105:
973:
117:
58:
93:
691:
968:{\displaystyle {\begin{aligned}V_{W}&=V_{\text{cylinder}}+2\cdot V_{\text{half-sphere}}\\&=V_{\text{cylinder}}+V_{\text{sphere}}\\&=\pi hr^{2}+{\frac {4}{3}}\pi r^{3}\\&=\pi 2r\cdot (n-1)\cdot r^{2}+{\frac {4}{3}}\pi r^{3}\\&=2\cdot \left(n-{\frac {1}{3}}\right)\pi r^{3}\end{aligned}}}
2067:
upwards it is always optimal to arrange the spheres along a straight line. That is, the sausage catastrophe no longer occurs once we go above 4 dimensions. The overall conjecture remains open. The best results so far are those of Ulrich Betke und Martin Henk, who proved the conjecture for dimensions
69:
refers to any arrangement of a set of spatially-connected, possibly differently-sized or differently-shaped objects in space such that none of them overlap. In the case of the finite sphere packing problem, these objects are restricted to equally-sized spheres. Such a packing of spheres determines a
2076:
While it may be proved that the sausage packing is not optimal for 56 spheres, and that there must be some other packing that is optimal, it is not known what the optimal packing looks like. It is difficult to find the optimal packing as there is no "simple" formula for the volume of an arbitrarily
454:
comes from the fact that the optimal packing shape suddenly shifts from the orderly sausage packing to the relatively unordered cluster packing and vice versa as one goes from one number to another, without a satisfying explanation as to why this happens. Even so, the transition in three dimensions
225:, which is defined as the ratio of the volume of the spheres to the volume of the total convex hull. The higher the packing density, the less empty space there is in the packing and thus the smaller the volume of the hull (in comparison to other packings with the same number and size of spheres).
228:
To pack the spheres efficiently, it might be asked which packing has the highest possible density. It is easy to see that such a packing should have the property that the spheres lie next to each other, that is, each sphere should touch another on the surface. A more exact phrasing is to form a
3113:
2385:
and others used it to formulate a unified theory of finite and infinite packings. In three dimensions, Wills gives a simple argument that such a unified theory is not possible based on this definition: The densest finite arrangement of coins in three dimensions is the sausage with
510:, the optimal packing is always either a sausage or a cluster, and never a pizza. It is an open problem whether this holds true for all dimensions. This result only concerns spheres and not other convex bodies; in fact Gritzmann and Arhelger observed that for any dimension
1196:
2089:. A crucial step towards a unified theory of both finite and infinite (lattice and non-lattice) sphere packings was the introduction of parametric densities by Jörg Wills in 1992. The parametric density takes into account the influence of the edges of the packing.
1679:
409:, a cluster packing exists that is more efficient that the sausage packing, as shown in 1992 by Jörg Wills and Pier Mario Gandini. It remains unknown what these most efficient cluster packings look like. For example, in the case
1833:
1445:
2898:
136:, as the convex hull has a sausage-like shape. An approximate example in real life is the packing of tennis balls in a tube, though the ends must be rounded for the tube to coincide with the actual convex hull.
2555:
2887:
2206:
1555:
237:
for each sphere and connects vertices with edges whenever the corresponding spheres if their surfaces touch. Then the highest-density packing must satisfy the property that the corresponding graph is
1028:
696:
213:
One or two spheres always make a sausage. With three, a pizza packing (that is not also a sausage) becomes possible, and with four or more, clusters (that are not also pizzas) become possible.
1287:
148:. Approximate real-life examples of this kind of packing include billiard balls being packed in a triangle as they are set up. This holds for packings in three-dimensional Euclidean space.
3283:
544:
In the following section it is shown that for 455 spheres the sausage packing is non-optimal, and that there instead exists a special cluster packing that occupies a smaller volume.
