20:
359:. Since an affine transformation preserves straight lines and ratios of areas, it sends equidissections to equidissections. This means that one is free to apply any affine transformation to a polygon that might give it a more manageable form. For example, it is common to choose coordinates such that three of the vertices of a polygon are (0, 1), (0, 0), and (1, 0).
858:-dissection, and that certain quadrilaterals arbitrarily close to being squares have odd equidissections. However, he did not solve the general problem of odd equidissections of squares, and he left it off the exam. Richman's friend John Thomas became interested in the problem; in his recollection,
211:
if the triangles meet only along common edges. Some authors restrict their attention to simplicial dissections, especially in the secondary literature, since they are easier to work with. For example, the usual statement of
Sperner's lemma applies only to simplicial dissections. Often simplicial
998:, a graduate student, "was looking for some algebraic topic she could slip into" the seminar. Sherman Stein suggested dissections of the square and the cube: "a topic that Chakerian grudgingly admitted was geometric." After her talk, Stein asked about regular pentagons. Kasimatis answered with
94:
Equidissections do not have many direct applications. They are considered interesting because the results are counterintuitive at first, and for a geometry problem with such a simple definition, the theory requires some surprisingly sophisticated algebraic tools. Many of the results rely upon
862:"Everyone to whom the problem was put (myself included) said something like 'that is not my area but the question surely must have been considered and the answer is probably well known.' Some thought they had seen it, but could not remember where. I was interested because it reminded me of
845:
Richman wanted to include a question on geometry in the exam, and he noticed that it was difficult to find (what is now called) an odd equidissection of a square. Richman proved to himself that it was impossible for 3 or 5, that the existence of an
929:
of the plane then implies that in all dissections of the square, at least one triangle has an area with what amounts to an even denominator, and therefore all equidissections must be even. The essence of the argument is found already in
828:
remark of Monsky's theorem, "one could have guessed that surely the answer must have been known for a long time (if not to the Greeks)." But the study of equidissections did not begin until 1965, when Fred
Richman was preparing a
66:
Much of the literature is aimed at generalizing Monsky's theorem to broader classes of polygons. The general question is: Which polygons can be equidissected into how many pieces? Particular attention has been given to
1117:. A decade later, Stein made what he describes as "a surprising breakthrough", conjecturing that no polyomino has an odd equidissection. He proved the result of a polyomino with an odd number of squares in
1318:
triangles, how close can the triangle areas be to equal? In particular, what is the smallest possible difference between the areas of the smallest and largest triangles? Let the smallest difference be
1963:
884:"The referee's reaction was predictable. He thought the problem might be fairly easy (although he could not solve it) and was possibly well-known (although he could find no reference to it)."
705:
594:
738:
polygons with more than three sides cannot be equidissected. Although most polygons cannot be cut into equal-area triangles, all polygons can be cut into equal-area quadrilaterals.
981:
535:
1034:
498:
421:
362:
The fact that affine transformations preserve equidissections also means that certain results can be easily generalized. All results stated for a regular polygon also hold for
323:
297:
2414:
874:
Thomas proved that an odd equidissection was impossible if the coordinates of the vertices are rational numbers with odd denominators. He submitted this proof to
325:. A simple example of a non-principal polygon is the quadrilateral with vertices (0, 0), (1, 0), (0, 1), (3/2, 3/2); its spectrum includes 2 and 3 but not 1.
2700:
Combinatorial
Geometry and Graph Theory: Indonesia-Japan Joint Conference, IJCCGGT 2003, Bandung, Indonesia, September 13-16, 2003, Revised Selected Papers
218:, although the vertices of the triangles are not restricted to the vertices or edges of the polygon. Simplicial equidissections are therefore also called
2365:
Labbé, Jean-Philippe; Rote, Günter; Ziegler, Günter M. (2018), "Area
Difference Bounds for Dissections of a Square into an Odd Number of Triangles",
711:
with rational coordinates can be equidissected, although not all of them are principal; see the above example of a kite with a vertex at (3/2, 3/2).
2985:
2715:
2502:
1981:
1911:
1889:
1859:
3050:
1066:
began the study of the spectra of two particular generalizations of squares: trapezoids and kites. Trapezoids have been further studied by
2845:
2296:
2738:
1973:
19:
2070:
773:
910:
then built on Thomas' argument to prove that there are no odd equidissections of a square, without any rationality assumptions.
2007:
Bekker, B. M.; Netsvetaev, N. Yu. (October 1998), "Generalized
Sperner lemma and subdivisions into simplices of equal volume",
991:
3121:
2335:
2227:
1130:
3159:
1969:
1881:
1139:
890:
834:
2257:
Jepsen, Charles H.; Sedberry, Trevor; Hoyer, Rolf (18 March 2009), "Equidissections of kite-shaped quadrilaterals",
3149:
2899:
Su, Zhanjun (2004), "关于一类特殊梯形的等面积三角形划分 (On
Cutting a Family of Special Trapezoids into Triangles of Equal Areas)",
646:
1048:. It is also the first proof to explicitly use an affine transformation to set up a convenient coordinate system.
3127:
731:
2172:
3154:
2773:
340:
1419:
560:
2493:, Raymond W. Brink selected mathematical papers, vol. 3, Mathematical Association of America, pp.
1087:
824:
The idea of an equidissection seems like the kind of elementary geometric concept that should be quite old.
765:
363:
332:
2733:
2219:
2684:
990:
Generalization to regular polygons arrived in 1985, during a geometry seminar run by G. D. Chakerian at
957:
719:
511:
460:
352:
328:
214:
1013:
477:
400:
302:
276:
1060:. They proved that almost all polygons lack equidissections, and that not all polygons are principal.
