1937:) since rotation in the meridional and longitudinal directions of a torus preserves the torus (as opposed to moving it to a different torus). Hence, there is actually a four-dimensional space of Clifford tori. In fact, there is a one-to-one correspondence between Clifford tori in the unit 3-sphere and pairs of polar great circles (i.e., great circles that are maximally separated). Given a Clifford torus, the associated polar great circles are the core circles of each of the two complementary regions. Conversely, given any pair of polar great circles, the associated Clifford torus is the locus of points of the 3-sphere that are equidistant from the two circles.
168:
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464:. The relationship is similar to that of projecting the edges of a cube onto a sheet of paper. Such a projection creates a lower-dimensional image that accurately captures the connectivity of the cube edges, but also requires the arbitrary selection and removal of one of the three fully symmetric and interchangeable axes of the cube.
1933:, we can move the "standard" Clifford torus defined above to other equivalent tori via rigid rotations. These are all called "Clifford tori". The six-dimensional group O(4) acts transitively on the space of all such Clifford tori sitting inside the 3-sphere. However, this action has a two-dimensional stabilizer (see
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593:"-shaped torus is positively curved on the outer rim and negatively curved on the inner. Although having a different geometry than the standard embedding of a torus in three-dimensional Euclidean space, the square torus can also be embedded into three-dimensional space, by the
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1039:{\displaystyle {\tfrac {1}{\sqrt {2}}}S^{1}\times {\tfrac {1}{\sqrt {2}}}S^{1}=\left\{\left.{\tfrac {1}{\sqrt {2}}}(\cos \theta ,\sin \theta ,\cos \varphi ,\sin \varphi )\,\right|\,0\leq \theta <2\pi ,0\leq \varphi <2\pi \right\}.}
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1637:{\displaystyle {\tfrac {1}{\sqrt {2}}}S^{1}\times {\tfrac {1}{\sqrt {2}}}S^{1}=\left\{\left.{\tfrac {1}{\sqrt {2}}}\left(e^{i\theta },e^{i\varphi }\right)\,\right|\,0\leq \theta <2\pi ,0\leq \varphi <2\pi \right\}.}
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2158:
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The
Clifford torus is "flat": Every point has a neighborhood that can be flattened out onto a piece of the plane without distortion, unlike the standard torus of revolution.
581:, are played on a square torus; anything that moves off one edge of the screen reappears on the opposite edge with the same orientation.) It is further known as a
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in contrast requires the highly asymmetric application of a rotation operator to the second circle, since that circle will only have one independent axis
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is the unit 3-sphere, and so the
Clifford torus sits inside this 3-sphere. In fact, the Clifford torus divides this 3-sphere into two congruent
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2859:{\displaystyle T_{r_{1},\ldots ,r_{n}}={\bigl \{}(z_{1},\ldots ,z_{n})\in \mathbf {C} ^{n}:|z_{k}|=r_{k},~1\leqslant k\leqslant n{\bigr \}}.}
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These generalized
Clifford tori are all disjoint from one another. We may once again conclude that the union of each one of these tori
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Borrelli, V.; Jabrane, S.; Lazarus, F.; Thibert, B. (April 2012), "Flat tori in three-dimensional space and convex integration",
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2349:{\displaystyle \operatorname {area} \left(T_{\theta }\right)=4\pi ^{2}\cos \theta \sin \theta =2\pi ^{2}\sin 2\theta ,}
1892:{\displaystyle {\sqrt {{\tfrac {1}{2}}\left|e^{i\theta }\right|^{2}+{\tfrac {1}{2}}\left|e^{i\varphi }\right|^{2}}}=1.}
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2636:{\displaystyle S^{2n-1}=\left\{(z_{1},\ldots ,z_{n})\in \mathbf {C} ^{n}:|z_{1}|^{2}+\cdots +|z_{n}|^{2}=1\right\}.}
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834:{\displaystyle S^{1}={\bigl \{}(\cos \varphi ,\sin \varphi )\,{\big |}\,0\leq \varphi <2\pi {\bigr \}}.}
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In the
Clifford torus as defined above, the distance of any point of the Clifford torus to the origin of
721:{\displaystyle S^{1}={\bigl \{}(\cos \theta ,\sin \theta )\,{\big |}\,0\leq \theta <2\pi {\bigr \}}.}
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is an asymmetric reduced-dimension projection of the maximally symmetric
Clifford torus embedded in
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Clifford tori and their images under conformal transformations are the global minimizers of the
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1444:{\displaystyle S^{1}=\left\{\left.e^{i\varphi }\,\right|\,0\leq \varphi <2\pi \right\}.}
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with the standard symplectic structure. (Of course, any product of embedded circles in
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that is most commonly called the "Clifford torus" – and it is also the only one of the
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422:. The historically popular view that the Cartesian product of two circles is an
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1770:{\displaystyle \left|z_{1}\right|^{2}=\left|z_{2}\right|^{2}={\tfrac {1}{2}}.}
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The
Clifford torus divides the 3-sphere into two congruent solid tori. (In a
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set of ripples running in two perpendicular directions along the surface.
