809:
2805:
2800:
2797:
2793:
2788:
2785:
2779:
2774:
2770:
2766:
2762:
2755:
2752:
2748:
2743:
2740:
2737:, CUP (1983).
2736:
2730:
2727:
2723:
2717:
2714:
2710:
2708:0-521-81496-0
2704:
2700:
2695:
2694:
2688:
2682:
2679:
2675:
2669:
2667:
2663:
2659:
2655:
2651:
2650:
2643:
2640:
2637:
2631:
2628:
2624:
2618:
2616:
2614:
2610:
2606:
2602:
2596:
2593:
2589:
2583:
2580:
2574:
2571:
2566:
2565:
2560:
2554:
2551:
2545:
2543:
2541:
2539:
2535:
2531:
2526:
2523:
2516:
2511:
2506:
2503:
2501:
2498:
2496:
2493:
2491:
2488:
2486:
2483:
2480:
2477:
2475:
2472:
2471:
2466:
2464:
2461:
2460:amplituhedron
2457:
2453:
2448:
2446:
2442:
2438:
2434:
2430:
2426:
2422:
2418:
2410:
2408:
2406:
2405:amplituhedron
2402:
2398:
2394:
2390:
2386:
2381:
2379:
2375:
2371:
2366:
2364:
2360:
2356:
2350:
2348:
2344:
2339:
2337:
2336:
2331:
2327:
2323:
2319:
2315:
2310:
2308:
2307:
2302:
2297:
2295:
2294:tessellations
2291:
2287:
2282:
2280:
2276:
2272:
2268:
2267:
2262:
2257:
2255:
2251:
2250:
2245:
2241:
2237:
2233:
2231:
2227:
2226:Arthur Cayley
2223:
2218:
2212:
2207:
2203:
2199:
2195:
2191:
2188:
2184:
2180:
2177:
2173:
2170:
2166:
2163:
2160:{3,5,3}, and
2159:
2155:
2152:
2148:
2147:square tiling
2144:
2140:
2136:
2133:
2129:
2125:
2121:
2117:
2113:
2109:
2108:
2107:
2104:
2097:
2092:
2085:
2083:
2080:
2078:
2073:
2071:
2066:
2064:
2060:
2056:
2053: −
2052:
2048:
2044:
2036:
2034:
2032:
2028:
2024:
2021:
2017:
1999:
1985:
1979:
1971:
1969:
1966:
1961:
1959:
1954:
1948:
1940:
1938:
1936:
1932:
1927:
1925:
1921:
1917:
1913:
1912:plane tilings
1907:
1899:
1894:
1876:
1873:
1870:
1862:
1859:
1853:
1850:
1847:
1840:
1839:
1838:
1824:
1821:
1814:
1810:
1802:
1800:
1798:
1778:
1756:
1752:
1729:
1726:
1723:
1715:
1712:
1706:
1703:
1700:
1695:
1692:
1689:
1685:
1681:
1678:
1675:
1670:
1666:
1662:
1657:
1653:
1649:
1644:
1640:
1636:
1633:
1626:
1625:
1624:
1610:
1603:
1599:
1583:
1575:
1567:
1562:
1560:
1558:
1552:
1551:Star polytope
1544:
1542:
1540:
1536:
1531:
1529:
1525:
1521:
1517:
1512:
1510:
1506:
1502:
1494:
1490:
1486:
1483:
1480:
1476:
1472:
1469:
1466:
1462:
1458:
1455:
1454:
1453:
1450:
1448:
1447:dual polytope
1445:; hence, the
1444:
1440:
1435:
1427:
1425:
1409:
1393:
1392:dual polytope
1327:
1277:
1274:
1271:
1245:
1242:
1232:
1229:
1207:
1197:
1187:
1184:
1179:
1169:
1164:
1149:
1146:
1143:
1066:
1055:
1050:
1040:
1029:
993:
952:
943:
938:
936:
932:
913:
910:
907:
904:
899:
889:
883:
880:
877:
863:
859:
855:
851:
847:
841:
833:
828:
826:
824:
820:
819:
814:
810:
806:
802:
798:
794:
790:
786:
785:
780:
776:
775:
770:
766:
756:
754:
750:
746:
742:
739:
736:
732:
728:
724:
721:
718:
714:
710:
706:
703:
700:
696:
681:
673:
659:
651:
648:
644:
640:
637:
635:
631:
616:
608:
594:
586:
583:
580:
576:
573:
570:
566:
563:
560:
556:
553:
550:
546:
542:
538:
534:
530:
524:
519:
518:
515:
512:
510:
506:
502:
498:
494:
490:
486:
482:
478:
474:
465:
459:
456:
452:
449:
446:
442:
439:
436:
432:
428:
424:
421:
418:
414:
411:
408:
404:
400:
395:
394:
391:
388:
386:
382:
378:
374:
370:
366:
362:
357:
355:
351:
347:
343:
338:
336:
332:
328:
324:
320:
316:
312:
308:
304:
300:
296:
294:
288:
284:
279:
277:
273:
269:
265:
261:
257:
253:
249:
244:
242:
238:
233:
231:
227:
222:
214:
212:
210:
206:
202:
198:
194:
190:
186:
181:
179:
175:
171:
167:
166:tessellations
163:
158:
154:
142:
136:
132:
117:
113:
112:
107:
103:
99:
90:
85:
76:
71:
67:
63:
60:
56:
53:
49:
46:
42:
39:
35:
32:
28:
27:
22:
3871:
3840:
3831:
3823:
3814:
3809:
3805:
3785:10-orthoplex
3521:Dodecahedron
3442:
3431:
3420:
3411:
3402:
3393:
3389:
3379:
3371:
3367:
3359:
3355:
3294:
3260:
3199:Hyperpyramid
3164:Hypersurface
3146:
3057:Affine space
3047:Vector space
2990:
2972:
2944:
2917:
2909:Klee, Victor
2885:, New York:
2880:
2869:Bibliography
2831:
2827:
2821:
2811:
2799:
2787:
2768:
2764:
2754:
2742:
2734:
2729:
2721:
2716:
2692:
2681:
2673:
2660:, MR 2271992
2647:
2642:
2635:
2630:
2622:
2595:
2587:
2582:
2573:
2563:
2559:Richeson, D.
2553:
2530:Coxeter 1973
2525:
2449:
2436:
2423:studies the
2417:optimization
2414:
2411:Applications
2389:optimization
2382:
2367:
2362:
2351:
2340:
2333:
2311:
2304:
2298:
2283:
2278:
2275:George Boole
2264:
2258:
2253:
2247:
2234:
2219:
2216:
2208:{5/2,5,5/2}.
2111:
2105:
2101:
2081:
2074:
2067:
2062:
2058:
2054:
2050:
2046:
2042:
2040:
2019:
2015:
1981:
1962:
1950:
1928:
1909:
1806:
1794:
1598:contractible
1573:
1571:
1554:
1532:
1520:dodecahedron
1513:
1508:
1504:
1498:
1451:
1437:
941:
939:
930:
861:
857:
845:
843:
816:
808:
804:
800:
796:
788:
782:
778:
772:
768:
764:
762:
752:
744:
734:
726:
716:
708:
698:
646:
642:
638:
633:
528:
513:
508:
504:
496:
492:
488:
484:
480:
476:
472:
469:
401:Description
398:of polytope
389:
368:
364:
360:
358:
344:) seen as a
339:
306:
292:
286:
280:
260:tessellation
245:
234:
225:
220:
218:
204:
196:
188:
182:
159:
152:
140:
127:
109:
101:
95:
3794:10-demicube
3755:9-orthoplex
3705:8-orthoplex
3655:7-orthoplex
3612:6-orthoplex
3582:5-orthoplex
3537:Pentachoron
3525:Icosahedron
3500:Tetrahedron
3284:Codimension
3263:-dimensions
3184:Hypersphere
3067:Free module
2771:: 117–138.
