362:
112:
97:
234:
1266:
951:
250:
In three dimensions, the kissing number is 12, but the correct value was much more difficult to establish than in dimensions one and two. It is easy to arrange 12 spheres so that each touches a central sphere, with a lot of space left over, and it is not obvious that there is no way to pack in a 13th
1719:
Note that in one dimension, "spheres" are just pairs of points separated by the unit distance. (The vertical dimension of one-dimensional illustration is merely evocative.) Unlike in higher dimensions, it is necessary to specify that the interior of the spheres (the unit-length open intervals) do
79:
Finding the kissing number when centers of spheres are confined to a line (the one-dimensional case) or a plane (two-dimensional case) is trivial. Proving a solution to the three-dimensional case, despite being easy to conceptualise and model in the physical world, eluded mathematicians until the
290:
In four dimensions, the kissing number is 24. This was proven in 2003 by Oleg Musin. Previously, the answer was thought to be either 24 or 25: it is straightforward to produce a packing of 24 spheres around a central sphere (one can place the spheres at the vertices of a suitably scaled
1012:
251:
sphere. (In fact, there is so much extra space that any two of the 12 outer spheres can exchange places through a continuous movement without any of the outer spheres losing contact with the center one.) This was the subject of a famous disagreement between mathematicians
761:
350:= 24 dimensions. For 5, 6, and 7 dimensions the arrangement with the highest known kissing number found so far is the optimal lattice arrangement, but the existence of a non-lattice arrangement with a higher kissing number has not been excluded.
1457:
80:
mid-20th century. Solutions in higher dimensions are considerably more challenging, and only a handful of cases have been solved exactly. For others, investigations have determined upper and lower bounds, but not exact solutions.
711:
where the approximation ratio depends on the kissing number. For example, there is a polynomial-time 10-approximation algorithm to find a maximum non-intersecting subset of a set of rotated unit squares.
377:. The base of exponential growth is not known. The grey area in the above plot represents the possible values between known upper and lower bounds. Circles represent values that are known exactly.
259:. Newton correctly thought that the limit was 12; Gregory thought that a 13th could fit. Some incomplete proofs that Newton was correct were offered in the nineteenth century, most notably one by
1261:{\displaystyle \exists xy\ \left\{\sum _{n}\left(x_{n}^{\textsf {T}}x_{n}-1\right)^{2}+\sum _{m,n:m<n}{\Big (}(x_{n}-x_{m})^{\textsf {T}}(x_{n}-x_{m})-1-(y_{nm})^{2}{\Big )}^{2}=0\right\}}
1538:
which must be solved for real values only. The two methods, being entirely equivalent, have various different uses. For example, in the second case one can randomly alter the values of the
3257:
1339:
2277:
2364:
1532:
2672:
2141:
2211:
695:, for example, it is known that both the lattice kissing number and the translative kissing number are equal to 18, whereas the congruent kissing number is at least 56.
687:
that touch a given copy of the body. There are different versions of the problem depending on whether the copies are only required to be congruent to the original body,
2165:
1326:
991:
358:
The following table lists some known bounds on the kissing number in various dimensions. The dimensions in which the kissing number is known are listed in boldface.
749:
946:{\displaystyle \exists x\ \left\{\forall _{n}\{x_{n}^{\textsf {T}}x_{n}=1\}\land \forall _{m,n:m\neq n}\{(x_{n}-x_{m})^{\textsf {T}}(x_{n}-x_{m})\geq 1\}\right\}}
1768:
Zong, Chuanming (2008). "The kissing number, blocking number and covering number of a convex body". In
Goodman, Jacob E.; Pach, János; Pollack, Richard (eds.).
2607:
1978:
Machado, Fabrício C.; Oliveira, Fernando M. (2018). "Improving the
Semidefinite Programming Bound for the Kissing Number by Exploiting Polynomial Symmetry".
758:-dimensional position vectors of the centres of the spheres. The condition that this set of spheres can lie round the centre sphere without overlapping is:
39:(arrangement of spheres) in a given space, a kissing number can also be defined for each individual sphere as the number of spheres it touches. For a
1770:
Surveys on
Discrete and Computational Geometry: Twenty Years Later (AMS-IMS-SIAM Joint Summer Research Conference, June 18ÔÇô22, 2006, Snowbird, Utah)
43:
packing the kissing number is the same for every sphere, but for an arbitrary sphere packing the kissing number may vary from one sphere to another.
3237:
2575:
3353:
1282: + 1 vectors would be equivalent to determining the existence of real solutions to a quartic polynomial in 1025 variables. For the
237:
A highly symmetrical realization of the kissing number 12 in three dimensions is by aligning the centers of outer spheres with vertices of a
3299:
2664:
3349:
2787:
3212:
3189:
2945:
2893:
295:
centered at the origin), but, as in the three-dimensional case, there is a lot of space left over — even more, in fact, than for
2797:
2535:
1649:
2824:
2923:
2634:
1731:
Marathe, M. V.; Breu, H.; Hunt, H. B.; Ravi, S. S.; Rosenkrantz, D. J. (1995). "Simple heuristics for unit disk graphs".
