1206:
122:
905:(mentioned below). Failing that, for a polyhedron with a circumscribed sphere, inscribed sphere, or midsphere (one with all edges as tangents), this can be used. However, it is possible to reciprocate a polyhedron about any sphere, and the resulting form of the dual will depend on the size and position of the sphere; as the sphere is varied, so too is the dual form. The choice of center for the sphere is sufficient to define the dual up to similarity.
945:
31:
932:, where lines and edges are interchanged. Projective polarity works well enough for convex polyhedra. But for non-convex figures such as star polyhedra, when we seek to rigorously define this form of polyhedral duality in terms of projective polarity, various problems appear. Because of the definitional issues for geometric duality of non-convex polyhedra,
1194:
912:
has a face plane, edge line, or vertex lying on the center of the sphere, the corresponding element of its dual will go to infinity. Since
Euclidean space never reaches infinity, the projective equivalent, called extended Euclidean space, may be formed by adding the required 'plane at infinity'. Some
1000:
Even when a pair of polyhedra cannot be obtained by reciprocation from each other, they may be called duals of each other as long as the vertices of one correspond to the faces of the other, and the edges of one correspond to the edges of the other, in an incidence-preserving way. Such pairs of
1035:(poset) of elements, such that incidences, or connections, between elements of the set correspond to incidences between elements (faces, edges, vertices) of a polyhedron. Every such poset has a dual poset, formed by reversing all of the order relations. If the poset is visualized as a
1071:
which is geometrically self-dual about its intersphere: all angles are congruent, as are all edges, so under duality these congruences swap. Similarly, every topologically self-dual convex polyhedron can be realized by an equivalent geometrically self-dual polyhedron, its
189:
about a sphere. Here, each vertex (pole) is associated with a face plane (polar plane or just polar) so that the ray from the center to the vertex is perpendicular to the plane, and the product of the distances from the center to each is equal to the square of the radius.
1095:(with the same number of sides). Adding a frustum (pyramid with the top cut off) below the prism generates another infinite family, and so on. There are many other convex self-dual polyhedra. For example, there are 6 different ones with 7 vertices and 16 with 8 vertices.
1023:
More generally, for any polyhedron whose faces form a closed surface, the vertices and edges of the polyhedron form a graph embedded on this surface, and the vertices and edges of the (abstract) dual polyhedron form the dual graph of the original graph.
971:
If we reciprocate such a canonical polyhedron about its midsphere, the dual polyhedron will share the same edge-tangency points, and thus will also be canonical. It is the canonical dual, and the two together form a canonical dual compound.
1042:
Every geometric polyhedron corresponds to an abstract polyhedron in this way, and has an abstract dual polyhedron. However, for some types of non-convex geometric polyhedra, the dual polyhedra may not be realizable geometrically.
1059:. Geometrically, it is not only topologically self-dual, but its polar reciprocal about a certain point, typically its centroid, is a similar figure. For example, the dual of a regular tetrahedron is another regular tetrahedron,
1066:
Every polygon is topologically self-dual, since it has the same number of vertices as edges, and these are switched by duality. But it is not necessarily self-dual (up to rigid motion, for instance). Every polygon has a
1098:
A self-dual non-convex icosahedron with hexagonal faces was identified by BrĂĽckner in 1900. Other non-convex self-dual polyhedra have been found, under certain definitions of non-convex polyhedra and their duals.
77:
of a polyhedron. Therefore, for many classes of polyhedra defined by their symmetries, the duals belong to a corresponding symmetry class. For example, the regular polyhedra – the (convex)
370:
618:
277:
704:
968:(or intersphere) exists tangent to every edge, and such that the average position of the points of tangency is the center of the sphere. This form is unique up to congruences.
892:
845:
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66:
of the other, and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other. Such dual figures remain combinatorial or
416:
517:
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818:
771:
724:
540:
488:
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390:
297:
211:
183:
70:, but not all can also be constructed as geometric polyhedra. Starting with any given polyhedron, the dual of its dual is the original polyhedron.
1642:
1169:
polytopes, the dual facets will be polar reciprocals of the original's vertex figure. For example, in four dimensions, the vertex figure of the
1821:
1731:
1551:
1161:
In general, the facets of a polytope's dual will be the topological duals of the polytope's vertex figures. For the polar reciprocals of the
147:
There are many kinds of duality. The kinds most relevant to elementary polyhedra are polar reciprocity and topological or abstract duality.