2688:
2672:
2494:
2486:
1841:
1837:
1311:
1148:
876:
769:
160:
3021:
2843:(December 2000), "A Generalized Conjecture about Cutting a Polygon into Triangles of Equal Areas",
444:
424:
394:
52:
366:; in particular, results concerning the unit square also apply to other parallelograms, including
3095:
3005:
2829:
2790:
2662:
2646:
2610:
2574:
2535:
2464:
2392:
2374:
2195:
2093:
2026:
1940:
336:
207:
36:
3070:
863:
830:
107:
2711:
2498:
1977:
1907:
1885:
1855:
809:
805:
749:
440:
56:
3122:
Sperner’s Lemma, Brouwer’s Fixed-Point
Theorem, And The Subdivision Of Squares Into Triangles
2515:
2291:
1105:
conjectured that no centrally symmetric polygon has an odd equidissection, and he proved the
3103:
3087:
3062:
3038:
3013:
2997:
2973:
2949:
2925:
2888:
2864:
2854:
2840:
2821:
2809:
2798:
2782:
2768:
2757:
2747:
2721:
2703:
2638:
2618:
2602:
2582:
2566:
2543:
2527:
2472:
2456:
2433:
2423:
2384:
2354:
2344:
2330:
2326:
2315:
2305:
2287:
2276:
2266:
2246:
2236:
2203:
2187:
2160:
2142:
2101:
2085:
2058:
2034:
2016:
1987:
1959:
1948:
1932:
1920:
1865:
1847:
1052:
then framed the problem of finding the spectrum of a general polygon, introducing the terms
1045:
995:
922:
746:
597:
432:
96:
2156:
3107:
3066:
3042:
3017:
2977:
2953:
2929:
2892:
2868:
2802:
2761:
2725:
2622:
2586:
2547:
2476:
2437:
2358:
2319:
2280:
2250:
2207:
2164:
2152:
2126:
2105:
2062:
2038:
1991:
1952:
1869:
1418:
obtain a superpolynomial upper bound, derived from an explicit construction that uses the
624:
383:
379:
76:
72:
938:
was the first to use a 2-adic valuation to cover dissections with arbitrary coordinates.
265:. In fact, often a polygon's spectrum consists precisely of the multiples of some number
2676:
1906:, Dolciani Mathematical Expositions, vol. 11, Mathematical Association of America,
3029:
Su, Zhanjun; Ding, Ren (20 September 2004), "Cutting a
Hyperpolyomino into Simplexes",
1900:
1379:
926:
708:
554:
2428:
2109:
1314:
asked the converse problem in 2003: Given a dissection of the whole of a polygon into
3143:
3009:
2961:
2937:
2913:
2876:
2794:
2539:
2349:
2046:
2030:
1944:
1833:
914:
348:
111:
2938:"关于多边形三角划分中的一个逼近问题 (An Approximation Problem About Cutting Polygons into Triangles)"
2097:
2396:
2388:
2707:
2702:, Lecture Notes in Computer Science, vol. 3330, Springer, pp. 146–158,
2130:
1851:
730:) has no equidissection. More generally, no polygon whose vertex coordinates are
2695:
2554:
2511:
2482:
2444:
2215:
448:
103:
2412:
Mead, David G. (September 1979), "Dissection of the hypercube into simplexes",
2241:
2593:
Praton, Iwan (November 2002), "Cutting
Polyominos into Equal-Area Triangles",
2271:
1217:) = 1; otherwise it is less than 1. The authors show that for a quadrilateral
735:
356:
60:
3133:
202:). A general theoretical goal is to compute the spectrum of a given polygon.
1128:
The topic of equidissections has recently been popularized by treatments in
777:
612:
542:
456:
428:
367:
88:
84:
68:
2147:
135:
is a finite set of triangles that do not overlap and whose union is all of
3051:"四边形的等积三角剖分 (Dissections of Quadrilaterals into Triangles of Equal Areas)"
2859:
1090:, chiefly by Professor Ding Ren and his students Du Yatao and Su Zhanjun.
867:
812:
may determine whether or not the spectrum is empty for algebraic numbers
344:
226:
170:
is a dissection in which every triangle has the same area. For a polygon
44:
28:
3099:
3001:
2833:
2786:
2650:
2614:
2578:
2531:
2468:
2310:
2199:
2089:
2021:
1936:
918:
452:
375:
371:
230:
80:
40:
2734:"On the area discrepancy of triangulations of squares and trapezoids"
1398:) with a better dissection, and he proves that there exist values of
1179:? The ratio of the area of the best possible coverage to the area of
898:). When nobody else submitted a solution, the proof was published in
3091:
2825:
2642:
2606:
2570:
2460:
2191:
397:
states that a square has no odd equidissections, so its spectrum is
2812:(March 1999), "Cutting a Polyomino into Triangles of Equal Areas",
2379:
2667:
59:
cannot be equidissected into an odd number of triangles. In fact,
18:
2752:
1923:(March 2004), "Cutting a Polygon into Triangles of Equal Areas",
1902:
Old and New
Unsolved Problems in Plane Geometry and Number Theory
1040:-adic valuation to the complex numbers for each prime divisor of
2771:(June 1989), "Equidissections of centrally symmetric octagons",
2220:"Constructing Equidissections for Certain Classes of Trapezoids"
439:
has an odd equidissection, where a special polygon is one whose
331:
of the plane are useful for studying equidissections, including
48:
2405:
Ungerade Triangulierungen eines Quadrats von kleiner Diskrepanz
2292:"Dissections of regular polygons into triangles of equal areas"
1962:; Szabó, Sándor (2008), "Tiling by Triangles of Equal Areas",
3128:Über die Zerlegung eines Quadrats in Dreiecke gleicher Fläche
2962:"关于Stein猜想的推广 (A Generalization About a Conjecture of Stein)"
888:
The question was instead given as an Advanced Problem in the
269:; in this case, both the spectrum and the polygon are called
51:. The study of equidissections began in the late 1960s with
2966:
Journal of Hebei Normal University (Natural Science Edition)
2918:
Journal of Hebei Normal University (Natural Science Edition)
2881:
Journal of Hebei Normal University (Natural Science Edition)
2051:
Journal of Hebei Normal University (Natural Science Edition)
1965:
Algebra and Tiling: Homomorphisms in the Service of Geometry
1752:
1750:
1455:
1453:
1451:
1449:
1167:
consider a variation of the problem: Given a convex polygon
1513:
1511:
1509:
1507:
2629:
Richman, Fred; Thomas, John (March 1967), "Problem 5471",
382:
coordinates, or polygons whose vertices fall on any other
2071:"More on cutting a polygon into triangles of equal areas"
2557:(June–July 1996), "Calculating a Trapezoidal Spectrum",
2447:(February 1970), "On Dividing a Square Into Triangles",
1044:
and applying some elementary results from the theory of
3078:
Thomas, John (September 1968), "A Dissection Problem",
2986:"Dissections of polygons into triangles of equal areas"
2877:"关于Stein猜想的局部证明 (A Local Proof on Stein's Conjectures)"
2047:"多边形的等积三角剖分 (Further Results about Odd Equidissection)"
808:. It is conjectured that a similar condition involving
1840:(2010), "One square and an odd number of triangles",
1494:
1492:
1016:
960:
649:
563:
514:
480:
403:
305:
279:
2942:
Journal of Hebei Normal University (Natural Science)
2912:
Su, Zhanjun; Wang, Xinke; Tian, Huizhu (July 2002),
1436:
1434:
1036:. Her proof builds on Monsky's proof, extending the
1079:
16:
Partition of a polygon into triangles of equal area
2960:Su, Zhanjun; Wei, Xianglin; Liu, Fuyi (May 2003),
2333:(1 December 1990), "Equidissections of polygons",
1899:
1028:
975:
699:
588:
529:
492:
415:
317:
291:
1093:Attempting to generalize the results for regular
946:The first generalization of Monsky's theorem was
2415:Proceedings of the American Mathematical Society
1772:
1415:
1164:
984:
225:The terms can be extended to higher-dimensional
1175:non-overlapping triangles of equal area inside
917:result that generalizes Sperner's lemma and an
3130:- Notes by Moritz W. Schmitt (German language)
1668:for more precise statements of this principle.