2836:
1980:
1667:
2814:
2626:
2397:
1785:
1613:
3222:
3047:
1665:
Mittelmann, Hans D.; Vallentin, Frank (2010). "High accuracy semidefinite programming bounds for kissing numbers".
3242:
3144:
3042:
2863:
2048:
1924:
956:
3486:
3418:
3292:
2388:
2225:
679:
The kissing number problem can be generalized to the problem of finding the maximum number of non-overlapping
3127:
1462:
256:
2314:
1772:. Contemporary Mathematics. Vol. 453. Providence, RI: American Mathematical Society. pp. 529–548.
274:
in which all atoms have the same size (as in a chemical element). A coordination number of 12 is found in a
158:
Assume by contradiction that there are more than six touching circles. Then at least two adjacent rays, say
3413:
3037:
1476:
3232:
3091:
2888:
2982:
3507:
3403:
3357:
3345:
3032:
3027:
2950:
2853:
2819:
2782:
2641:
2528:
1892:
1641:
1294: + 1, the quartic would have 19,322,732,544 variables. An alternative statement in terms of
963:
which is why this problem has only been solved up to four dimensions. By adding additional variables,
3512:
3393:
3285:
2997:
2913:
2567:
2287:
are needed. Or, equivalent, the matrix can be assumed to be antisymmetric. Anyway the matrix has just
1886:(1979). "О границах для упаковок в n-мерном евклидовом пространстве" [On bounds for packings in
3408:
2868:
2809:
2792:
2777:
2722:
2402:
2100:
721:
1452:{\displaystyle \exists R\ \{\forall _{n}\{R_{0n}=1\}\land \forall _{m,n:m<n}\{R_{mn}\geq 1\}\}}
2846:
2599:
2498:
2420:; Vallentin, Frank (2008). "New upper bounds for kissing numbers from semidefinite programming".
1570:
704:
2178:
35:
that can be arranged in that space such that they each touch a common unit sphere. For a given
3367:
3217:
3207:
2972:
2935:
2908:
2802:
688:
279:
263:, but the first correct proof (according to Brass, Moser, and Pach) did not appear until 1953.
2150:
3398:
3264:
3171:
3057:
3052:
2841:
2754:
2729:
2521:
1605:
1555:
680:
76:. Ordinary spheres correspond to two-dimensional closed surfaces in three-dimensional space.
1301:
966:
3178:
2898:
2655:
2459:
2439:
1883:
1822:
1795:
1686:
727:
1857:
8:
3460:
3322:
2987:
2967:
2883:
2873:
2734:
692:
267:
238:
2443:
1826:
1690:
3202:
2918:
2831:
2559:
2463:
2429:
2068:
2026:
2007:
1989:
1920:"New bounds on the number of unit spheres that can touch a unit sphere in n dimensions"
1838:
1740:
1702:
1676:
1593:
708:
370:
275:
28:
3465:
3362:
3161:
3133:
3015:
2992:
2940:
2858:
2744:
2393:
1954:
1938:
1919:
1861:
1842:
1781:
1645:
1609:
1565:
1295:
2011:
1834:
720:
The kissing number problem can be stated as the existence of a solution to a set of
3308:
3121:
3097:
3073:
2903:
2649:
2447:
2417:
2072:
2060:
1999:
1957:
1933:
1830:
1773:
1750:
1698:
1694:
994:
960:
2467:
2451:
2003:
1706:
3109:
2878:
2772:
2455:
1791:
339:
271:
73:
40:
3444:
3385:
3372:
3337:
3152:
2977:
2930:
2717:
1911:
1777:
1560:
260:
36:
2483:
2064:
241:. This leaves slightly more than 0.1 of the radius between two nearby spheres.
3501:
3481:
3428:
3197:
3079:
3021:
2406:
2147:
positional vectors. As the condition behind the second universal quantifier (
1915:
1291:
666:
313:
1633:
3085:
2955:
2764:
2739:
2702:
2544:
2384:
1754:
252:
959:. However, general methods of solving problems in this form take at least
266:
The twelve neighbors of the central sphere correspond to the maximum bulk
3327:
3103:
2712:
2591:
2494:
2410:
1597:
684:
338:
arrangements, in which the centres of the spheres all lie on points in a
2434:
1542:
by small amounts to try to minimise the polynomial in terms of the
3227:
3115:
1279:
489:
474:
461:
448:
438:
426:
414:
303:
2175:
are exchanged, it is sufficient to let this quantor extend just over
1962:
1745:
328:
210:
3277:
2749:
2690:
2513:
1994:
62:
20:
1681:
2707:
2695:
2583:
361:
292:
2295: − 1)/2 free scalar variables. In addition, there are
3261:
2213:. For simplification the sphere radiuses are assumed to be 1/2.
32:
1862:"Kissing numbers, sphere packings, and some unexpected proofs"
1813:
O. R. Musin (2003). "The problem of the twenty-five spheres".
1720:
not overlap in order for there to be a finite packing density.
230:. Therefore, the circles 1 and 2 intersect – a contradiction.
233:
111:
96:
955:
Thus the problem for each dimension can be expressed in the
178:, are separated by an angle of less than 60°. The segments
691:
of the original body, or translated by a lattice. For the
31:
is defined as the greatest number of non-overlapping unit
1534:
gives a set of simultaneous polynomial equations in just
2025:
Lagarias, Jeffrey C.; Zong, Chuanming (December 2012).