2794:
2732:
1333:
2436:
1941:, which would involve subtracting the excess volume at the corners and edges of the tetrahedron. This allows the sausage packing to be proved non-optimal for smaller values of
27:
can be most efficiently packed. The question of packing finitely many spheres has only been investigated in detail in recent decades, with much of the groundwork being laid by
381:
2438:, so the infinite value cannot be obtained as a limit of finite values. To solve this issue, Wills introduces a modification to the definition by adding a positive parameter
636:
3355:
331:
168:
By the given definitions, any sausage packing is technically also a pizza packing, and any pizza packing is technically also a cluster packing. In the more general case of
3319:
2690:
2654:
1730:
1587:
2376:
2410:
3199:
508:
407:
2618:
534:
3143:
2577:
2456:
2310:
2283:
2236:
1916:
1889:
1364:
683:
1863:
433:
3225:
2065:
1918:
is about 2845 for the tetrahedral packing and 2856 for the sausage packing, which implies that for this number of spheres the tetrahedron is more closely packed.
479:
287:
3747:
3163:
2814:
2752:
2330:
2256:
2110:
2039:
2019:
1979:
1959:
1939:
1703:
1579:
1488:
1468:
1243:
1223:
1020:
1000:
656:
585:
565:
267:
207:
186:
2332:). For a linear arrangement (sausage), the convex hull is a line segment through all the midpoints of the spheres. The plus sign in the formula refers to
3693:
1738:
2579:
allows the influence of the edges to be considered (giving the convex hull a certain thickness). This is then combined with methods from the theory of
1372:
3743:
49:
Sphere packing problems are distinguished between packings in given containers and free packings. This article primarily discusses free packings.
2001:
in 1975, which concerns a generalized version of the problem to spheres, convex hulls, and volume in higher dimensions. A generalized sphere in
978:
Similarly, it is possible to find the volume of the convex hull of a tetrahedral packing, in which the spheres are arranged so that they form a
2041:-dimensional body in which every boundary point lies equally far away from the midpoint. Fejes Tóth's sausage conjecture then states that from
3108:{\displaystyle {\frac {V(B^{d})}{2V(B^{d-1})}}{\rho _{c}(d)}^{1-d}\leq \delta (B^{d})\leq {\frac {V(B^{d})}{2V(B^{d-1})}}{\rho _{s}(d)}^{1-d}}
3515:
3452:
2464:
2819:
86:
There are many possible ways to arrange spheres, which can be classified into three basic groups: sausage, pizza, and cluster packing.
3901:
2118:
1496:
1191:{\displaystyle n=\sum _{i=1}^{x}\sum _{j=1}^{i}j=\sum _{i=1}^{x}{\frac {i\cdot (i+1)}{2}}={\frac {x\cdot (x+1)\cdot (x+2)}{6}}}
3880:
3812:
3686:
2077:
shaped cluster. Optimality (and non-optimality) is shown through appropriate estimates of the volume, using methods from
3807:
230:
104:
2082:
1251:
221:
The empty space between spheres varies depending on the type of packing. The amount of empty space is measured in the
1921:
It is also possible with some more effort to derive the exact formula for the volume of the tetrahedral convex hull
3797:
3751:
3739:
1994:
28:
3787:
3679:
2092:
The definition of density used earlier concerns the volume of the convex hull of the spheres (or convex bodies)
3802:
3230:
3469:
2757:
2695:
1299:
3761:
43:
2415:
2382:
249:
With three or four spheres, the sausage packing is optimal. It is believed that this holds true for any
234:
982:
shape, which only leads to completely filled tetrahedra for specific numbers of spheres. If there are
336:
34:
The similar problem for infinitely many spheres has a longer history of investigation, from which the
3628:
2086:
1674:{\displaystyle V<{\frac {2\cdot \left(x-1+{\sqrt {6}}\right)^{3}\cdot {\sqrt {2}}\cdot r^{3}}{3}}}
594:
3495:
3387:
Wills, J. M. (1998). "Spheres and
Sausages, crystals and catastrophes- and a joint packing theory".