1459:
1082:. General quadrilaterals have been studied in
1063:
1049:
1717:
1153:
1113:= 8 cases. The full conjecture was proved by
895:
825:
700:{\displaystyle S(T(r/s))=\langle r+s\rangle }
431:have no odd equidissections. A conjecture by
299:. For example, the spectrum of a triangle is
8:
2990:Journal of Applied Mathematics and Computing
2914:"关于Stein猜想的研究 (Study on Stein's conjecture)"
2516:"A conjecture of Stein on plane dissections"
2078:Journal of Applied Mathematics and Computing
1705:
1689:
1075:
1023:
1017:
970:
961:
850:-equidissection implies the existence of an
694:
682:
583:
564:
524:
515:
487:
481:
410:
404:
312:
306:
286:
280:
1677:
3134:Tiling Polygons by Triangles of Equal Area
2936:Su, Zhanjun; Wang, Xinke (November 2002),
1756:
1741:
1640:
1565:
1553:
1517:
1483:
1143:
619:is the ratio of parallel side lengths. If
2984:Su, Zhanjun; Ding, Ren (September 2003),
2858:
2751:
2666:
2427:
2378:
2348:
2309:
2270:
2240:
2146:
2020:
1878:Continuous Symmetry: From Euclid to Klein
1616:
1171:, how much of its area can be covered by
1015:
999:
959:
665:
648:
571:
562:
513:
479:
402:
304:
278:
63:polygons cannot be equidissected at all.
1876:Barker, William H.; Howe, Roger (2007),
913:Monsky's proof relies on two pillars: a
378:coordinates also apply to polygons with
147:-dissection, and it is classified as an
3031:Southeast Asian Bulletin of Mathematics
1811:
1529:
1430:
1383:
1257:is affinely congruent to the trapezoid
1086:. Several papers have been authored at
734:has an equidissection. This means that
589:{\displaystyle \langle 2^{n-1}\rangle }
374:. All results stated for polygons with
2698:; Edy Tri Baskoro; Mikio Kano (eds.),
2689:"Equal Area Polygons in Convex Bodies"
2659:On equidissection of balanced polygons
1800:
1796:
1768:
1729:
1665:
1652:
1593:
1589:
1440:
1363:
1122:
1121:. The full conjecture was proved when
1114:
1083:
1071:
1067:
935:
931:
907:
903:
600:from the proof for the octahedron in
249:-equidissection of a triangle for all
2846:Discrete & Computational Geometry
2487:"On Dividing a Square Into Triangles"
2297:Discrete & Computational Geometry
2171:Jepsen, Charles H. (June–July 1996),
1784:
1701:
1604:
1577:
1498:
1471:
1135:
1118:
1102:
1078:. Kites have been further studied by
950:, who proved that the spectrum of an
906:), three years after it was written.
870:, which has a clever odd-even proof."
257:-equidissection, then it also has an
7:
1628:
1541:
947:
643:is a fraction in lowest terms, then
2739:Electronic Journal of Combinatorics
2069:Du, Yatao; Ding, Ren (March 2005),
1974:Mathematical Association of America
1080:Jepsen, Sedberry & Hoyer (2009)
792:)) contains all sufficiently large
423:. More generally, it is known that
253:. As a result, if a polygon has an
2901:Mathematics in Practice and Theory
2407:(Diplomarbeit), Germany: TU Berlin
1898:Klee, Víctor; Wagon, Stan (1991),
1846:(4th ed.), pp. 131–138,
1006:> 5, the spectrum of a regular
976:{\displaystyle \langle n!\rangle }
530:{\displaystyle \langle n!\rangle }
470:> 4, the spectrum of a regular
14:
2559:The American Mathematical Monthly
2449:The American Mathematical Monthly
2429:10.1090/S0002-9939-1979-0537093-6
2180:The American Mathematical Monthly
1414:) decreases arbitrarily quickly.
1029:{\displaystyle \langle n\rangle }
493:{\displaystyle \langle n\rangle }
416:{\displaystyle \langle 2\rangle }
318:{\displaystyle \langle 1\rangle }
292:{\displaystyle \langle m\rangle }
2009:Journal of Mathematical Sciences
1416:Labbé, Rote & Ziegler (2018)
1366:gave the asymptotic upper bound
1165:Sakai, Nara & Urrutia (2005)
3049:Su, Zhanjun; Ding, Ren (2005),
2173:"Equidissections of Trapezoids"
2732:Schulze, Bernd (1 July 2011),
2135:Pacific Journal of Mathematics
2129:; Straus, E. G. (March 1982),
1925:The Mathematical Intelligencer
1131:The Mathematical Intelligencer
985:Bekker & Netsvetaev (1998)
925:on the real numbers. A clever
676:
673:
659:
653:
229:: an equidissection is set of
23:A 6-equidissection of a square
1:
2875:Su, Zhanjun (November 2002),
2814:American Mathematical Monthly
2631:American Mathematical Monthly
2595:American Mathematical Monthly
2389:10.1080/10586458.2018.1459961
1970:Carus Mathematical Monographs
1882:American Mathematical Society
1146:), and the fourth edition of
1140:Carus Mathematical Monographs
891:American Mathematical Monthly
2708:10.1007/978-3-540-30540-8_17
2350:10.1016/0012-365X(90)90384-T
1852:10.1007/978-3-642-00856-6_20
1773:Bekker & Netsvetaev 1998
1064:Kasimatis & Stein (1990)
1050:Kasimatis & Stein (1990)
983:. The proof is revisited by
635:) is principal. In fact, if
453:centrally symmetric polygons
273:and the spectrum is denoted
81:centrally symmetric polygons
1358:and greater than 0 for odd
921:result, the existence of a
835:New Mexico State University
826:Aigner & Ziegler (2010)
504:> 1, the spectrum of an
3176:
2491:Selected Papers on Algebra
2242:10.1016/j.disc.2007.10.031
1460:Kasimatis & Stein 1990
1076:Jepsen & Monsky (2008)
880:, but it was put on hold:
463:are all special polygons.