1952:
46:
Other names for kissing number that have been used are
1592:
1461:
This must be supplemented with the condition that the
155:, so the sum of angles between adjacent rays is 360°.
2317:
2228:
2181:
2153:
2103:
1479:
1342:
1304:
1015:
969:
764:
730:
2403:
Table of the
Highest Kissing Numbers Presently Known
1473:
dimensions, since that volume must be zero. Setting
1730:
1664:
1604:(3rd ed.). New York: Springer-Verlag. p.
365:Rough volume estimates show that kissing number in
342:, then this restricted kissing number is known for
2358:
2271:
2205:
2159:
2135:
2049:"Approximation Algorithms for Intersection Graphs"
1526:
1451:
1320:
1260:
985:
945:
743:
324:= 24 (where it is 196,560). The kissing number in
3238:Statal Institute of Higher Education Isaac Newton
1236:
1126:
83:
3499:
2416:
1977:
2392:(Institute Of Physics Publishing London 2000)
1855:
1631:
151:. These rays all emanate from the same center
61:seeks the maximum possible kissing number for
3293:
2529:
1910:
1465:is zero for any set of points which forms a (
2422:Journal of the American Mathematical Society
2047:Kammer, Frank; Tholey, Torsten (July 2012).
2046:
2034:Notices of the American Mathematical Society
2024:
1869:Notices of the American Mathematical Society
1446:
1443:
1421:
1387:
1365:
1352:
935:
859:
825:
789:
320:= 8 (where the kissing number is 240), and
299:= 3 — so the situation was even less clear.
108:In two dimensions, the kissing number is 6:
1882:
1812:
93:In one dimension, the kissing number is 2:
50:(after the originator of the problem), and
3300:
3286:
2536:
2522:
2433:
2027:"Mysteries in packing regular tetrahedra"
1993:
1937:
1744:
1680:
1658:
1163:
1060:
894:
803:
715:
2272:{\displaystyle y=(y_{mn})_{N\times {N}}}
1588:
1586:
360:
302:The existence of the highly symmetrical
232:
125:that is touched by circles with centers
1971:
1808:
1806:
1627:
1625:
3500:
2359:{\displaystyle x=(x_{nd})_{N\times D}}
1638:Research problems in discrete geometry
3281:
2788:Newton's law of universal gravitation
2517:
1953:
1583:
1527:{\displaystyle R_{mn}=1+{y_{mn}}^{2}}
3307:
2946:Newton's theorem of revolving orbits
2543:
1803:
1767:
1622:
1602:Sphere Packings, Lattices and Groups
353:
285:
2894:Leibniz–Newton calculus controversy
2635:standing on the shoulders of giants
1890:-dimensional Euclidean space].
245:
13:
2311:, which correspondent to a matrix
2154:
1394:
1356:
1343:
1298:is given by the distances squared
1016:
993:this can be converted to a single
832:
780:
765:
334:If arrangements are restricted to
226:– has a side length of less than 2
110:
95:
14:
3524:
2481:
2475:
674:
331:is unknown for other dimensions.
103:
3223:Isaac Newton Group of Telescopes
1270:Therefore, to solve the case in
121:: Consider a circle with center
88:
3243:Newton International Fellowship
2924:generalized Gauss–Newton method
2837:Newton's method in optimization
2501:from the original on 2021-12-12
2216:
2079:
2040:
2018:
1946:
1925:Journal of Combinatorial Theory
1904:
1835:10.1070/RM2003v058n04ABEH000651
1632:Brass, Peter; Moser, W. O. J.;
957:existential theory of the reals
3429:Sphere-packing (Hamming) bound
2389:The Pursuit of Perfect Packing
2341:
2324:
2252:
2235:
2124:
2110:
1876:
1849:
1761:
1723:
1713:
1699:10.1080/10586458.2010.10129070
1274: = 5 dimensions and
1224:
1207:
1195:
1169:
1158:
1131:
926:
900:
889:
862:
316:has allowed known results for
84:Known greatest kissing numbers
1:
2452:10.1090/S0894-0347-07-00589-9
2377:
2136:{\displaystyle x=(x_{n})_{N}}
2004:10.1080/10586458.2017.1286273
698:
2864:Newton's theorem about ovals
1939:10.1016/0097-3165(79)90074-8
7:
3233:Sir Isaac Newton Sixth Form
2889:Corpuscular theory of light
2815:Schrödinger–Newton equation
1549:
1469: + 1) simplex in
10:
3529:
2642:Notes on the Jewish Temple
2206:{\displaystyle m,n:m<n}
1893:Doklady Akademii Nauk SSSR
193:. Therefore, the triangle
3474:
3453:
3437:
3384:
3336:
3315:
3251:
3188:
3143:
3066:
3008:
2763:
2683:
2618:
2551:
2065:10.1007/s00453-012-9671-1
1463:Cayley–Menger determinant
665:
488:
437:
425:
413:
403:
139:, .... Consider the rays
3354:isosceles right triangle
2793:post-Newtonian expansion
2673:Corruptions of Scripture
2665:Ancient Kingdoms Amended
2279:only the entries having
2160:{\displaystyle \forall }
1981:Experimental Mathematics
1884:Levenshtein, Vladimir I.