3324:
3854:
3716:
2584:
292:
3288:
2659:
2623:
1708:
132:
An arrangement in which the midpoint of all the spheres lie on a single straight line is called a
116:
3658:
3565:
3412:
2333:
440:
2339:
2389:
446:
The sudden transition in optimal packing shape is jokingly known by some mathematicians as the
92:
3859:
3756:
3650:
3606:
3557:
3511:
3448:
3404:
3168:
2588:
487:
386:
39:
35:
2597:
1561:
Substituting this value into the volume formula for the tetrahedron, we know that the volume
513:
3702:
3640:
3596:
3549:
3503:
3440:
3396:
3121:
2562:
2441:
2288:
2261:
2214:
1894:
1868:
1342:
661:
2078:
1842:
412:
238:
222:
210:-dimensional arrangements, and "pizzas" to those with an in-between number of dimensions.
156:
If the midpoints of the spheres are arranged throughout 3D space, the packing is termed a
3204:
2044:
458:
272:
3838:
3817:
3779:
3766:
3731:
3507:
3148:
2799:
2737:
2315:
2241:
2095:
2024:
2004:
1964:
1944:
1924:
1688:
1564:
1473:
1453:
1450:
In the case of many spheres being arranged inside a tetrahedron, the length of an edge
1228:
1208:
1005:
985:
641:
570:
550:
252:
192:
171:
3895:
3875:
3822:
3662:
3569:
3434:
3416:
2816:
the cluster is densest. These parameters are dimension-specific. In two dimensions,
3285:
and it is predicted that this holds for all dimensions, in which case the value of
2580:
160:. Real-life approximations include fruit being packed in multiple layers in a box.
1828:{\displaystyle V_{\text{W}}={\frac {x\cdot (x+1)\cdot (x+2)-2}{3}}\cdot \pi r^{3}}
1732:
of the convex hull of a sausage packing with the same number of spheres, we have
1440:{\displaystyle V_{T}={\frac {\sqrt {2}}{12}}\cdot a^{3}={\sqrt {192}}\cdot r^{3}}
481:
dimensions the sudden transition is conjectured to happen around 377000 spheres.
3721:
979:
436:
71:
2889:
so that there is a transition from sausages to clusters (sausage catastrophe).
1470:
increases by twice the radius of a sphere for each new layer, meaning that for
439:
packing like the classical packing of cannon balls, but is likely some kind of
3537:
3444:
75:
57:
3654:
3610:
3561:
3408:
2412:. However, the optimal infinite arrangement is a hexagonal arrangement with
188:
dimensions, "sausages" refer to one-dimensional arrangements, "clusters" to
3645:
1203:
588:
587:
is calculable with elementary geometry. The middle part of the hull is a
1581:
of the convex hull must be smaller than the tetrahedron itself, so that
3601:
3584:
3553:
3400:
1002:
spheres along one edge of the tetrahedron, the total number of spheres
2312:(instead of the sphere, we can also take an arbitrary convex body for
1705:
layers and substituting into the earlier expression to get the volume
536:
there exists a convex shape for which the closest packing is a pizza.
46:
and treated as infinite sphere packings thanks to their large number.
24:
3671:
2381:
This definition works in two dimensions, where Laszlo Fejes-Toth,
56:
2550:{\displaystyle \delta (K,C_{n})={\frac {nV(K)}{V(C_{n}+\rho K)}}}
2882:{\displaystyle \rho _{c}(2)=\rho _{s}(2)={\frac {\sqrt {3}}{2}}}
3675:
23:
concerns the question of how a finite number of equally-sized
2201:{\displaystyle \delta (K,C_{n})={\frac {nV(K)}{V(C_{n}+K)}}}
1550:{\displaystyle a=2\cdot \left(x-1+{\sqrt {6}}\right)\cdot r}
2378:
refers to the volume of the convex hull of the spheres.