335:, uniform and non-uniform
3055:Acta Mathematica Scientia
2520:Mathematische Zeitschrift
2272:10.2140/involve.2009.2.89
1718:Aigner & Ziegler 2010
1568:, pp. 121, 128, 131.
1154:Aigner & Ziegler 2010
896:Richman & Thomas 1967
760:) is a trickier case. If
732:algebraically independent
714:At the other extreme, if
549:. and the spectrum of an
220:equal-area triangulations
2774:Aequationes Mathematicae
2657:Rudenko, Daniil (2012),
2367:Experimental Mathematics
1706:Jepsen & Monsky 2008
1690:Jepsen & Monsky 2008
261:-equidissection for all
3124:- Notes by Akhil Mathew
1678:Hales & Straus 1982
1261:(2/3). For a pentagon,
1253:) = 8/9 if and only if
1125:treated the even case.
1088:Hebei Normal University
840:
364:affine-regular polygons
212:dissections are called
205:A dissection is called
143:triangles is called an
2148:10.2140/pjm.1982.99.31
2131:"Projective colorings"
2045:Du, Yatao (May 2003),
1757:Stein & Szabó 2008
1742:Stein & Szabó 2008
1641:Stein & Szabó 2008
1566:Stein & Szabó 2008
1554:Stein & Szabó 2008
1518:Stein & Szabó 2008
1484:Stein & Szabó 2008
1386:improves the bound to
1204:-equidissection, then
1144:Stein & Szabó 2008
1030:
977:
707:. More generally, all
701:
590:
531:
494:
447:edges each sum to the
417:
329:Affine transformations
319:
293:
245:It is easy to find an
55:, which states that a
24:
3136:- Notes by AlexGhitza
2860:10.1007/s004540010021
2683:Sakai, T.; Nara, C.;
1031:
978:
954:-dimensional cube is
720:transcendental number
702:
596:. The latter follows
591:
532:
508:-dimensional cube is
495:
418:
320:
294:
186:exists is called the
155:according to whether
22:
3160:Geometric dissection
3080:Mathematics Magazine
2336:Discrete Mathematics
2327:Kasimatis, Elaine A.
2288:Kasimatis, Elaine A.
2228:Discrete Mathematics
2214:Jepsen, Charles H.;
1976:, pp. 107–134,
1843:Proofs from THE BOOK
1666:Su & Ding (2003)
1338:) for the trapezoid
1149:Proofs from THE BOOK
1084:Su & Ding (2003)
1014:
958:
900:Mathematics Magazine
877:Mathematics Magazine
647:
561:
512:
478:
401:
303:
277:
139:. A dissection into
2677:2012arXiv1206.4591R
2403:Mansow, K. (2003),
2218:(6 December 2008),
1486:, pp. 108–109.
1420:Thue–Morse sequence
1326:) for a square and
1138:), a volume of the
1002:, proving that for
772:or cubic), and its
441:equivalence classes
425:centrally symmetric
182:-equidissection of
3002:10.1007/BF02936072
2787:10.1007/BF01836454
2532:10.1007/BF02571264
2514:(September 1990),
2311:10.1007/BF02187738
2090:10.1007/BF02936053
2022:10.1007/BF02434927
1937:10.1007/BF02985395
1838:Ziegler, Günter M.
1801:Du & Ding 2005
1797:Su & Ding 2003
1653:Su & Ding 2003
1026:
973:
810:stable polynomials
776:all have positive
697:
586:
527:
490:
413:
315:
289:
25:
3150:Discrete geometry
2841:Stein, Sherman K.
2810:Stein, Sherman K.
2769:Stein, Sherman K.
2717:978-3-540-24401-1
2504:978-0-88385-203-3
2331:Stein, Sherman K.
2290:(December 1989),
2235:(23): 5672–5681,
1983:978-0-88385-041-1
1960:Stein, Sherman K.
1921:Stein, Sherman K.
1913:978-0-88385-315-3
1891:978-0-8218-3900-3
1861:978-3-642-00855-9
1826:Secondary sources
1580:, pp. 12–20.
1312:Günter M. Ziegler
1046:cyclotomic fields
806:algebraic integer
750:irrational number
435:proposes that no
174:, the set of all
3167:
3110:
3074:
3069:, archived from
3045:
3025:
3020:, archived from
2980:
2956:
2932:
2908:
2895:
2871:
2862:
2836:
2805:
2781:(2–3): 313–318,
2764:
2755:
2728:
2693:
2679:
2670:
2653:
2625:
2589:
2550:
2507:
2479:
2440:
2431:
2408:
2399:
2382:
2361:
2352:
2322:
2313:
2283:
2274:
2253:
2244:
2224:
2210:
2177:
2167:
2150:
2122:
2121:
2120:
2114:
2108:, archived from
2084:(1–2): 259–267,
2075:
2065:
2041:
2024:
2015:(6): 3492–3498,
1994:
1972:, vol. 25,
1955:
1916:
1905:
1894:
1872:
1815:
1809:
1803:
1794:
1788:
1782:
1776:
1766:
1760:
1754:
1745:
1739:
1733:
1727:
1721:
1715:
1709:
1699:
1693:
1687:
1681:
1675:
1669:
1662:
1656:
1650:
1644:
1638:
1632:
1626:
1620:
1614:
1608:
1602:
1596:
1587:
1581:
1575:
1569:
1563:
1557:
1551:
1545:
1539:
1533:
1527:
1521:
1515:
1502:
1496:
1487:
1481:
1475:
1469:
1463:
1457:
1444:
1438:
1354:) is 0 for even
1160:Related problems
1035:
1033:
1032:
1027:
1000:Kasimatis (1989)
996:Elaine Kasimatis
982:
980:
979:
974:
923:2-adic valuation
857:
841:Monsky's theorem
816:of all degrees.