1668:Experimental Mathematics
1576:
705:approximation algorithms
185:have the same length – 2
2983:Absolute space and time
2847:truncated Newton method
2820:Newton's laws of motion
2783:Newton's law of cooling
2143:is the sequence of the
1571:Cylinder sphere packing
213:, and its third side –
3368:Circle packing theorem
3218:Isaac Newton Telescope
3208:Isaac Newton Institute
2978:Newton–Puiseux theorem
2973:Parallelogram of force
2961:kissing number problem
2951:Newton–Euler equations
2854:Gauss–Newton algorithm
2803:gravitational constant
2360:
2273:
2222:Concerning the matrix
2207:
2161:
2137:
1778:10.1090/conm/453/08812
1755:10.1002/net.3230250205
1729:See also Lemma 3.1 in
1528:
1453:
1322:
1321:{\displaystyle R_{mn}}
1262:
987:
986:{\displaystyle y_{nm}}
947:
745:
716:Mathematical statement
378:
280:hexagonal close-packed
242:
115:
100:
59:kissing number problem
3172:Isaac Newton Gargoyle
3082: (nephew-in-law)
3058:Copernican Revolution
3053:Scientific Revolution
2914:Newton–Cotes formulas
2778:Newton's inequalities
2755:Structural coloration
2361:
2302:-dimensional vectors
2274:
2208:
2167:) does not change if
2162:
2138:
1556:Equilateral dimension
1529:
1454:
1323:
1263:
988:
948:
746:
744:{\displaystyle x_{n}}
364:
236:
114:
99:
3350:equilateral triangle
3179:Astronomers Monument
2869:Newton–Pepys problem
2842:Apollonius's problem
2810:Newton–Cartan theory
2723:Newton–Okounkov body
2656:hypotheses non fingo
2645: (c. 1680)
2315:
2226:
2179:
2151:
2101:
1640:. Springer. p.
1477:
1340:
1302:
1286:= 24 dimensions and
1013:
1005: − 1)/2 +
967:
762:
728:
66:-dimensional spheres
3487:Slothouber–Graatsma
2988:Luminiferous aether
2936:Newton's identities
2909:Newton's cannonball
2884:Classical mechanics
2874:Newtonian potential
2735:Newtonian telescope
2444:2008JAMS...21..909B
1827:2003RuMaS..58..794M
1691:2009arXiv0902.1105M
1065:
808:
709:intersection graphs
693:regular tetrahedron
371:grows exponentially
268:coordination number
239:regular icosahedron
3213:Isaac Newton Medal
3018: (birthplace)
2832:Newtonian dynamics
2730:Newton's reflector
2356:
2269:
2203:
2157:
2133:
1955:Weisstein, Eric W.
1860:(September 2004).
1858:Ziegler, Günter M.
1856:Pfender, Florian;
1524:
1449:
1318:
1258:
1123:
1049:
1042:
983:
943:
792:
741:
703:There are several
379:
276:cubic close-packed
243:
116:
101:
29:mathematical space
3508:Discrete geometry
3495:
3494:
3454:Other 3-D packing
3438:Other 2-D packing
3363:Apollonian gasket
3275:
3274:
3167: (sculpture)
3134:Abraham de Moivre
3088: (professor)
3016:Woolsthorpe Manor
2968:Newton's quotient
2941:Newton polynomial
2899:Newton's notation
2630: (1661–1665)
2484:"Kissing Numbers"
2418:Bachoc, Christine
1651:978-0-387-23815-9
1351:
1296:distance geometry
1165:
1096:
1062:
1033:
1027:
896:
805:
773:
672:
671:
354:Some known bounds
286:Larger dimensions
72:+ 1)-dimensional
16:Geometric concept
3520:
3513:Packing problems
3376:
3316:Abstract packing
3309:Packing problems
3302:
3295:
3288:
3279:
3278:
3263:
3158: (monotype)
3122:William Stukeley
3118: (disciple)
3098:Benjamin Pulleyn
3074:Catherine Barton
2993:Newtonian series
2904:Rotating spheres
2650:General Scholium
2545:Sir Isaac Newton
2538:
2531:
2524:
2515:
2514:
2510:
2508:
2506:
2488:
2471:
2437:
2371:
2365:
2363:
2362:
2357:
2355:
2354:
2339:
2338:
2283: <
2278:
2276:
2275:
2270:
2268:
2267:
2266:
2250:
2249:
2220:
2214:
2212:
2210:
2209:
2204:
2166:
2164:
2163:
2158:
2142:
2140:
2139:
2134:
2132:
2131:
2122:
2121:
2083:
2077:
2076:
2044:
2038:
2037:
2031:
2022:
2016:
2015:
1997:
1975:
1969:
1968:
1967:
1958:"Kissing Number"
1950:
1944:
1943:
1941:
1916:Sloane, N. J. A.