1535:
ICGG 2018 - Proceedings of the 18th
International Conference on Geometry and Graphics: 40th Anniversary - Milan, Italy, August 3-7, 2018
1373:
1851:
1349:
1210:
113:, a geometric transformation that, when applied to a convex polyhedron, realizes the dual polyhedron as another convex polyhedron.
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of the polyhedron), embedded on the surface of the polyhedron (a topological sphere). This graph can be projected to form a
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1972:
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1967:
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if its dual has exactly the same connectivity between vertices, edges, and faces. Abstractly, they have the same
917:
found a way to represent these infinite duals, in a manner suitable for making models (of some finite portion).
1342:
1284:
493:
Typically when no sphere is specified in the construction of the dual, then the unit sphere is used, meaning
216:
1797:
989:
902:
1532:
Wohlleben, Eva (2019), "Duality in Non-Polyhedral Bodies Part I: Polyliner", in
Cocchiarella, Luigi (ed.),
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argues that any proper definition of a non-convex polyhedron should include a notion of a dual polyhedron.
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polyhedron (one in which any two vertices are equivalent under symmetries of the polyhedron) is an
1977:
1962:
1677:
1538:, Advances in Intelligent Systems and Computing, vol. 809, Springer, p. 485–486,
1084:
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There are infinitely many geometrically self-dual polyhedra. The simplest infinite family is the
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984:, each face of the dual polyhedron may be derived from the original polyhedron's corresponding
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of cuboctahedron (light) and rhombic dodecahedron (dark). Pairs of edges meet on their common
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on a flat plane. The graph formed by the vertices and edges of the dual polyhedron is the
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polyhedron (one in which any two faces are equivalent ), and vice versa. The dual of an
1280:
1091:, consists of polyhedra that can be roughly described as a pyramid sitting on top of a
913:
theorists prefer to stick to
Euclidean space and say that there is no dual. Meanwhile,
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42:. Vertices of one correspond to faces of the other, and edges correspond to each other.
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1039:, the dual poset can be visualized simply by turning the Hasse diagram upside down.
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1806:
Shaping Space: Exploring polyhedra in nature, art, and the geometrical imagination
1708:, Algorithms and Combinatorics, vol. 25, Berlin: Springer, pp. 461–488,
129:
can be constructed by connecting the face centers. In general this creates only a
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polyhedron (one in which any two edges are equivalent ) is also isotoxal.
1662:
International
Journal of Mathematical Education in Science and Technology
1590:
Anthony M. Cutler and Egon
Schulte; "Regular Polyhedra of Index Two", I;
1461:, Pages 3-5. (Note, Wenninger's discussion includes nonconvex polyhedra.)
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also discusses some issues on the way to deriving his infinite duals.
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and is centered at the origin (so that it is defined by the equation
136:
1706:
Discrete and
Computational Geometry: The Goodman–Pollack Festschrift
1193:
1204:
1192:
1130:− 1)-dimensional elements, or facets, of the other, and the
943:
29:
901:, it is common to use a sphere centered on this point, as in the
27:
Polyhedron associated with another by swapping vertices for faces
35:
800:. The correspondence between the vertices, edges, and faces of
1577:
1611:, Vol. A 30, Part 4 July 1974, Fig. 3c and accompanying text.
1201:, {4,4}, is self-dual, as shown by these red and blue tilings
1692:(2003), "Are your polyhedra the same as my polyhedra?", in
1434:, "Basic notions about stellation and duality", p. 1.
365:{\displaystyle P^{\circ }=\{q~{\big |}~q\cdot p\leq r^{2}}
1660:
Gailiunas, P.; Sharp, J. (2005), "Duality of polyhedra",
1213:, {∞,∞} in red, and its dual position in blue
1573:, based on paper by Gunnar Brinkmann, Brendan D. McKay,
1001:
polyhedra are still topologically or abstractly dual.
1004:
The vertices and edges of a convex polyhedron form a
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86: – form dual pairs, where the regular
1752:(2007), "Graphs of polyhedra; polyhedra as graphs",
1138:− 1)-dimensional element will correspond to
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1126:The vertices of one polytope correspond to the (
620:the corresponding vertex of the dual polyhedron
1622:Vielecke und Vielflache: Theorie und Geschichte
847:reverses inclusion. For example, if an edge of
1487:
960:Any convex polyhedron can be distorted into a
279:), then the polar dual of a convex polyhedron
1217:The primary class of self-dual polytopes are
894:will be contained in the corresponding face.