638:
while the caps at the end are half-spheres with radius
2734:
the sausage is the densenst packing (for all integers
547:
The volume of a convex hull of a sausage packing with
144:
If all the midpoints lie on a plane, the packing is a
3327:
3291:
3233:
3207:
3171:
3151:
3124:
2901:
2822:
2802:
2760:
2740:
2698:
2662:
2626:
2600:
2565:
2467:
2444:
2418:
2392:
2342:
2318:
2291:
2264:
2244:
2217:
2121:
2098:
2047:
2027:
2007:
1967:
1947:
1927:
1897:
1871:
1845:
1741:
1711:
1691:
1590:
1567:
1499:
1476:
1456:
1375:
1345:
1302:
1254:
1231:
1211:
1031:
1008:
988:
694:
664:
644:
597:
573:
553:
516:
490:
461:
415:
389:
339:
295:
275:
255:
195:
174:
3868:
3847:
3831:
3778:
3730:
3709:
3633:
Mitteilungen der
Deutschen Mathematiker-Vereinigung
3349:
3313:
3277:
3219:
3193:
3157:
3137:
3107:
2881:
2808:
2788:
2746:
2726:
2684:
2648:
2612:
2571:
2549:
2450:
2430:
2404:
2370:
2324:
2304:
2277:
2250:
2230:
2200:
2104:
2059:
2033:
2013:
1973:
1953:
1933:
1910:
1883:
1857:
1827:
1724:
1697:
1673:
1573:
1549:
1482:
1462:
1439:
1358:
1327:
1281:
1237:
1217:
1190:
1014:
994:
967:
677:
650:
630:
579:
559:
528:
502:
473:
427:
401:
375:
325:
281:
261:
201:
180:
1685:Taking the number of spheres in a tetrahedron of
1366:of the tetrahedron is then given by the formula
1282:{\displaystyle r={\frac {\sqrt {6}}{12}}\cdot a}
540:Example of the sausage packing being non-optimal
435:, it is known that the optimal packing is not a
3439:(in German). Wiesbaden: Vieweg+Teubner Verlag.
3428:
3426:
3687:
8:
3622:
3620:
3694:
3680:
3672:
3644:
3600:
3494:Gritzmann, Peter; Wills, Jörg M. (1993),
3338:
3326:
3296:
3290:
3278:{\displaystyle \rho _{s}(d)=\rho _{c}(d)}
3260:
3238:
3232:
3206:
3182:
3170:
3150:
3129:
3123:
3093:
3077:
3072:
3053:
3029:
3016:
3004:
2979:
2963:
2958:
2939:
2915:
2902:
2900:
2867:
2849:
2827:
2821:
2801:
2771:
2759:
2739:
2709:
2697:
2667:
2661:
2631:
2625:
2599:
2564:
2526:
2496:
2484:
2466:
2443:
2417:
2391:
2353:
2341:
2317:
2296:
2290:
2269:
2263:
2243:
2222:
2216:
2180:
2150:
2138:
2120:
2097:
2046:
2026:
2006:
1966:
1946:
1926:
1902:
1896:
1870:
1844:
1819:
1755:
1746:
1740:
1716:
1710:
1690:
1659:
1645:
1636:
1624:
1597:
1589:
1566:
1529:
1498:
1475:
1455:
1431:
1417:
1408:
1389:
1380:
1374:
1350:
1344:
1312:
1301:
1261:
1253:
1230:
1210:
1140:
1107:
1101:
1090:
1074:
1063:
1053:
1042:
1030:
1007:
987:
955:
933:
900:
883:
874:
824:
807:
798:
772:
759:
739:
720:
703:
695:
693:
669:
663:
643:
596:
572:
552:
515:
489:
460:
414:
388:
338:
294:
274:
254:
194:
173:
74:of the packing, defined as the smallest
3368:
88:
3382:
3380:
3378:
3376:
3374:
3372:
2789:{\displaystyle \rho \geq \rho _{c}(d)}
2727:{\displaystyle \rho \leq \rho _{s}(d)}
2072:Parametric density and related methods
164:Relationships between types of packing
3589:Discrete & Computational Geometry
3468:Gandini, P. M.; Wills, J. M. (1992).