1908:
1902:
1901:
1880:
1874:
1872:
1866:
1853:
1847:
1846:
1815:Russ. Math. Surv
1810:
1801:
1799:
1765:
1759:
1758:
1748:
1727:
1721:
1717:
1711:
1710:
1684:
1662:
1656:
1655:
1629:
1620:
1619:
1598:Neil J.A. Sloane
1590:
1533:
1531:
1530:
1525:
1523:
1522:
1517:
1516:
1515:
1492:
1491:
1458:
1456:
1455:
1450:
1436:
1435:
1420:
1419:
1380:
1379:
1364:
1363:
1349:
1327:
1325:
1324:
1319:
1317:
1316:
1267:
1265:
1264:
1259:
1257:
1253:
1246:
1245:
1240:
1239:
1232:
1231:
1222:
1221:
1194:
1193:
1181:
1180:
1168:
1167:
1166:
1156:
1155:
1143:
1142:
1130:
1129:
1122:
1092:
1091:
1086:
1082:
1075:
1074:
1064:
1063:
1057:
1041:
1025:
995:quartic equation
992:
990:
989:
984:
982:
981:
961:exponential time
952:
950:
949:
944:
942:
938:
925:
924:
912:
911:
899:
898:
897:
887:
886:
874:
873:
858:
857:
818:
817:
807:
806:
800:
788:
787:
771:
750:
748:
747:
742:
740:
739:
381:
380:
270:of an atom in a
246:Three dimensions
57:In general, the
3528:
3527:
3523:
3522:
3521:
3519:
3518:
3517:
3498:
3497:
3496:
3491:
3470:
3449:
3433:
3380:
3374:
3373:Tammes problem
3332:
3311:
3306:
3276:
3271:
3270:
3269:
3268:
3267:
3260:
3247:
3203:Newton's cradle
3184:
3139:
3112: (student)
3110:William Whiston
3106: (student)
3062:
3043:Religious views
3004:
2919:Newton's method
2879:Newtonian fluid
2773:Bucket argument
2759:
2679:
2614:
2547:
2542:
2504:
2502:
2486:
2478:
2435:math.MG/0608426
2380:
2375:
2374:
2370:column vectors.
2344:
2340:
2331:
2327:
2316:
2313:
2312:
2310:
2262:
2255:
2251:
2242:
2238:
2227:
2224:
2223:
2221:
2217:
2180:
2177:
2176:
2152:
2149:
2148:
2127:
2123:
2117:
2113:
2102:
2099:
2098:
2084:
2080:
2045:
2041:
2029:
2023:
2019:
1976:
1972:
1951:
1947:
1909:
1905:
1900:(6): 1299–1303.
1881:
1877:
1864:
1854:
1850:
1811:
1804:
1788:
1766:
1762:
1728:
1724:
1718:
1714:
1663:
1659:
1652:
1630:
1623:
1616:
1594:Conway, John H.
1591:
1584:
1579:
1552:
1518:
1508:
1504:
1503:
1502:
1484:
1480:
1478:
1475:
1474:
1428:
1424:
1397:
1393:
1372:
1368:
1359:
1355:
1341:
1338:
1337:
1309:
1305:
1303:
1300:
1299:
1241:
1235:
1234:
1233:
1227:
1223:
1214:
1210:
1189:
1185:
1176:
1172:
1162:
1161:
1157:
1151:
1147:
1138:
1134:
1125:
1124:
1100:
1087:
1070:
1066:
1059:
1058:
1053:
1048:
1044:
1043:
1037:
1032:
1028:
1014:
1011:
1010:
974:
970:
968:
965:
964:
920:
916:
907:
903:
893:
892:
888:
882:
878:
869:
865:
835:
831:
813:
809:
802:
801:
796:
783:
779:
778:
774:
763:
760:
759:
735:
731:
729:
726:
725:
718:
701:
677:
393:
388:
356:
309:
288:
272:crystal lattice
248:
225:
219:
208:
202:
183:
177:
167:
150:
138:
131:
106:
91:
86:
74:Euclidean space
17:
12:
11:
5:
3526:
3516:
3515:
3510:
3493:
3492:
3490:
3489:
3484:
3478:
3476:
3472:
3471:
3469:
3468:
3463:
3457:
3455:
3451:
3450:
3448:
3447:
3445:Square packing
3441:
3439:
3435:
3434:
3432:
3431:
3426:
3424:Kissing number
3421:
3416:
3411:
3406:
3401:
3396:
3390:
3388:
3386:Sphere packing
3382:
3381:
3379:
3378:
3370:
3365:
3360:
3342:
3340:
3338:Circle packing
3334:
3333:
3331:
3330:
3325:
3319:
3317:
3313:
3312:
3305:
3304:
3297:
3290:
3282:
3273:
3272:
3259:
3258:
3256:
3255:
3253:
3249:
3248:
3246:
3245:
3240:
3235:
3230:
3225:
3220:
3215:
3210:
3205:
3200:
3194:
3192:
3186:
3185:
3183:
3182:
3175:
3168:
3159:
3149:
3147:
3141:
3140:
3138:
3137:
3136: (friend)
3131:
3130: (friend)
3125:
3124: (friend)
3119:
3113:
3107:
3101:
3095:
3094: (mentor)
3092:William Clarke
3089:
3083:
3077:
3070:
3068:
3064:
3063:
3061:
3060:
3055:
3050:
3048:Occult studies
3045:
3040:
3035:
3030:
3025:
3019:
3012:
3010:
3006:
3005:
3003:
3002:
3001:
3000:
2990:
2985:
2980:
2975:
2970:
2965:
2964:
2963:
2953:
2948:
2943:
2938:
2933:
2931:Newton fractal
2928:
2927:
2926:
2916:
2911:
2906:
2901:
2896:
2891:
2886:
2881:
2876:
2871:
2866:
2861:
2859:Newton's rings
2856:
2851:
2850:
2849:
2844:
2834:
2829:
2828:
2827:
2817:
2812:
2807:
2806:
2805:
2800:
2795:
2785:
2780:
2775:
2769:
2767:
2761:
2760:
2758:
2757:
2752:
2747:
2745:Newton's metal
2742:
2737:
2732:
2727:
2726:
2725:
2718:Newton polygon
2715:
2710:
2705:
2700:
2699:
2698:
2687:
2685:
2681:
2680:
2678:
2677:
2669:
2661:
2652:" (1713;
2646:
2638:
2631:
2622:
2620:
2619:Other writings
2616:
2615:
2613:
2612:
2604:
2596:
2588:
2580:
2572:
2564:
2555:
2553:
2549:
2548:
2541:
2540:
2533:
2526:
2518:
2512:
2511:
2482:Grime, James.