867:contains a vertex, the corresponding edge of
332:
8:
1515:
1454:
402:
321:
1607:N. J. Bridge; "Faceting the Dodecahedron",
1598:April 2011, Volume 52, Issue 1, pp 133–161.
1571:Symmetries of Canonical Self-Dual Polyhedra
1291:The self-dual (infinite) regular Euclidean
1228:. All regular polygons, {a} are self-dual,
613:{\displaystyle x_{0}x+y_{0}y+z_{0}z=r^{2},}
1873:, Providence: American Mathematical Soc.,
1051:Topologically, a polyhedron is said to be
1775:
1713:
1647:(2nd ed.), Oxford: Clarendon Press,
1519:
1491:
1458:
1431:
1185:, which are the dual of the icosahedron.
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1808:, New York: Springer, pp. 211–216,
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1424:
272:{\displaystyle x^{2}+y^{2}+z^{2}=r^{2}}
1150:)-dimensional element. The dual of an
1142:hyperplanes that intersect to give a (
1596:Contributions to Algebra and Geometry
1243:The self-dual regular polytopes are:
1189:Self-dual polytopes and tessellations
1076:, reciprocal about the center of the
7:
1800:(2013), "Duality of polyhedra", in
1368:Paracompact hyperbolic honeycombs:
1177:; the dual of the 600-cell is the
699:{\displaystyle (x_{0},y_{0},z_{0})}
1592:Beiträge zur Algebra und Geometrie
1119:in two dimension these are called
25:
1329:The self-dual (infinite) regular
1211:Infinite-order apeirogonal tiling
542:described by the linear equation
1575:Fast generation of planar graphs
1103:Dual polytopes and tessellations
1457:, 3.2 Duality, pp. 78–79;
1354:Compact hyperbolic honeycombs:
1348:Paracompact hyperbolic tiling:
924:here is closely related to the
773:corresponds to an edge line of
726:corresponds to a face plane of
1846:, Cambridge University Press,
1484:GrĂĽnbaum & Shephard (2013)
1276:in 4 dimensions, {3,4,3}.
1107:Duality can be generalized to
693:
654:
109:Duality is closely related to
1:
1154:-dimensional tessellation or
185:is often defined in terms of
1869:Barvinok, Alexander (2002),
1814:10.1007/978-0-387-92714-5_15
1724:10.1007/978-3-642-55566-4_21
1488:Gailiunas & Sharp (2005)
1337:Compact hyperbolic tilings:
1240:of the form {a,b,b,a}, etc.
1061:reflected through the origin
706:. Similarly, each vertex of
54:is associated with a second
1506:, Theorem 3.1, p. 449.
1087:. Another infinite family,
193:When the sphere has radius
165:, the dual of a polyhedron
1994:
1768:10.1016/j.disc.2005.09.037
1516:Cundy & Rollett (1961)
1455:Cundy & Rollett (1961)
1394:Conway polyhedron notation
1158:can be defined similarly.
887:{\displaystyle P^{\circ }}
840:{\displaystyle P^{\circ }}
793:{\displaystyle P^{\circ }}
746:{\displaystyle P^{\circ }}
640:{\displaystyle P^{\circ }}
519:in the above definitions.
154:
1674:10.1080/00207390500064049
1641:; Rollett, A. P. (1961),
1624:, Teubner, Leipzig, 1900.
1544:10.1007/978-3-319-95588-9
62:of one correspond to the
1317:In general, all regular
1285:grand stellated 120-cell
1261:In general, all regular
990:Dorman Luke construction
976:Dorman Luke construction
903:Dorman Luke construction
897:For a polyhedron with a
753:, and each edge line of
439:{\displaystyle q\cdot p}
84:Kepler–Poinsot polyhedra
1321:-dimensional Euclidean
1111:-dimensional space and
1020:of the original graph.