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1220:{\displaystyle x_{1}^{2}+y_{1}^{2}=x_{2}^{2}+y_{2}^{2}={\tfrac {1}{2}}.}
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must be a
Clifford torus. A proof of this conjecture was published by
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It is easy to verify that the
Clifford torus is a minimal surface in
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Still more general definition of
Clifford tori in higher dimensions
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may be expressed in terms of the complex coordinates as follows:
2153:{\displaystyle T_{\theta }=S(\cos \theta )\times S(\sin \theta )}
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585:(the "2" is its topological dimension); figures drawn on it obey
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As before, this is an embedded submanifold, in the unit sphere
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The same circles may be thought of as having radii that are
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The set of all points at a distance of 1 from the origin of
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the
Clifford torus is a submanifold of the unit 3-sphere
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It is also common to consider the Clifford torus as an
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as if it were flat, whereas the surface of a common "
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3118:Brendle, Simon (2013), "Embedded minimal tori in
2227:, each of which corresponds to a great circle of
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437:available to it after the first circle consumes
3044:Proceedings of the National Academy of Sciences
269:of a torus, with opposite edges sewn together.
3151:; see reviews by João Lucas Marques Barbosa (
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1065:, the Clifford torus is an embedded torus in
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624:can be parameterized by an angle coordinate:
1255:Alternative derivation using complex numbers
16:Geometrical object in four-dimensional space
3101:The Australian Mathematical Society Gazette
1981:are sometimes also called "Clifford tori".
53:Learn how and when to remove these messages
1951:that are the product of circles of radius
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150:Learn how and when to remove this message
1941:More general definition of Clifford tori
452:Stated another way, a torus embedded in
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175:This article includes a list of general
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2969:, so these need not be Clifford tori.)
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1686:, then the Clifford torus is given by
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88:adding citations to reliable sources
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18:
2646:Then, for any non-negative numbers
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2174:denotes the circle in the plane
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3124:and the Lawson conjecture",
3092:Norbs, P. (September 2005),
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844:Then the Clifford torus is
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1931:orthogonal transformations
3187:Four-dimensional geometry
3148:10.1007/s11511-013-0101-2
2180:defined by having center
3024:William Kingdom Clifford
2932:stereographic projection
1668:is given by coordinates
1088:is given by coordinates
539:complex coordinate space
326:William Kingdon Clifford
248:stereographic projection
3057:10.1073/pnas.1118478109
196:more precise citations.