1328:{\displaystyle a=2{\sqrt {6}}\cdot r}
7:
1891:spheres the coefficient in front of
3536:Hajnal, A.; Tóth, L. Fejes (1975).
3436:Kugelpackungen von Kepler bis heute
3508:10.1016/b978-0-444-89597-4.50008-1
3118:where the volume of the unit ball
2431:{\displaystyle \delta \approx 0.9}
1225:of a tetrahedral with side length
14:
3629:"Kugelpackungen- Altes und Neues"
376:{\displaystyle n=56,59,60,61,62}
115:
103:
91:
42:can be simplistically viewed as
3542:Periodica Mathematica Hungarica
1490:layers the side length becomes
631:{\displaystyle h=2r\cdot (n-1)}
450:(Wills, 1985). The designation
78:that includes all the spheres.
3823:Sphere-packing (Hamming) bound
3502:, Elsevier, pp. 861–897,
3389:The Mathematical Intelligencer
3350:{\displaystyle \delta (B^{d})}
3344:
3331:
3308:
3302:
3272:
3266:
3250:
3244:
3188:
3175:
3089:
3083:
3065:
3046:
3035:
3022:
3010:
2997:
2975:
2969:
2951:
2932:
2921:
2908:
2861:
2855:
2839:
2833:
2783:
2777:
2721:
2715:
2679:
2673:
2643:
2637:
2541:
2519:
2511:
2505:
2490:
2471:
2365:
2346:
2192:
2173:
2165:
2159:
2144:
2125:
2085:, mixed Minkowski volumes and
1794:
1782:
1776:
1764:
1179:
1167:
1161:
1149:
1128:
1116:
864:
852:
625:
613:
19:In mathematics, the theory of
1:
3627:Wills, Jörg M. (1995-01-01).
3496:"Finite Packing and Covering"
1993:comes from the mathematician
326:{\displaystyle n=57,58,63,64}
70:specific volume known as the
38:is most well-known. Atoms in
3585:"Finite Packings of Spheres"
3583:Betke, U.; Henk, M. (1998).
3314:{\displaystyle \rho _{c}(d)}
2685:{\displaystyle \rho _{c}(d)}
2649:{\displaystyle \rho _{s}(d)}
1725:{\displaystyle V_{\text{W}}}
3500:Handbook of Convex Geometry
3470:"On finite sphere-packings"
2892:There holds an inequality:
2620:there are parameter values
3918:
3321:can be found from that of
2371:{\displaystyle V(C_{n}+K)}
2238:is the convex hull of the
2083:Brunn-Minkowski inequality
3445:10.1007/978-3-322-80299-6
2405:{\displaystyle \delta =1}
3902:Euclidean solid geometry
3748:isosceles right triangle
3194:{\displaystyle V(B^{d})}
2796:and suffiricently large
503:{\displaystyle d\leq 10}
402:{\displaystyle n\geq 65}
53:Packing and convex hulls
3433:Leppmeier, Max (1997).
2613:{\displaystyle d\geq 2}
685:is therefore given by.