764:is algebraic of
706:
704:
703:
698:
669:
598:mutatis mutandis
595:
593:
592:
587:
582:
581:
536:
534:
533:
528:
499:
497:
496:
491:
433:Sherman K. Stein
422:
420:
419:
414:
395:Monsky's theorem
324:
322:
321:
316:
298:
296:
295:
290:
233:having the same
110:to more general
100:-adic valuations
77:regular polygons
53:Monsky's theorem
3175:
3174:
3170:
3169:
3168:
3166:
3165:
3164:
3155:Affine geometry
3140:
3139:
3118:
3113:
3092:10.2307/2689143
3077:
3048:
3028:
2983:
2959:
2935:
2911:
2898:
2874:
2839:
2826:10.2307/2589681
2808:
2767:
2731:
2718:
2691:
2682:
2656:
2643:10.2307/2316055
2628:
2607:10.2307/3072370
2592:
2571:10.2307/2974718
2553:
2510:
2505:
2481:
2461:10.2307/2317329
2443:
2411:
2402:
2364:
2325:
2286:
2256:
2222:
2213:
2192:10.2307/2974717
2175:
2170:
2125:
2118:
2116:
2112:
2073:
2068:
2044:
2006:
2000:Primary sources
1997:
1984:
1958:
1919:
1914:
1897:
1892:
1875:
1862:
1832:
1823:
1818:
1810:
1806:
1795:
1791:
1783:
1779:
1771:, p. 251;
1767:
1763:
1755:
1748:
1740:
1736:
1728:
1724:
1716:
1712:
1700:
1696:
1688:
1684:
1676:
1672:
1663:
1659:
1651:
1647:
1639:
1635:
1627:
1623:
1615:
1611:
1603:
1599:
1588:
1584:
1576:
1572:
1564:
1560:
1552:
1548:
1540:
1536:
1528:
1524:
1516:
1505:
1497:
1490:
1482:
1478:
1470:
1466:
1458:
1447:
1439:
1432:
1428:
1291:
1278:
1267:
1248:
1229:
1212:
1191:
1162:
1097:-gons for even
1012:
1011:
956:
955:
944:
942:Generalizations
864:Sperner's Lemma
851:
843:
831:master's degree
822:
709:convex polygons
645:
644:
625:rational number
567:
559:
558:
510:
509:
476:
475:
437:special polygon
399:
398:
392:
301:
300:
275:
274:
243:
149:even dissection
125:
120:
108:Sperner's lemma
17:
12:
11:
5:
3173:
3171:
3163:
3162:
3157:
3152:
3142:
3141:
3138:
3137:
3131:
3125:
3117:
3116:External links
3114:
3112:
3111:
3086:(4): 187–190,
3075:
3061:(5): 718–721,
3057:(in Chinese),
3046:
3037:(3): 573–576,
3026:
2996:(1–2): 29–36,
2981:
2972:(3): 223–224,
2968:(in Chinese),
2957:
2944:(in Chinese),
2933:
2924:(4): 341–342,
2920:(in Chinese),
2909:
2903:(in Chinese),
2896:
2887:(6): 559–560,
2883:(in Chinese),
2872:
2853:(1): 141–145,
2837:
2820:(3): 255–257,
2806:
2765:
2729:
2716:
2680:
2654:
2637:(3): 328–329,
2626:
2601:(9): 818–826,
2590:
2565:(6): 500–501,
2551:
2526:(1): 583–592,
2508:
2503:
2455:(2): 161–164,
2441:
2422:(2): 302–304,
2409:
2400:
2362:
2343:(3): 281–294,
2323:
2304:(1): 375–381,
2284:
2254:
2211:
2186:(6): 498–500,
2168:
2123:
2066:
2057:(3): 220–222,
2042:
2003:
2002:
2001:
1996:
1995:
1982:
1956:
1917:
1912:
1895:
1890:
1873:
1860:
1834:Aigner, Martin
1829:
1828:
1827:
1822:
1819:
1817:
1816:
1804:
1789:
1777:
1775:, p. 3492
1761:
1759:, p. 108.
1746:
1744:, p. 107.
1734:
1732:, p. 187.
1722:
1720:, p. 131.
1710:
1704:, p. 21;
1694:
1682:
1670:
1657:
1645:
1643:, p. 122.
1633:
1621:
1617:Kasimatis 1989
1609:
1597:
1582:
1570:
1558:
1556:, p. 126.
1546:
1544:, p. 302.
1534:
1522:
1520:, p. 120.