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1269:. In two copies of
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84:improve this article
2993:Willmore functional
2945:symplectic geometry
2939:Uses in mathematics
1975:in another 2-plane
1230:This shows that in
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3182:Geometric topology
3094:"The 12th problem"
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3015:Clifford parallel
2974:Lawson conjecture
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252:simple rotation
236:
225:
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207:
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198:
188:Please help to
187:
171:
167:
156:
145:
139:
136:
93:
91:
81:
69:
28:
24:
17:
12:
11:
5:
3200:
3190:
3189:
3184:
3169:
3168:
3132:(2): 177–190,
3110:
3081:
3033:
3031:
3028:
3027:
3026:
3021:
3012:
3010:Hopf fibration
3007:
3000:
2997:
2940:
2937:
2936:
2935:
2928:
2923:
2920:
2911:
2885:
2878:
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2710:
2705:
2701:
2696:
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2669:
2658:
2651:
2644:
2643:
2632:
2628:
2624:
2621:
2616:
2611:
2604:
2600:
2595:
2591:
2588:
2585:
2580:
2575:
2568:
2564:
2559:
2555:
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2545:
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2524:
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2497:
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2449:
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2396:
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2299:
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2283:
2279:
2274:
2270:
2266:
2262:
2259:
2241:
2161:
2160:
2149:
2146:
2143:
2140:
2137:
2134:
2131:
2128:
2125:
2122:
2119:
2116:
2113:
2110:
2105:
2101:
2063:The union for
1942:
1939:
1900:
1899:
1888:
1885:
1878:
1873:
1868:
1865:
1861:
1857:
1849:
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1825:
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1818:
1814:
1806:
1803:
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1426:
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1417:
1412:
1405:
1402:
1398:
1393:
1389:
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1201:
1196:
1191:
1187:
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1137:
1133:
1115:
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1101:
1094:
1047:
1046:
1035:
1031:
1027:
1024:
1021:
1018:
1015:
1012:
1009:
1006:
1003:
1000:
997:
994:
991:
986:
981:
978:
975:
972:
969:
966:
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960:
957:
954:
951:
948:
945:
942:
939:
936:
933:
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902:
898:
890:
886:
880:
875:
871:
863:
859:
842:
841:
830:
825:
820:
817:
814:
811:
808:
805:
799:
793:
790:
787:
784:
781:
778:
775:
772:
769:
764:
759:
754:
750:
729:
728:
717:
712:
707:
704:
701:
698:
695:
692:
686:
680:
677:
674:
671:
668:
665:
662:
659:
656:
651:
646:
641:
637:
606:
603:
491:
475:
402:
386:
370:
354:
316:
300:
279:Clifford torus
238:
237:
220:
219:
174:
172:
165:
158:
157:
72:
70:
63:
58:
32:
31:
29:
22:
15:
9:
6:
4:
3:
2:
3199:
3188:
3185:
3183:
3180:
3179:
3177:
3165:
3161:
3157:
3154:
3149:
3145:
3140:
3135:
3131:
3127:
3122:
3114:
3106:
3102:
3095:
3088:
3086:
3077:
3073:
3068:
3063:
3058:
3053:
3049:
3045:
3038:
3034:
3025:
3022:
3020:
3016:
3013:
3011:
3008:
3006:
3003:
3002:
2996:
2994:
2989:
2987:
2986:Simon Brendle
2983:
2979:
2975:
2970:
2967:
2961:
2955:
2950:
2946:
2933:
2929:
2926:
2925:
2919:
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2897:
2888:
2884:
2877:
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2853:
2843:
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2834:
2831:
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2820:
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2812:
2802:
2798:
2789:
2784:
2774:
2766:
2762:
2758:
2755:
2752:
2747:
2743:
2729:
2722:
2718:
2714:
2711:
2708:
2703:
2699:
2694:
2686:
2685:
2684:
2679:
2675:
2668:
2661:
2657:
2650:
2630:
2626:
2622:
2619:
2614:
2602:
2598:
2589:
2586:
2583:
2578:
2566:
2562:
2553:
2548:
2538:
2530:
2526:
2522:
2519:
2516:
2511:
2507:
2499:
2495:
2490:
2487:
2484:
2481:
2477:
2469:
2468:
2467:
2464:
2460:
2445:
2442:
2435:
2431:
2424:
2420:
2416:is the torus
2403:
2395:
2391:. This torus
2389:
2371:
2363:
2343:
2340:
2337:
2334:
2331:
2326:
2322:
2318:
2315:
2312:
2309:
2306:
2303:
2300:
2297:
2292:
2288:
2284:
2281:
2277:
2272:
2268:
2264:
2260:
2257:
2250:
2249:
2248:
2244:
2240:
2234:
2231:
2215:
2207:
2200:
2194:
2188:
2178:
2171:
2167:
2144:
2141:
2138:
2132:
2129:
2123:
2120:
2117:
2111:
2108:
2103:
2099:
2091:
2090:
2089:
2076:
2068:
2061:
2048:
2040:
2033:
2017:
2009:
2004:in the range
2002:
1996:
1989:
1982:
1979:
1971:
1961:
1955:
1949:
1938:
1936:
1932:
1927:
1922:
1917:
1915:
1911:
1906:
1886:
1883:
1876:
1871:
1866:
1863:
1859:
1855:
1847:
1844:
1838:
1833:
1828:
1823:
1820:
1816:
1812:
1804:
1801:
1789:
1788:
1787:
1784:
1764:
1758:
1755:
1749:
1744:
1739:
1734:
1730:
1726:
1721:
1716:
1711:
1706:
1702:
1698:
1689:
1688:
1687:
1680:
1673:
1666:
1660:
1657:
1651:
1631:
1627:
1623:
1620:
1617:
1614:
1611:
1608:
1605:
1602:
1599:
1596:
1593:
1590:
1587:
1582:
1576:
1570:
1567:
1563:
1559:
1554:
1551:
1547:
1542:
1534:
1530:
1519:
1515:
1510:
1506:
1498:
1494:
1488:
1483:
1479:
1471:
1467:
1457:
1456:
1455:
1438:
1434:
1430:
1427:
1424:
1421:
1418:
1415:
1410:
1403:
1400:
1396:
1387:
1383:
1378:
1374:
1366:
1365:
1364:
1346:
1342:
1339:
1336:
1333:
1330:
1327:
1322:
1315:
1312:
1308:
1299:
1295:
1290:
1286:
1278:
1277:
1276:
1273:
1267:
1262:
1252:
1249:
1243:
1240:
1234:
1214:
1208:
1205:
1199:
1194:
1189:
1185:
1181:
1176:
1171:
1167:
1163:
1158:
1153:
1149:
1145:
1140:
1135:
1131:
1123:
1122:
1121:
1114:
1107:
1100:
1093:
1086:
1080:
1077:
1073:
1069:
1063:
1058:
1053:
1033:
1029:
1025:
1022:
1019:
1016:
1013:
1010:
1007:
1004:
1001:
998:
995:
992:
989:
984:
976:
973:
970:
967:
964:
961:
958:
955:
952:
949:
946:
943:
940:
937:
934:
924:
920:
909:
905:
900:
896:
888:
884:
878:
873:
869:
861:
857:
847:
846:
845:
828:
818:
815:
812:
809:
806:
803:
788:
785:
782:
779:
776:
773:
770:
757:
752:
748:
740:
739:
738:
735:
715:
705:
702:
699:
696:
693:
690:
675:
672:
669:
666:
663:
660:
657:
644:
639:
635:
627:
626:
625:
622:
616:
612:
602:
600:
596:
592:
588:
584:
580:
576:
572:
568:
564:
559:
556:
550:
544:
540:
535:
529:
525:
494:
487:
478:
471:
465:
462:
456:
450:
447:
441:
435:
430:
426:
420:
414:
405:
398:
389:
382:
373:
366:
357:
350:
344:
340:. To see why
338:
332:
327:
319:
312:
303:
296:
292:
288:
284:
280:
276:
268:
264:
259:
253:
249:
244:
234:
231:
216:
213:
205:
202:November 2019
195:
191:
185:
184:
178:
173:
164:
163:
154:
151:
143:
140:November 2019
132:
129:
125:
122:
118:
115:
111:
108:
104:
101: –
100:
96:
95:Find sources:
89:
85:
79:
78:
73:This article
71:
67:
62:
61:
56:
54:
47:
46:
41:
40:
35:
30:
21:
20:
3129:
3125:
3120:
3113:
3107:(4): 244–246
3104:
3100:
3047:
3043:
3037:
2990:
2982:round metric
2971:
2965:
2959:
2953:
2942:
2912:
2908:
2902:
2895:
2892:is the unit
2886:
2882:
2875:
2871:
2868:
2677:
2673:
2666:
2659:
2655:
2648:
2645:
2462:
2458:
2451:
2440:
2433:
2429:
2422:
2418:
2401:
2393:
2387:
2369:
2361:
2358:
2242:
2238:
2235:
2229:
2213:
2205:
2198:
2192:
2186:
2176:
2169:
2165:
2162:
2074:
2066:
2062:
2046:
2038:
2031:
2015:
2007:
2000:
1994:
1987:
1983:
1977:
1969:
1959:
1953:
1947:
1944:
1935:group action
1925:
1918:
1904:
1901:
1782:
1779:
1678:
1671:
1664:
1661:
1655:
1649:
1646:
1453:
1362:
1271:
1265:
1258:
1247:
1244:
1238:
1232:
1229:
1112:
1105:
1098:
1091:
1084:
1081:
1075:
1071:
1067:
1061:
1051:
1048:
843:
733:
730:
620:
614:
608:
582:
577:, including
563:square torus
562:
560:
554:
548:
542:
533:
527:
492:
485:
476:
469:
466:
460:
454:
451:
445:
439:
433:
424:
418:
416:rather than
412:
403:
396:
387:
380:
371:
364:
355:
348:
342:
336:
330:
317:
310:
301:
294:
278:
272:
226:
208:
199:
180:
146:
137:
127:
120:
113:
106:
94:
82:Please help
77:verification
74:
50:
43:
37:
36:Please help
33:
3005:Duocylinder
2236:This torus
2184:and radius
1963:and radius
1057:submanifold
611:unit circle
575:video games
194:introducing
3176:Categories
3164:1305.53061
3030:References
2922:Properties
2664:such that
1910:solid tori
177:references
110:newspapers
39:improve it
3139:1203.6597
2988:in 2013.