522:For each face plane of
1940:"Self-dual polyhedron"
1609:Acta Crystallographica
1214:
1202:
1134:points that define a (
957:
888:
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747:
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647:will have coordinates
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293:
273:
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179:
144:
73:Duality preserves the
43:
1871:A course in convexity
1323:hypercubic honeycombs
1236:of the form {a,b,a},
1208:
1196:
1033:partially ordered set
1031:is a certain kind of
947:
889:
862:
842:
815:
795:
768:
748:
721:
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446:denotes the standard
441:
413:
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367:
294:
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208:
180:
124:
58:structure, where the
33:
1755:Discrete Mathematics
1074:canonical polyhedron
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851:
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424:
411:{\displaystyle P\},}
396:
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305:
283:
217:
197:
169:
1973:Self-dual polyhedra
1921:"Dual tessellation"
1644:Mathematical Models
1232:of the form {a,a},
1181:, whose facets are
1047:Self-dual polyhedra
1029:abstract polyhedron
996:Topological duality
930:projective geometry
908:If a polyhedron in
512:{\displaystyle r=1}
187:polar reciprocation
157:Polar reciprocation
151:Polar reciprocation
1936:Weisstein, Eric W.
1917:Weisstein, Eric W.
1898:Weisstein, Eric W.
1802:Senechal, Marjorie
1283:{5,5/2,5} and the
1215:
1203:
1089:elongated pyramids
982:uniform polyhedron
964:, in which a unit
958:
899:center of symmetry
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68:abstract polyhedra
44:
18:Self-dual polytope
1902:"Dual polyhedron"
1840:Wenninger, Magnus
1823:978-0-387-92713-8
1733:978-3-642-62442-1
1696:; Basu, Saugata;
1553:978-3-319-95588-9
1409:Self-dual polygon
1350:{∞,∞}
1219:regular polytopes
860:{\displaystyle P}
813:{\displaystyle P}
766:{\displaystyle P}
719:{\displaystyle P}
535:{\displaystyle P}
483:{\displaystyle p}
463:{\displaystyle q}
385:{\displaystyle p}
339:
329:
292:{\displaystyle P}
206:{\displaystyle r}
178:{\displaystyle P}
111:polar reciprocity
94:. The dual of an
16:(Redirected from
1985:
1968:Duality theories
1949:
1948:
1930:
1929:
1911:
1910:
1883:
1864:
1834:
1794:GrĂĽnbaum, Branko
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1762:(3–5): 445–463,
1750:GrĂĽnbaum, Branko
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1690:GrĂĽnbaum, Branko
1684:
1655:
1639:Cundy, H. Martyn
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1520:Wenninger (1983)
1518:, p. 117;
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1492:Wenninger (1983)
1482:See for example
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1459:Wenninger (1983)
1452:
1446:
1441:
1435:
1432:Wenninger (1983)
1429:
1333:honeycombs are:
1325:: {4,3,...,3,4}.
1249:regular polygons
1226:Schläfli symbols
1014:Schlegel diagram
915:Wenninger (1983)
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141:Harmonices Mundi
131:topological dual
117:Kinds of duality
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1444:GrĂĽnbaum (2003)
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1404:Self-dual graph
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1312:Cubic honeycomb
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940:Canonical duals
934:GrĂĽnbaum (2007)
920:The concept of
910:Euclidean space
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1890:External links
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1715:10.1.1.102.755
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1668:(6): 617–642,
1657:
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1620:BrĂĽckner, M.;
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1389:
1386:
1385:
1384:
1366:
1352:
1346:
1327:
1326:
1315:
1309:
1303:
1289:
1288:
1281:great 120-cell
1277:
1270:
1259:
1252:
1190:
1187:
1104:
1101:
1048:
1045:
997:
994:
977:
974:
962:canonical form
941:
938:
881:
877:
856:
834:
830:
809:
787:
783:
762:
740:
736:
715:
695:
690:
686:
682:
677:
673:
669:
664:
660:
656:
634:
630:
609:
604:
600:
596:
593:
588:
584:
580:
577:
572:
568:
564:
561:
556:
552:
531:
508:
505:
502:
479:
459:
435:
432:
429:
407:
404:
401:
381:
359:
355:
351:
348:
345:
342:
334:
326:
323:
320:
315:
311:
301:
299:is defined as
288:
266:
262:
258:
253:
249:
245:
240:
236:
232:
227:
223:
202:
174:
152:
149:
127:Platonic solid
125:The dual of a
118:
115:
34:The dual of a
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
1990:
1979:
1976:
1974:
1971:
1969:
1966:
1964:
1961:
1960:
1958:
1947:
1946:
1941:
1937:
1932:
1928:
1927:
1922:
1918:
1913:
1909:
1908:
1903:
1899:
1894:
1893:
1889:
1882:
1876:
1872:
1867:
1863:
1859:
1855:
1853:0-521-54325-8
1849:
1845:
1841:
1837:
1833:
1829:
1825:
1819:
1815:
1811:
1807:
1803:
1799:
1795:
1791:
1787:
1783:
1778:
1773:
1769:
1765:
1761:
1757:
1756:
1751:
1747:
1743:
1739:
1735:
1729:
1725:
1721:
1716:
1711:
1707:
1703:
1702:Sharir, Micha
1699:
1695:
1694:Aronov, Boris
1691:
1687:
1683:
1679:
1675:
1671:
1667:
1663:
1658:
1654:
1650:
1646:
1645:
1640:
1636:
1635:
1630:
1623:
1617:
1614:
1610:
1604:
1601:
1597:
1593:
1587:
1584:
1581:
1579:
1576:
1572:
1568:
1562:
1559:
1555:
1549:
1545:
1541:
1537:
1536:
1528:
1525:
1522:, p. 30.