1503:
1488:
1476:
1464:
1445:
1429:
1427:
1424:
1384:Schulze (2011)
1380:Big O notation
1287:
1276:
1265:
1246:
1225:
1208:
1187:
1161:
1158:
1025:
1022:
1019:
972:
969:
966:
963:
943:
940:
886:
885:
872:
871:
842:
839:
821:
818:
696:
693:
690:
687:
684:
681:
678:
675:
672:
668:
664:
661:
658:
655:
652:
585:
580:
577:
574:
570:
566:
555:cross-polytope
526:
523:
520:
517:
489:
486:
483:
412:
409:
406:
391:
388:
314:
311:
308:
288:
285:
282:
242:
239:
215:triangulations
168:equidissection
153:odd dissection
124:
121:
119:
116:
112:colored graphs
106:and extending
33:equidissection
15:
13:
10:
9:
6:
4:
3:
2:
3172:
3161:
3158:
3156:
3153:
3151:
3148:
3147:
3145:
3135:
3132:
3129:
3126:
3123:
3120:
3119:
3115:
3109:
3105:
3101:
3097:
3093:
3089:
3085:
3081:
3076:
3073:on 2015-04-02
3072:
3068:
3064:
3060:
3056:
3052:
3047:
3044:
3040:
3036:
3032:
3027:
3024:on 2005-01-18
3023:
3019:
3015:
3011:
3007:
3003:
2999:
2995:
2991:
2987:
2982:
2979:
2975:
2971:
2967:
2963:
2958:
2955:
2951:
2947:
2943:
2939:
2934:
2931:
2927:
2923:
2919:
2915:
2910:
2906:
2902:
2897:
2894:
2890:
2886:
2882:
2878:
2873:
2870:
2866:
2861:
2856:
2852:
2848:
2847:
2842:
2838:
2835:
2831:
2827:
2823:
2819:
2815:
2811:
2807:
2804:
2800:
2796:
2792:
2788:
2784:
2780:
2776:
2775:
2770:
2766:
2763:
2759:
2754:
2749:
2745:
2741:
2740:
2735:
2730:
2727:
2723:
2719:
2713:
2709:
2705:
2701:
2697:
2690:
2686:
2681:
2678:
2674:
2669:
2664:
2660:
2655:
2652:
2648:
2644:
2640:
2636:
2632:
2627:
2624:
2620:
2616:
2612:
2608:
2604:
2600:
2596:
2591:
2588:
2584:
2580:
2576:
2572:
2568:
2564:
2560:
2556:
2552:
2549:
2545:
2541:
2537:
2533:
2529:
2525:
2521:
2517:
2513:
2509:
2506:
2500:
2496:
2492:
2488:
2485:(July 1977),
2484:
2480:Reprinted as
2478:
2474:
2470:
2466:
2462:
2458:
2454:
2450:
2446:
2442:
2439:
2435:
2430:
2425:
2421:
2417:
2416:
2410:
2406:
2401:
2398:
2394:
2390:
2386:
2381:
2376:
2372:
2368:
2363:
2360:
2356:
2351:
2346:
2342:
2338:
2337:
2332:
2328:
2324:
2321:
2317:
2312:
2307:
2303:
2299:
2298:
2293:
2289:
2285:
2282:
2278:
2273:
2268:
2264:
2260:
2255:
2252:
2248:
2243:
2238:
2234:
2230:
2229:
2221:
2217:
2212:
2209:
2205:
2201:
2197:
2193:
2189:
2185:
2181:
2174:
2169:
2166:
2162:
2158:
2154:
2149:
2144:
2140:
2136:
2132:
2128:
2124:
2115:on 2015-04-02
2111:
2107:
2103:
2099:
2095:
2091:
2087:
2083:
2079:
2072:
2067:
2064:
2060:
2056:
2052:
2048:
2043:
2040:
2036:
2032:
2028:
2023:
2018:
2014:
2010:
2005:
2004:
1999:
1998:
1993:
1989:
1985:
1979:
1975:
1971:
1967:
1966:
1961:
1957:
1954:
1950:
1946:
1942:
1938:
1934:
1930:
1926:
1922:
1918:
1915:
1909:
1904:
1903:
1896:
1893:
1887:
1883:
1879:
1874:
1871:
1867:
1863:
1857:
1853:
1849:
1845:
1844:
1839:
1835:
1831:
1830:
1825:
1824:
1820:
1813:
1808:
1805:
1802:
1798:
1793:
1790:
1787:, p. 18.
1786:
1781:
1778:
1774:
1770:
1765:
1762:
1758:
1753:
1751:
1747:
1743:
1738:
1735:
1731:
1726:
1723:
1719:
1714:
1711:
1707:
1703:
1698:
1695:
1691:
1686:
1683:
1680:, p. 42.
1679:
1674:
1671:
1667:
1661:
1658:
1654:
1649:
1646:
1642:
1637:
1634:
1630:
1625:
1622:
1618:
1613:
1610:
1607:, p. 20.
1606:
1601:
1598:
1595:
1591:
1586:
1583:
1579:
1574:
1571:
1567:
1562:
1559:
1555:
1550:
1547:
1543:
1538:
1535:
1531:
1526:
1523:
1519:
1514:
1512:
1510:
1508:
1504:
1501:, p. 17.
1500:
1495:
1493:
1489:
1485:
1480:
1477:
1473:
1468:
1465:
1461:
1456:
1454:
1452:
1450:
1446:
1442:
1437:
1435:
1431:
1425:
1423:
1421:
1417:
1413:
1409:
1405:
1401:
1397:
1393:
1389:
1385:
1381:
1377:
1373:
1369:
1365:
1364:Mansow (2003)
1361:
1357:
1353:
1349:
1345:
1341:
1337:
1333:
1329:
1325:
1321:
1317:
1313:
1309:
1307:
1303:
1299:
1295:
1290:
1286:
1283:) ≥ 3/4, and
1282:
1275:
1271:
1264:
1260:
1256:
1252:
1245:
1241:
1237:
1233:
1228:
1224:
1220:
1216:
1211:
1207:
1203:
1199:
1195:
1190:
1186:
1182:
1178:
1174:
1170:
1166:
1159:
1157:
1155:
1151:
1150:
1145:
1141:
1137:
1133:
1132:
1126:
1124:
1123:Praton (2002)
1120:
1116:
1115:Monsky (1990)
1112:
1108:
1104:
1100:
1096:
1091:
1089:
1085:
1081:
1077:
1073:
1072:Monsky (1996)
1069:
1068:Jepsen (1996)
1065:
1061:
1059:
1055:
1051:
1047:
1043:
1039:
1020:
1009:
1005:
1001:
997:
993:
988:
986:
967:
964:
953:
949:
941:
939:
937:
936:Monsky (1970)
933:
932:Thomas (1968)
928:
924:
920:
916:
915:combinatorial
911:
909:
908:Monsky (1970)
905:
901:
897:
893:
892:
883:
882:
881:
879:
878:
869:
865:
861:
860:
859:
855:
849:
838:
836:
832:
827:
819:
817:
815:
811:
807:
803:
799:
795:
791:
787:
783:
779:
775:
771:
767:
763:
759:
755:
751:
748:
744:
739:
737:
733:
729:
725:
721:
717:
712:
710:
691:
688:
685:
679:
670:
666:
662:
656:
650:
642:
638:
634:
630:
626:
622:
618:
614:
610:
606:
601:
599:
578:
575:
572:
568:
556:
553:-dimensional
552:
548:
544:
540:
521:
518:
507:
503:
484:
473:
469:
464:
462:
458:
454:
450:
446:
442:
438:
434:
430:
427:polygons and
426:
407:
396:
389:
387:
385:
381:
377:
373:
369:
365:
360:
358:
354:
350:
346:
342:
338:
334:
330:
326:
309:
283:
272:
268:
264:
260:
256:
252:
248:
241:Preliminaries
240:
238:
236:
232:
228:
223:
221:
217:
216:
210:
209:
203:
201:
197:
193:
189:
185:
181:
178:for which an
177:
173:
169:
164:
162:
158:
154:
150:
146:
142:
138:
134:
131:of a polygon
130:
122:
117:
115:
113:
109:
105:
101:
99:
92:
90:
86:
82:
78:
74:
70:
64:
62:
58:
54:
50:
46:
42:
38:
34:
30:
21:
3083:
3079:
3071:the original
3058:
3054:
3034:
3030:
3022:the original
2993:
2989:
2969:
2965:
2948:(4): 95–97,
2945:
2941:
2921:
2917:
2907:(1): 145–149
2904:
2900:
2884:
2880:
2850:
2844:
2817:
2813:
2778:
2772:
2753:10.37236/624
2746:(1): #P137,
2743:
2737:
2699:
2658:
2634:
2630:
2598:
2594:
2562:
2558:
2555:Monsky, Paul
2523:
2519:
2512:Monsky, Paul
2490:
2483:Monsky, Paul
2452:
2448:
2445:Monsky, Paul
2419:
2413:
2404:
2370:
2366:
2340:
2334:
2301:
2295:
2265:(1): 89–93,
2262:
2258:
2232:
2226:
2216:Monsky, Paul
2183:
2179:
2141:(2): 31–43,
2138:
2134:
2127:Hales, A. W.