2841:⩽
2835:⩽
2775:∈
2756:…
2712:…
2587:⋯
2539:∈
2520:…
2488:−
2341:θ
2335:
2323:π
2313:θ
2310:
2304:θ
2301:
2289:π
2273:θ
2261:
2145:θ
2142:
2130:×
2124:θ
2121:
2104:θ
1867:φ
1824:θ
1624:π
1615:φ
1612:≤
1603:π
1594:θ
1591:≤
1571:φ
1555:θ
1489:×
1431:π
1422:φ
1419:≤
1404:φ
1343:π
1334:θ
1331:≤
1316:θ
1263:torus in
1026:π
1017:φ
1014:≤
1005:π
996:θ
993:≤
977:φ
974:
965:φ
962:
953:θ
950:
941:θ
938:
879:×
819:π
810:φ
807:≤
789:φ
786:
777:φ
774:
706:π
697:θ
694:≤
676:θ
673:
664:θ
661:
579:Asteroids
567:isometric
443:and
263:rectangle
45:talk page
3076:22523238
2999:See also
2900:-sphere
2672:+ ... +
1923:acts on
1261:embedded
591:doughnut
546:, since
524:3-sphere
3156:3143888
3067:3358891
2881:, ...,
2654:, ...,
2412:
2398:
2380:
2366:
2224:
2210:
2163:(where
2085:
2071:
2057:
2043:
2026:
2012:
1966:√
599:fractal
519:
512:√
502:
291:circles
289:of two
265:is the
190:improve
124:scholar
3162:
3074:
3064:
2829:
2182:(0, 0)
1919:Since
571:square
277:, the
179:, but
126:
119:
112:
105:
97:
3134:arXiv
3097:(PDF)
1912:(see
569:to a
429:torus
131:JSTOR
117:books
3072:PMID
3017:and
2972:The
2898:− 1)
2258:area
2203:and
2065:0 ≤
2036:and
2006:0 ≤
1993:sin
1991:and
1986:cos
1968:1 −
1921:O(4)
1618:<
1597:<
1425:<
1363:and
1337:<
1020:<
999:<
813:<
700:<
609:The
483:and
394:and
362:and
308:and
283:flat
103:news
3160:Zbl
3144:doi
3130:211
3062:PMC
3052:doi
3048:109
2951:of
2943:In
2918:).
2916:= 0
2681:= 1
2332:sin
2307:sin
2298:cos
2201:= 0
2139:sin
2118:cos
2060:).
2034:= 0
1929:by
1916:).
1786:is
1662:If
1653:in
1082:If
1059:of
971:sin
959:cos
947:sin
935:cos
783:sin
771:cos
670:sin
658:cos
618:in
467:If
273:In
86:by
3178::
3153:MR
3142:,
3128:,
3105:32
3103:,
3099:,
3084:^
3070:,
3060:,
3046:,
2995:.
2894:(2
2461:=
2444:.
2208:=
2069:≤
2041:=
2010:≤
1887:1.
1677:,
1659:.
1251:.
1242:.
1111:,
1104:,
1097:,
1074:=
1070:×
558:.
449:.
246:A
48:.
3166:)
3146::
3136::
3121:S
3079:.
3054::
2966:C
2960:C
2954:C
2913:k
2909:r
2903:S
2896:n
2887:n
2883:r
2879:1
2876:r
2872:T
2854:.