1521:
1517:
1512:
1509:
1505:
1500:
1497:
1493:
1489:
1485:
1479:
1476:
1472:
1467:
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1456:
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1440:
1437:
1433:
1428:
1425:
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1407:
1405:
1402:
1400:
1397:
1395:
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1391:
1387:
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1375:
1371:
1367:
1365:
1361:
1357:
1353:
1351:
1347:
1344:
1340:
1336:
1335:
1334:
1332:
1324:
1320:
1316:
1313:
1310:
1307:
1306:Square tiling
1304:
1301:
1298:
1297:
1296:
1294:
1286:
1282:
1278:
1275:
1271:
1269:, {3,3,...,3}
1268:
1264:
1260:
1257:
1253:
1250:
1246:
1245:
1244:
1241:
1239:
1235:
1231:
1227:
1224:
1220:
1212:
1207:
1200:
1199:square tiling
1195:
1188:
1186:
1184:
1180:
1176:
1172:
1168:
1164:
1159:
1157:
1153:
1149:
1145:
1141:
1137:
1133:
1129:
1124:
1122:
1121:dual polygons
1118:
1116:
1110:
1102:
1100:
1096:
1094:
1090:
1086:
1081:
1079:
1075:
1070:
1064:
1062:
1058:
1057:Hasse diagram
1054:
1046:
1044:
1040:
1038:
1037:Hasse diagram
1034:
1030:
1025:
1021:
1019:
1015:
1011:
1007:
1002:
995:
993:
991:
988:by using the
987:
986:vertex figure
983:
975:
973:
969:
967:
963:
955:
951:
950:dual compound
946:
939:
937:
935:
931:
927:
923:
918:
916:
911:
906:
904:
900:
895:
879:
875:
854:
832:
828:
807:
785:
781:
760:
738:
734:
713:
688:
684:
680:
675:
671:
667:
662:
658:
632:
628:
607:
602:
598:
594:
591:
586:
582:
578:
575:
570:
566:
562:
559:
554:
550:
529:
520:
506:
503:
500:
491:
477:
457:
449:
433:
430:
427:
405:
399:
379:
357:
353:
349:
346:
343:
340:
324:
318:
313:
309:
300:
286:
264:
260:
256:
251:
247:
243:
238:
234:
230:
225:
221:
200:
191:
188:
172:
164:
158:
150:
148:
142:
138:
132:
128:
123:
116:
114:
112:
107:
105:
101:
97:
93:
89:
85:
81:
76:
71:
69:
65:
61:
57:
53:
49:
41:
37:
32:
19:
1943:
1924:
1905:
1870:
1843:
1805:
1759:
1753:
1705:
1665:
1661:
1643:
1631:Bibliography
1621:
1616:
1608:
1603:
1595:
1591:
1586:
1574:
1561:
1534:
1527:
1511:
1499:
1478:
1466:
1450:
1439:
1427:
1399:Dual polygon
1345:, ... {p,p}.
1328:
1318:
1290:
1272:The regular
1262:
1242:
1216:
1160:
1151:
1147:
1143:
1139:
1135:
1131:
1127:
1125:
1112:
1108:
1106:
1097:
1082:
1069:regular form
1065:
1052:
1050:
1041:
1026:
1022:
1003:
999:
979:
970:
959:
921:
919:
907:
896:
521:
492:
419:
192:
160:
146:
135:Images from
110:
108:
72:
55:
45:
1844:Dual Models
1698:Pach, János
1473:, Page 143.