2117:, retrieved
2110:the original
2081:
2077:
2054:
2050:
2012:
2008:
1964:
1931:(1): 17–21,
1928:
1924:
1901:
1877:
1842:
1821:Bibliography
1814:, p. 2.
1812:Schulze 2011
1807:
1792:
1780:
1764:
1737:
1725:
1713:
1697:
1685:
1673:
1660:
1648:
1636:
1624:
1612:
1600:
1585:
1573:
1561:
1549:
1537:
1530:Schulze 2011
1525:
1479:
1467:
1411:
1407:
1403:
1399:
1395:
1391:
1387:
1375:
1371:
1367:
1359:
1355:
1351:
1347:
1343:
1339:
1335:
1331:
1327:
1323:
1319:
1315:
1310:
1305:
1301:
1297:
1293:
1288:
1284:
1280:
1273:
1269:
1262:
1258:
1254:
1250:
1243:
1239:
1235:
1231:
1226:
1222:
1218:
1214:
1209:
1205:
1201:
1197:
1193:
1188:
1184:
1180:
1176:
1172:
1168:
1163:
1147:
1129:
1127:
1119:Stein (1999)
1110:
1106:
1103:Stein (1989)
1098:
1094:
1092:
1062:
1057:
1053:
1041:
1037:
1007:
1003:
989:
951:
945:
912:
899:
889:
887:
875:
873:
853:
847:
844:
823:
813:
801:
797:
793:
789:
785:
781:
761:
757:
753:
742:
740:
727:
723:
715:
713:
640:
636:
632:
628:
620:
616:
608:
604:
602:
550:
546:
538:
505:
501:
471:
467:
465:
436:
393:
390:Best results
361:
353:similarities
351:, and other
333:translations
327:
270:
266:
262:
258:
254:
250:
246:
244:
234:
224:
219:
213:
206:
204:
199:
195:
194:and denoted
191:
187:
183:
179:
175:
171:
167:
165:
156:
152:
148:
144:
140:
136:
132:
128:
126:
104:real numbers
97:
93:
65:
32:
26:
2696:Jin Akiyama
2685:Urrutia, J.
2373:(3): 1–23,
1769:Monsky 1970
1730:Thomas 1968
1708:, p. 3
1594:Praton 2002
1590:Monsky 1990
1441:Monsky 1970
1242:+ 1), with
1183:is denoted
948:Mead (1979)
904:Thomas 1968
451:. Squares,
449:zero vector
357:linear maps
341:reflections
161:even or odd
123:Definitions
3144:Categories
3108:0164.51502
3067:1098.52004
3043:1067.52017
3018:1048.52011
2978:1036.52020
2954:1040.52002
2930:1024.52002
2893:1038.52002
2869:0968.52011
2803:0681.52008
2762:1222.52017
2726:1117.52010
2623:1026.05027
2587:0856.51008
2548:0693.51008
2477:0187.19701
2438:0423.51012
2380:1708.02891
2359:0736.05028
2320:0675.52005
2281:1176.52003
2251:1156.51304
2208:0856.51007
2165:0451.51010
2119:2012-08-06
2106:1066.52017
2063:1036.52019
2039:0891.51013
1992:0930.52003
1953:1186.52015
1870:1185.00001
1785:Stein 2004
1702:Stein 2004
1605:Stein 2004
1578:Stein 2004
1499:Stein 2004
1472:Stein 2004
1426:References
1402:for which
1136:Stein 2004
796:such that
778:real parts
774:conjugates
736:almost all
457:polyominos
429:polyominos
368:rectangles
208:simplicial
129:dissection
95:extending
89:hypercubes
85:polyominos
69:trapezoids
3010:121587469
2795:120042596
2668:1206.4591
2540:122009844
2031:123203936
1945:117930135
1629:Mead 1979
1542:Mead 1979
1304:+ 1) for
1272:) ≥ 2/3,
1058:principal
1024:⟩
1018:⟨
971:⟩
962:⟨
919:algebraic
770:quadratic
747:algebraic
695:⟩
683:⟨
613:trapezoid
584:⟩
576:−
565:⟨
543:factorial
541:! is the
525:⟩
516:⟨
488:⟩
482:⟨
461:polyhexes
411:⟩
405:⟨
372:rhombuses
345:rotations
313:⟩
307:⟨
287:⟩
281:⟨
271:principal
237:-volume.
231:simplexes
227:polytopes
47:of equal
45:triangles
37:partition
2687:(2005),
2098:16100898
1394:) = O(1/
1374:) = O(1/
1346:). Then
1109:= 6 and
1054:spectrum
1010:-gon is
992:UC Davis
927:coloring
868:topology
833:exam at
804:) is an
768:2 or 3 (
537:, where
474:-gon is
445:parallel
380:rational
188:spectrum
118:Overview
29:geometry
3100:2689143
2834:2589681
2673:Bibcode
2651:2316055
2615:3072370
2579:2974718
2495:249–251
2469:2317329
2397:3995120
2259:Involve
2200:2974717
2157:0651484
1378:) (see
1200:has an
820:History
780:, then
752:, then
722:, then
627:, then
611:) be a
384:lattice
376:integer
337:scaling
102:to the
41:polygon
3106:
3098:
3065:
3041:
3016:
3008:
2976:
2952:
2928:
2891:
2867:
2832:
2801:
2793:
2760:
2724:
2714:
2649:
2621:
2613:
2585:
2577:
2546:
2538:
2501:
2475:
2467:
2436:
2395:
2357:
2318:
2279:
2249:
2206:
2198:
2163:
2155:
2104:
2096:
2061:
2037:
2029:
1990:
1980:
1951:
1943:
1910:
1888:
1868:
1858:
1196:). If
1074:, and
934:, but
800:/(1 +
766:degree
745:is an
615:where
500:. For
459:, and
349:shears
151:or an
87:, and
57:square
3096:JSTOR
3006:S2CID
2830:JSTOR
2791:S2CID
2694:, in
2692:(PDF)
2663:arXiv
2647:JSTOR
2611:JSTOR
2575:JSTOR
2536:S2CID
2465:JSTOR
2393:S2CID
2375:arXiv
2223:(PDF)
2196:JSTOR
2176:(PDF)
2113:(PDF)
2094:S2CID
2074:(PDF)
2027:S2CID
1941:S2CID
1308:≥ 5.