2849:}
2844:n
2838:k
2832:1
2826:,
2821:k
2817:r
2813:=
2809:|
2803:k
2799:z
2794:|
2790::
2785:n
2780:C
2772:)
2767:n
2763:z
2759:,
2753:,
2748:1
2744:z
2740:(
2735:{
2730:=
2723:n
2719:r
2715:,
2709:,
2704:1
2700:r
2695:T
2678:n
2674:r
2670:1
2667:r
2660:n
2656:r
2652:1
2649:r
2631:.
2627:}
2623:1
2620:=
2615:2
2610:|
2603:n
2599:z
2594:|
2590:+
2584:+
2579:2
2574:|
2567:1
2563:z
2558:|
2554::
2549:n
2544:C
2536:)
2531:n
2527:z
2523:,
2517:,
2512:1
2508:z
2504:(
2500:{
2496:=
2491:1
2485:n
2482:2
2478:S
2463:C
2459:R
2454:S
2441:S
2434:θ
2430:T
2423:θ
2419:T
2409:4
2406:/
2402:π
2394:T
2388:π
2386:2
2377:4
2374:/
2370:π
2362:T
2344:,
2338:2
2327:2
2319:2
2316:=
2293:2
2285:4
2282:=
2278:)
2269:T
2265:(
2243:θ
2239:T
2230:S
2221:2
2218:/
2214:π
2206:θ
2199:θ
2193:S
2187:r
2177:R
2172:)
2170:r
2168:(
2166:S
2148:)
2136:(
2133:S
2127:)
2115:(
2112:S
2109:=
2100:T
2082:2
2079:/
2075:π
2067:θ
2054:2
2051:/
2047:π
2039:θ
2032:θ
2023:2
2020:/
2016:π
2008:θ
2001:θ
1995:θ
1988:θ
1978:R
1970:r
1960:R
1954:r
1948:S
1926:R
1905:C
1884:=
1877:2
1872:|
1864:i
1860:e
1856:|
1848:2
1845:1
1839:+
1834:2
1829:|
1821:i
1817:e
1813:|
1805:2
1802:1
1783:C
1765:.
1759:2
1756:1
1750:=
1745:2
1740:|
1735:2
1731:z
1727:|
1722:=
1717:2
1712:|
1707:1
1703:z
1699:|
1684:)
1682:2
1679:z
1675:1
1672:z
1670:(
1665:C
1656:C
1650:S
1632:.
1628:}
1621:2
1609:0
1606:,
1600:2
1588:0
1583:|
1577:)
1568:i
1564:e
1560:,
1552:i
1548:e
1543:(
1535:2
1531:1
1520:{
1516:=
1511:1
1507:S
1499:2
1495:1
1484:1
1480:S
1472:2
1468:1
1439:.
1435:}
1428:2
1416:0
1411:|
1401:i
1397:e
1388:{
1384:=
1379:1
1375:S
1347:}
1340:2
1328:0
1323:|
1313:i
1309:e
1300:{
1296:=
1291:1
1287:S
1272:C
1266:C
1248:S
1239:S
1233:R
1215:.
1209:2
1206:1
1200:=
1195:2
1190:2
1186:y
1182:+
1177:2
1172:2
1168:x
1164:=
1159:2
1154:1
1150:y
1146:+
1141:2
1136:1
1132:x
1118:)
1116:2
1113:y
1109:2
1106:x
1102:1
1099:y
1095:1
1092:x
1090:(
1085:R
1078:.
1076:R
1072:R
1068:R
1062:R
1052:S
1034:.
1030:}
1023:2
1011:0
1008:,
1002:2
990:0
985:|
980:)
968:,
956:,
944:,
932:(
925:2
921:1
910:{
906:=
901:1
897:S
889:2
885:1
874:1
870:S
862:2
858:1
829:.
824:}
816:2
804:0
798:|
792:)
780:,
768:(
763:{
758:=
753:1
749:S
734:R
716:.
711:}
703:2
691:0
685:|
679:)
667:,
655:(
650:{
645:=
640:1
636:S
621:R
615:S
555:R
549:C
543:C
534:R
528:S
514:2
508:/
505:1
493:b
486:S
477:a
470:S
461:R
455:R
446:y
440:x
434:z
425:R
419:R
413:R
404:b
397:R
388:a
381:R
372:b
365:S
356:a
349:S
343:R
337:R
331:R
318:b
311:S
302:a
295:S
233:)
227:(
215:)
209:(
204:)
200:(
186:.
153:)
147:(
142:)
138:(
128:·
121:·
114:·
107:·
80:.
55:)
51:(
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