1382:{3,3,4,3,3}
1302:: {∞}
1287:{5/2,5,5/2}
1256:tetrahedron
1238:5-polytopes
1234:4-polytopes
1223:palindromic
1183:dodecahedra
1175:icosahedron
448:dot product
88:tetrahedron
82:and (star)
1957:Categories
1880:0821829688
1569:models at
1415:References
1331:hyperbolic
1293:honeycombs
1018:dual graph
1010:1-skeleton
948:Canonical
155:See also:
75:symmetries
52:polyhedron
40:octahedron
1978:Polytopes
1963:Polyhedra
1945:MathWorld
1926:MathWorld
1907:MathWorld
1777:1773/2276
1710:CiteSeerX
1682:120818796
1364:{5,3,3,5}
1314:: {4,3,4}
1300:Apeirogon
1267:simplexes
1230:polyhedra
1156:honeycomb
1115:polytopes
1078:midsphere
1053:self-dual
966:midsphere
954:midsphere
880:∘
833:∘
786:∘
739:∘
633:∘
431:⋅
350:≤
344:⋅
314:∘
100:isohedral
92:self-dual
1842:(1983),
1704:(eds.),
1388:See also
1254:Regular
1179:120-cell
1171:600-cell
1146:−
1085:pyramids
372:for all
104:isotoxal
96:isogonal
60:vertices
50:, every
48:geometry
1862:0730208
1832:3077226
1804:(ed.),
1786:2287486
1742:2038487
1653:0124167
1378:{4,4,4}
1374:{6,3,6}
1370:{3,6,3}
1360:{5,3,5}
1356:{3,5,3}
1308:: {4,4}
1274:24-cell
1258:: {3,3}
1173:is the
1167:uniform
1163:regular
926:duality
922:duality
1877:
1860:
1850:
1830:
1820:
1784:
1740:
1730:
1712:
1680:
1651:
1550:
1486:, and
1380:, and
1362:, and
1251:, {a}.
980:For a
420:where
338:
328:
143:(1619)
137:Kepler
38:is an
1678:S2CID
1420:Notes
1343:{6,6}
1339:{5,5}
1295:are:
1221:with
1113:dual
1093:prism
1008:(the
1006:graph
64:faces
1875:ISBN
1848:ISBN
1818:ISBN
1728:ISBN
1567:Java
1548:ISBN
1279:The
1247:All
1209:The
1197:The
1165:and
820:and
470:and
56:dual
36:cube
1810:doi
1772:hdl
1764:doi
1760:307
1720:doi
1670:doi
1578:PDF
1565:3D
1540:doi
1027:An
928:in
450:of
392:in
161:In
139:'s
90:is
46:In
1959::
1942:,
1938:,
1923:,
1919:,
1904:,
1900:,
1858:MR
1856:,
1828:MR
1826:,
1816:,
1796:;
1782:MR
1780:,
1770:,
1758:,
1738:MR
1736:,
1726:,
1718:,
1700:;
1676:,
1666:36
1664:,
1649:MR
1594:/
1546:,
1490:.
1376:,
1372:,
1358:,
1341:,
1123:.
1080:.
1063:.
992:.
490:.
1884:.
1865:.
1835:.
1812::
1789:.
1774::
1766::
1745:.
1722::
1685:.
1672::
1656:.
1542::
1319:n
1265:-
1263:n
1152:n
1148:j
1144:n
1140:j
1136:j
1132:j
1128:n
1117:;
1109:n
956:.
876:P
855:P
829:P
808:P
782:P
761:P
735:P
714:P
694:)
689:0
685:z
681:,
676:0
672:y
668:,
663:0
659:x
655:(
629:P
608:,
603:2
599:r
595:=
592:z
587:0
583:z
579:+
576:y
571:0
567:y
563:+
560:x
555:0
551:x
530:P
507:1
504:=
501:r
478:p
458:q
434:p
428:q
406:,
403:}
400:P
380:p
358:2
354:r
347:p
341:q
333:|
325:q
322:{
319:=
310:P
287:P
265:2
261:r
257:=
252:2
248:z
244:+
239:2
235:y
231:+
226:2
222:x
201:r
173:P
133:.
20:)
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