2735:Dual Models
2378:Michael Guy
2374:John Conway
2318:topological
2130:{3,3}, and
1924:apeirotopes
1524:icosahedron
1485:Orthoplexes
850:half-spaces
531:-polytope)
522:of element
385:convex hull
162:apeirotopes
3903:Categories
3780:10-simplex
3764:9-demicube
3714:8-demicube
3664:7-demicube
3621:6-demicube
3591:5-demicube
3505:Octahedron
3279:Hyperspace
3159:Hyperplane
2969:"Polytope"
2512:References
2322:CW-complex
2149:{4,4} and
1916:honeycombs
1906:Apeirotope
1563:Properties
1471:Hypercubes
823:polyhedron
747:− 1)-face
729:− 2)-face
711:− 3)-face
458:Polychoron
448:Polyhedron
381:halfspaces
361:polyhedron
346:1-polytope
331:4-polytope
319:polyhedron
301:, flat or
295:−1)-sphere
256:CW-complex
172:including
75:polyhedron
3909:Polytopes
3828:orthoplex
3750:9-simplex
3700:8-simplex
3650:7-simplex
3607:6-simplex
3577:5-simplex
3546:Tesseract
3169:Hypercube
3147:Polytopes
3127:Minkowski
3122:Hausdorff
3117:Inductive
3082:Spacetime
3033:Dimension
2974:MathWorld
2841:1312.2007
2588:Polyhedra
2517:Citations
2397:cosmology
2365:in 2002.
2338:in 1967.
2284:In 1895,
2176:canonical
2143:apeirogon
2023:imaginary
1935:apeirogon
1874:−
1860:−
1851:φ
1848:∑
1825:φ
1822:∑
1727:−
1713:−
1693:−
1682:±
1679:⋯
1676:−
1650:−
1634:χ
1611:χ
1457:Simplices
1410:∗
1243:≥
1233:∈
1198:∩
1170:∩
1165:∘
1067:≤
1041:∈
965:-polytope
942:reflexive
911:≥
905:∣
890:∈
682:⋮
660:⋮
617:⋮
595:⋮
520:Dimension
487:-face or
410:Nullitope
396:Dimension
272:simplices
226:polytopes
189:polyschem
170:manifolds
131:-polytope
116:polyhedra
21:Polytrope
3882:Topics:
3845:demicube
3810:polytope
3804:Uniform
3565:600-cell
3561:120-cell
3514:Demicube
3488:Pentagon
3468:Triangle
3296:Category
3272:See also
3072:Manifold
2987:Internet
2943:(1995),
2915:(eds.),
2879:(1973),
2561:(2008).
2467:See also
2433:boundary
2341:In 1952
2326:manifold
2303:'s book
2279:polytope
2259:In 1882
2153:{4,3,4}.
2134:{3,3,3}.
1744:, where
1477:and the
1463:and the
1222:for all
1087:, where
543:theory)
541:abstract
466:Elements
371:means a
369:polytope
365:polytope
327:polygons
303:toroidal
299:elliptic
264:manifold
252:topology
221:polytope
205:polytope
102:polytope
98:geometry
92:regions.
3819:simplex
3789:10-cube
3556:24-cell
3542:16-cell
3483:Hexagon
3337:regular
3194:Simplex
3132:Fractal
2846:Bibcode
2505:Opetope
2324:) of a
2266:polytop
2213:History
2196:, with
2194:24-cell
2116:simplex
2037:Duality
1958:11-cell
1340:-dilate
1293:-dilate
862:pointed
858:bounded
813:polygon
527:(in an
438:Polygon
373:bounded
197:polytop
135:polygon
108:sides (
89:polygon
3759:9-cube
3709:8-cube
3659:7-cube
3616:6-cube
3586:5-cube
3473:Square
3350:Family
3151:shapes
2929:
2893:
2724:, 1974
2705:
2656:
2603:
2435:of an
2429:linear
2185:, and
2164:{5,5}.
2137:Every
2132:5-cell
2096:5-cell
2041:Every
2025:ones.