2477:
2476:External links
2474:
2473:
2472:
2428:(3): 909–924.
2414:
2413:(lower bounds)
2405:maintained by
2400:
2379:
2376:
2373:
2372:
2353:
2350:
2347:
2343:
2337:
2334:
2330:
2326:
2323:
2320:
2306:
2265:
2261:
2258:
2254:
2248:
2245:
2241:
2237:
2234:
2231:
2215:
2202:
2199:
2196:
2193:
2190:
2187:
2184:
2156:
2130:
2126:
2120:
2116:
2112:
2109:
2106:
2093:run from 1 to
2078:
2059:(2): 312–336.
2039:
2017:
1988:(3): 362–369.
1970:
1945:
1932:(2): 210–214.
1912:Odlyzko, A. M.
1903:
1896:(in Russian).
1875:
1848:
1821:(4): 794–795.
1802:
1786:
1760:
1722:
1712:
1675:(2): 174–178.
1657:
1650:
1621:
1614:
1581:
1580:
1578:
1575:
1574:
1573:
1568:
1566:Soddy's hexlet
1563:
1561:Spherical code
1558:
1551:
1548:
1521:
1514:
1511:
1507:
1501:
1498:
1495:
1490:
1487:
1483:
1448:
1445:
1442:
1439:
1434:
1431:
1427:
1423:
1418:
1415:
1412:
1409:
1406:
1403:
1400:
1396:
1392:
1389:
1386:
1383:
1378:
1375:
1371:
1367:
1362:
1358:
1354:
1348:
1345:
1315:
1312:
1308:
1256:
1252:
1249:
1244:
1238:
1230:
1226:
1220:
1217:
1213:
1209:
1206:
1203:
1200:
1197:
1192:
1188:
1184:
1179:
1175:
1171:
1160:
1154:
1150:
1146:
1141:
1137:
1133:
1128:
1121:
1118:
1115:
1112:
1109:
1106:
1103:
1099:
1095:
1090:
1085:
1081:
1078:
1073:
1069:
1056:
1052:
1047:
1040:
1036:
1031:
1024:
1021:
1018:
980:
977:
973:
941:
937:
934:
931:
928:
923:
919:
915:
910:
906:
902:
891:
885:
881:
877:
872:
868:
864:
861:
856:
853:
850:
847:
844:
841:
838:
834:
830:
827:
824:
821:
816:
812:
799:
795:
791:
786:
782:
777:
770:
767:
738:
734:
717:
714:
700:
697:
683:copies of any
676:
675:Generalization
673:
670:
669:
664:
658:
657:
654:
651:
647:
646:
643:
640:
636:
635:
632:
629:
625:
624:
621:
618:
614:
613:
610:
607:
603:
602:
599:
596:
592:
591:
588:
585:
581:
580:
577:
574:
570:
569:
566:
563:
559:
558:
555:
552:
548:
547:
544:
541:
537:
536:
533:
530:
526:
525:
522:
519:
515:
514:
511:
508:
504:
503:
500:
497:
493:
492:
487:
481:
480:
477:
472:
468:
467:
464:
459:
455:
454:
451:
446:
442:
441:
436:
430:
429:
424:
418:
417:
412:
406:
405:
402:
396:
395:
390:
385:
355:
352:
307:
287:
284:
261:Reinhold Hoppe
247:
244:
223:
217:
206:
200:
181:
175:
165:
146:
136:
129:
105:
104:Two dimensions
102:
90:
87:
85:
82:
52:contact number
37:sphere packing
25:kissing number
15:
9:
6:
4:
3:
2:
3525:
3514:
3511:
3509:
3506:
3505:
3503:
3488:
3485:
3483:
3480:
3479:
3477:
3473:
3467:
3464:
3462:
3459:
3458:
3456:
3452:
3446:
3443:
3442:
3440:
3436:
3430:
3427:
3425:
3422:
3420:
3419:Close-packing
3417:
3415:
3414:In a cylinder
3412:
3410:
3407:
3405:
3402:
3400:
3397:
3395:
3392:
3391:
3389:
3387:
3383:
3377:
3371:
3369:
3366:
3364:
3361:
3359:
3355:
3351:
3347:
3344:
3343:
3341:
3339:
3335:
3329:
3326:
3324:
3321:
3320:
3318:
3314:
3310:
3303:
3298:
3296:
3291:
3289:
3284:
3283:
3280:
3266:
3262:
3254:
3250:
3244:
3241:
3239:
3236:
3234:
3231:
3229:
3226:
3224:
3221:
3219:
3216:
3214:
3211:
3209:
3206:
3204:
3201:
3199:
3198:Newton (unit)
3196:
3195:
3193:
3191:
3187:
3181:
3180:
3176:
3174:
3173:
3169:
3166:
3164:
3160:
3157:
3155:
3151:
3150:
3148:
3146:
3142:
3135:
3132:
3129:
3128:William Jones
3126:
3123:
3120:
3117:
3114:
3111:
3108:
3105:
3102:
3100: (tutor)
3099:
3096:
3093:
3090:
3087:
3084:
3081:
3080:John Conduitt
3078:
3076: (niece)
3075:
3072:
3071:
3069:
3065:
3059:
3056:
3054:
3051:
3049:
3046:
3044:
3041:
3039:
3036:
3034:
3031:
3029:
3026:
3023:
3022:Cranbury Park
3020:
3017:
3014:
3013:
3011:
3009:Personal life
3007:
2999:
2996:
2995:
2994:
2991:
2989:
2986:
2984:
2981:
2979:
2976:
2974:
2971:
2969:
2966:
2962:
2959:
2958:
2957:
2956:Newton number
2954:
2952:
2949:
2947:
2944:
2942:
2939:
2937:
2934:
2932:
2929:
2925:
2922:
2921:
2920:
2917:
2915:
2912:
2910:
2907:
2905:
2902:
2900:
2897:
2895:
2892:
2890:
2887:
2885:
2882:
2880:
2877:
2875:
2872:
2870:
2867:
2865:
2862:
2860:
2857:
2855:
2852:
2848:
2845:
2843:
2840:
2839:
2838:
2835:
2833:
2830:
2826:
2825:Kepler's laws
2823:
2822:
2821:
2818:
2816:
2813:
2811:
2808:
2804:
2801:
2799:
2798:parameterized
2796:
2794:
2791:
2790:
2789:
2786:
2784:
2781:
2779:
2776:
2774:
2771:
2770:
2768:
2766:
2762:
2756:
2753:
2751:
2748:
2746:
2743:
2741:
2738:
2736:
2733:
2731:
2728:
2724:
2721:
2720:
2719:
2716:
2714:
2711:
2709:
2706:
2704:
2701:
2697:
2694:
2693:
2692:
2689:
2688:
2686:
2684:Contributions
2682:
2675:
2674:
2670:
2667:
2666:
2662:
2659:
2657:
2651:
2647:
2644:
2643:
2639:
2637:" (1675)
2636:
2632:
2629:
2628:
2624:
2623:
2621:
2617:
2610:
2609:
2605:
2602:
2601:
2597:
2594:
2593:
2589:
2586:
2585:
2581:
2578:
2577:
2573:
2570:
2569:
2565:
2562:
2561:
2557:
2556:
2554:
2550:
2546:
2539:
2534:
2532:
2527:
2525:
2520:
2519:
2516:
2500:
2496:
2492:
2485:
2480:
2479:
2469:
2465:
2461:
2457:
2453:
2449:
2445:
2441:
2436:
2431:
2427:
2423:
2419:
2415:
2412:
2408:
2407:Gabriele Nebe
2404:
2401:
2399:
2398:0-7503-0648-3
2395:
2391:
2390:
2386:
2382:
2381:
2369:
2351:
2348:
2345:
2335:
2332:
2328:
2321:
2318:
2309:
2305:
2301:
2298:
2294:
2290:
2286:
2282:
2263:
2259:
2256:
2246:
2243:
2239:
2232:
2229:
2219:
2200:
2197:
2194:
2191:
2188:
2185:
2182:
2174:
2170:
2146:
2128:
2118:
2114:
2107:
2104:
2096:
2092:
2088:
2082:
2074:
2070:
2066:
2062:
2058:
2054:
2050:
2043:
2035:
2028:
2021:
2013:
2009:
2005:
2001:
1996:
1991:
1987:
1983:
1982:
1974:
1965:
1964:
1959:
1956:
1949:
1940:
1935:
1931:
1927:
1926:
1921:
1917:
1913:
1907:
1899:
1895:
1894:
1889:
1885:
1879:
1870:
1863:
1859:
1852:
1844:
1840:
1836:
1832:
1828:
1824:
1820:
1816:
1809:
1807:
1797:
1793:
1789:
1787:9780821842393
1783:
1779:
1775:
1771:
1764:
1756:
1752:
1747:
1742:
1738:
1734:
1726:
1716:
1708:
1704:
1700:
1696:
1692:
1688:
1683:
1678:
1674:
1670:
1669:
1661:
1653:
1647:
1643:
1639:
1635:
1628:
1626:
1617:
1615:0-387-98585-9
1611:
1607:
1603:
1599:
1595:
1589:
1587:
1582:
1572:
1569:
1567:
1564:
1562:
1559:
1557:
1554:
1553:
1547:
1545:
1541:
1537:
1519:
1512:
1509:
1505:
1499:
1496:
1493:
1488:
1485:
1481:
1472:
1468:
1464:
1459:
1440:
1437:
1432:
1429:
1425:
1416:
1413:
1410:
1407:
1404:
1401:
1398:
1390:
1384:
1381:
1376:
1373:
1369:
1360:
1346:
1335:
1331:
1313:
1310:
1306:
1297:
1293:
1290: =
1289:
1285:
1281:
1278: =
1277:
1273:
1268:
1254:
1250:
1247:
1242:
1228:
1218:
1215:
1211:
1204:
1201:
1198:
1190:
1186:
1182:
1177:
1173:
1152:
1148:
1144:
1139:
1135:
1119:
1116:
1113:
1110:
1107:
1104:
1101:
1097:
1093:
1088:
1083:
1079:
1076:
1071:
1067:
1054:
1050:
1045:
1038:
1034:
1029:
1022:
1019:
1008:
1004:
1000:
996:
978:
975:
971:
962:
958:
953:
939:
932:
929:
921:
917:
913:
908:
904:
883:
879:
875:
870:
866:
854:
851:
848:
845:
842:
839:
836:
828:
822:
819:
814:
810:
797:
793:
784:
775:
768:
757:
754:
736:
732:
723:
713:
710:
706:
696:
694:
690:
686:
682:
668:
663:
660:
659:
655:
652:
649:
648:
644:
641:
638:
637:
633:
630:
627:
626:
622:
619:
616:
615:
611:
608:
605:
604:
600:
597:
594:
593:
589:
586:
583:
582:
578:
575:
572:
571:
567:
564:
561:
560:
556:
553:
550:
549:
545:
542:
539:
538:
534:
531:
528:
527:
523:
520:
517:
516:
512:
509:
506:
505:
501:
498:
495:
494:
491:
486:
483:
482:
478:
476:
473:
470:
469:
465:
463:
460:
457:
456:
452:
450:
447:
444:
443:
440:
435:
432:
431:
428:
423:
420:
419:
416:
411:
408:
407:
401:
398:
397:
391:
386:
383:
382:
376:
372:
368:
363:
359:
351:
349:
346:= 1 to 9 and
345:
341:
337:
332:
330:
327:
323:
319:
315:
314:Leech lattice
311:
306:
300:
298:
294:
283:
281:
277:
273:
269:
264:
262:
258:
257:David Gregory
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3265:Isaac Newton
3177:
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3086:Isaac Barrow
3024: (home)
2960:
2765:Newtonianism
2740:Newton scale
2703:Impact depth
2676: (1754)
2671:
2668: (1728)
2663:
2653:
2640:
2625:
2611: (1711)
2606:
2603: (1707)
2598:
2595: (1704)
2590:
2587: (1704)
2582:
2579: (1687)
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2571: (1684)
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2552:Publications
2503:. Retrieved
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2383:T. Aste and
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2036:: 1540–1549.
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3404:In a sphere
3375:(on sphere)
3346:In a circle
3165:by Paolozzi
3104:Roger Cotes
2713:Newton disc
2627:Quaestiones
2600:Arithmetica
2495:Brady Haran
2411:Neil Sloane
1634:Pach, János
1009:variables:
685:convex body
369:dimensions
282:structure.
3502:Categories
3394:Apollonian
3252:Categories
3228:XMM-Newton
3145:Depictions
3116:John Keill
3038:Apple tree
3033:Later life
3028:Early life
2608:De Analysi
2505:11 October
2378:References
1995:1609.05167
1871:: 873–883.
699:Algorithms
689:translates
384:Dimension
329:dimensions
189:– for all
3466:Ellipsoid
3409:In a cube
3067:Relations
2576:Principia
2385:D. Weaire
2349:×
2260:×
2155:∀
1963:MathWorld
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3190:Namesake
3156:by Blake
2750:Spectrum
2691:Calculus
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2560:Fluxions
2499:Archived
2085:Numbers
2012:52903026
1918:(1979).
1733:Networks
1636:(2005).
1600:(1999).
1550:See also
1336:sphere:
656:122,351
21:geometry
3475:Puzzles
2708:Inertia
2696:fluxion
2592:Queries
2584:Opticks
2568:De Motu
2491:youtube
2487:(video)
2460:2393433
2440:Bibcode
2073:3065780
1823:Bibcode
1796:2405694
1687:Bibcode
667:196,560
653:93,150
645:80,810
642:49,896
634:53,524
631:27,720
623:36,195
620:17,400
612:24,417
609:10,668
601:16,406
590:10,978
340:lattice
336:lattice
310:lattice
293:24-cell
41:lattice
33:spheres
3482:Conway
3399:Finite
3358:square
3163:Newton
3154:Newton
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543:1,154
535:1,355
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389:bound
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2998:table
2464:S2CID
2430:arXiv
2069:S2CID
2030:(PDF)
2008:S2CID
1990:arXiv
1865:(PDF)
1839:S2CID
1741:arXiv
1703:S2CID
1677:arXiv
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387:Lower
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