1296:) ≥ 2
1234:) ≥ 4
718:is a
623:is a
73:kites
43:into
39:of a
35:is a
31:, an
2712:ISBN
2499:ISBN
1978:ISBN
1908:ISBN
1886:ISBN
1856:ISBN
1664:See
1056:and
856:+ 2)
603:Let
466:For
370:and
355:and
61:most
49:area
3104:Zbl
3088:doi
3063:Zbl
3039:Zbl
3014:Zbl
2998:doi
2974:Zbl
2950:Zbl
2926:Zbl
2889:Zbl
2865:Zbl
2855:doi
2822:doi
2818:106
2799:Zbl
2783:doi
2758:Zbl
2748:doi
2722:Zbl
2704:doi
2639:doi
2619:Zbl
2603:doi
2599:109
2583:Zbl
2567:doi
2563:103
2544:Zbl
2528:doi
2524:205
2473:Zbl
2457:doi
2434:Zbl
2424:doi
2385:doi
2355:Zbl
2345:doi
2316:Zbl
2306:doi
2277:Zbl
2267:doi
2247:Zbl
2237:doi
2233:308
2204:Zbl
2188:doi
2184:103
2161:Zbl
2143:doi
2102:Zbl
2086:doi
2059:Zbl
2035:Zbl
2017:doi
1988:Zbl
1949:Zbl
1933:doi
1866:Zbl
1848:doi
1382:).
1300:/(2
1238:/(4
1156:).
866:in
741:If
557:is
545:of
443:of
190:of
166:An
159:is
27:In
3146::
3102:,
3094:,
3084:41
3082:,
3059:25
3053:,
3035:28
3033:,
3012:,
3004:,
2994:13
2992:,
2988:,
2970:27
2964:,
2946:30
2940:,
2922:26
2916:,
2905:34
2885:26
2879:,
2863:,
2851:24
2849:,
2828:,
2816:,
2797:,
2789:,
2779:37
2777:,
2756:,
2744:18
2742:,
2736:,
2720:,
2710:,
2671:,
2661:,
2645:,
2635:74
2633:,
2617:,
2609:,
2597:,
2581:,
2573:,
2561:,
2542:,
2534:,
2522:,
2518:,
2497:,
2489:,
2471:,
2463:,
2453:77
2451:,
2432:,
2420:76
2418:,
2391:,
2383:,
2371:29
2369:,
2353:,
2341:85
2339:,
2329:;
2314:,
2300:,
2294:,
2275:,
2261:,
2245:,
2231:,
2225:,
2202:,
2194:,
2182:,
2178:,
2159:,
2153:MR
2151:,
2139:99
2137:,
2133:,
2100:,
2092:,
2082:17
2080:,
2076:,
2055:27
2053:,
2049:,
2033:,
2025:,
2013:91
2011:,
1986:,
1968:,
1947:,
1939:,
1929:26
1927:,
1884:,
1880:,
1864:,
1854:,
1836:;
1799:;
1749:^
1592:;
1506:^
1491:^
1448:^
1433:^
1422:.
1410:,
1362:.
1334:,
1221:,
1101:,
1070:,
994:.
987:.
837:.
455:,
386:.
347:,
343:,
339:,
259:mn
222:.
163:.
127:A
114:.
91:.
83:,
79:,
75:,
71:,
3090::
3000::
2857::
2824::
2785::
2750::
2706::
2675::
2665::
2641::
2605::
2569::
2530::
2459::
2426::
2387::
2377::
2347::
2308::
2302:4
2269::
2263:2
2239::
2190::
2145::
2088::
2019::
1935::
1850::
1692:.
1655:.
1631:.
1619:.
1532:.
1474:.
1462:.
1443:.
1412:n
1408:a
1406:(
1404:M
1400:a
1396:n
1392:n
1390:(
1388:M
1376:n
1372:n
1370:(
1368:M
1360:n
1356:n
1352:n
1350:(
1348:M
1344:a
1342:(
1340:T
1336:n
1332:a
1330:(
1328:M
1324:n
1322:(
1320:M
1316:n
1306:n
1302:n
1298:n
1294:K
1292:(
1289:n
1285:t
1281:K
1279:(
1277:3
1274:t
1270:K
1268:(
1266:2
1263:t
1259:T
1255:K
1251:K
1249:(
1247:2
1244:t
1240:n
1236:n
1232:K
1230:(
1227:n
1223:t
1219:K
1215:K
1213:(
1210:n
1206:t
1202:n
1198:K
1194:K
1192:(
1189:n
1185:t
1181:K
1177:K
1173:n
1169:K
1152:(
1142:(
1134:(
1111:n
1107:n
1099:n
1095:n
1042:n
1038:p
1021:n
1008:n
1004:n
968:!
965:n
952:n
902:(
894:(
854:n
852:(
848:n
814:a
802:a
798:n
794:n
790:a
788:(
786:T
784:(
782:S
762:a
758:a
756:(
754:T
743:a
728:a
726:(
724:T
716:a
692:s
689:+
686:r
680:=
677:)
674:)
671:s
667:/
663:r
660:(
657:T
654:(
651:S
641:s
639:/
637:r
633:a
631:(
629:T
621:a
617:a
609:a
607:(
605:T
579:1
573:n
569:2
551:n
547:n
539:n
522:!
519:n
506:n
502:n
485:n
472:n
468:n
408:2
310:1
284:m
267:m
263:n
255:m
251:n
247:n
235:n
200:P
198:(
196:S
192:P
184:P
180:n
176:n
172:P
157:n
145:n
141:n
137:P
133:P
98:p
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