2014:where
1918:) and
1489:square
1475:square
931:finite
846:convex
805:vertex
784:ridges
774:facets
674:
609:
552:Vertex
377:convex
350:vertex
193:German
191:. The
122:as an
3478:p-gon
3255:Eight
3250:Seven
3230:Three
3107:Krull
2836:arXiv
2145:{∞},
2126:{3},
1443:flags
793:faces
741:Facet
723:Ridge
503:uses
429:Dion
420:Monon
335:cells
323:faces
195:term
111:faces
3836:cube
3509:Cube
3339:and
3240:Five
3235:Four
3215:Zero
3149:and
2927:ISBN
2891:ISBN
2832:2014
2703:ISBN
2654:ISBN
2601:ISBN
2450:In
2443:and
2376:and
2228:and
2181:and
2094:The
1807:The
1522:and
1491:and
1479:cube
818:cell
809:edge
737:− 1
719:− 2
705:Peak
701:− 3
582:Cell
572:Face
562:Edge
525:Term
505:cell
497:edge
481:face
473:face
342:edge
329:, a
325:are
317:. A
313:and
164:and
155:– 1)
143:+ 1)
106:flat
100:, a
3385:(p)
3245:Six
3225:Two
3220:One
2989:– (
2854:doi
2773:doi
2427:of
2246:'s
1576:in
1503:of
1342:of
1295:of
763:An
743:– (
707:– (
536:−1
406:−1
207:by
3905::
3890:•
3886:•
3866:21
3862:•
3859:k1
3855:•
3852:k2
3830:•
3787:•
3757:•
3735:21
3731:•
3728:41
3724:•
3721:42
3707:•
3685:21
3681:•
3678:31
3674:•
3671:32
3657:•
3635:21
3631:•
3628:22
3614:•
3584:•
3563:•
3544:•
3523:•
3507:•
3439:/
3428:/
3418:/
3409:/
3387:/
2971:.
2925:,
2911:;
2889:,
2852:.
2844:.
2834:.
2830:.
2810::
2769:72
2767:.
2763:.
2701:,
2665:^
2612:^
2537:^
2447:.
2419:,
2399:,
2395:,
2391:,
2387:,
2328:.
2033:.
1960:.
1799:.
1015:,
825:.
578:3
568:2
558:1
548:0
454:4
444:3
434:2
426:1
416:0
356:.
239:,
211:.
180:.
87:A
73:A
3874:-
3872:n
3864:k
3857:2
3850:1
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3841:n
3834:-
3832:n
3826:-
3824:n
3817:-
3815:n
3808:-
3806:n
3733:4
3726:2
3719:1
3683:3
3676:2
3669:1
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3455:n
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3446:2
3443:G
3435:4
3432:F
3424:8
3421:E
3415:7
3412:E
3406:6
3403:E
3394:n
3390:D
3383:2
3380:I
3372:n
3368:B
3360:n
3356:A
3328:e
3321:t
3314:v
3261:n
3025:e
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2977:.
2952:.
2936:.
2900:.
2860:.
2856::
2848::
2838::
2781:.
2775::
2437:n
2254:n
2189:.
2114:-
2112:n
2063:n
2059:j
2055:j
2051:n
2047:j
2043:n
2020:n
2016:n
2000:n
1995:C
1877:1
1871:d
1867:)
1863:1
1857:(
1854:=
1779:j
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1716:1
1710:(
1707:+
1704:1
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1696:1
1690:d
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1637:=
1584:d
1574:P
1509:n
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1481:.
1467:.
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1272:t
1269:(
1246:0
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1185:=
1180:d
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1153:)
1150:1
1147:+
1144:t
1141:(
1119:P
1096:1
1075:}
1071:1
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1060:A
1056::
1051:d
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1033:{
1030:=
1025:P
1002:A
976:P
953:d
917:}
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900:2
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887:)
884:y
881:,
878:x
875:(
872:{
801:j
797:j
789:n
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765:n
753:n
745:n
735:n
727:n
717:n
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699:n
647:n
643:j
639:j
634:j
529:n
509:n
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489:j
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477:n
307:p
305:(
293:p
287:p
153:k
151:(
147:k
141:k
139:(
129:n
124:n
120:n
23:.
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