529:{\displaystyle d\geq 3}
455:is relatively tame; in
3762:Circle packing theorem
3646:10.1515/dmvm-1995-0407
3351:
3315:
3279:
3221:
3195:
3159:
3139:
3109:
2883:
2810:
2790:
2748:
2728:
2686:
2650:
2614:
2573:
2551:
2452:
2432:
2406:
2372:
2326:
2306:
2279:
2252:
2232:
2202:
2106:
2061:
2035:
2015:
1975:
1955:
1935:
1912:
1885:
1865:, which translates to
1859:
1829:
1726:
1699:
1675:
1575:
1551:
1484:
1464:
1441:
1360:
1329:
1283:
1239:
1219:
1192:
1106:
1079:
1058:
1016:
996:
969:
679:
652:
632:
581:
561:
530:
504:
475:
429:
403:
377:
327:
283:
263:
203:
182:
62:
44:closely-packed spheres
3477:Mathematica Pannonica
3352:
3316:
3280:
3222:
3196:
3160:
3140:
3138:{\displaystyle B^{d}}
3110:
2884:
2811:
2791:
2749:
2729:
2687:
2651:
2615:
2574:
2572:{\displaystyle \rho }
2552:
2453:
2451:{\displaystyle \rho }
2433:
2407:
2373:
2327:
2307:
2305:{\displaystyle K_{i}}
2280:
2278:{\displaystyle c_{i}}
2253:
2233:
2231:{\displaystyle C_{n}}
2203:
2107:
2062:
2036:
2016:
1976:
1956:
1936:
1913:
1911:{\displaystyle r^{3}}
1886:
1884:{\displaystyle n=455}
1860:
1830:
1727:
1700:
1676:
1576:
1552:
1485:
1465:
1442:
1361:
1359:{\displaystyle V_{T}}
1330:
1284:
1240:
1220:
1193:
1086:
1059:
1038:
1017:
997:
970:
680:
678:{\displaystyle V_{W}}
653:
633:
582:
562:
531:
505:
476:
430:
404:
378:
328:
284:
264:
204:
183:
60:
21:finite sphere packing
3744:equilateral triangle
3325:
3289:
3231:
3205:
3169:
3149:
3122:
2899:
2820:
2800:
2758:
2738:
2696:
2660:
2624:
2598:
2563:
2465:
2442:
2416:
2390:
2340:
2316:
2289:
2262:
2242:
2215:
2119:
2096:
2045:
2025:
2005:
1965:
1945:
1925:
1895:
1869:
1858:{\displaystyle x=13}
1843:
1739:
1709:
1689:
1588:
1565:
1497:
1474:
1454:
1373:
1343:
1300:
1252:
1229:
1209:
1029:
1006:
986:
692:
662:
642:
595:
571:
551:
514:
488:
459:
428:{\displaystyle n=56}
413:
387:
337:
293:
273:
253:
193:
172:
3881:Slothouber–Graatsma
3538:"Research problems"
3220:{\displaystyle d=2}
2594:For each dimension
2585:geometry of numbers
2060:{\displaystyle d=5}
658:. The total volume
474:{\displaystyle d=4}
448:sausage catastrophe
245:Sausage catastrophe
61:Convex hull in blue
16:Mathematical theory
3602:10.1007/PL00009341
3554:10.1007/BF02018822
3401:10.1007/bf03024394
3347:
3311:
3275:
3217:
3191:
3155:
3135:
3105:
2879:
2806:
2786:
2744:
2724:
2682:
2646:
2610:
2569:
2547:
2448:
2428:
2402:
2368:
2334:Minkowski addition
2322:
2302:
2275:
2248:
2228:
2198:
2102:
2057:
2031:
2011:
1999:sausage conjecture
1997:, who posited the
1985:Sausage conjecture
1971:
1951:
1931:
1908:
1881:
1855:
1825:
1722:
1695:
1671:
1571:
1547:
1480:
1460:
1437:
1356:
1325:
1293:From this we have
1279:
1235:
1215:
1188:
1012:
992:
965:
963:
675:
648:
628:
577:
567:spheres of radius
557:
526:
500:
471:
425:
399:
373:
323:
282:{\displaystyle 55}
279:
259:
199:
178:
63:
40:crystal structures
3889:
3888:
3848:Other 3-D packing
3832:Other 2-D packing
3757:Apollonian gasket
3517:978-0-444-89597-4
3454:978-3-528-06792-2
3158:{\displaystyle d}
3069:
2955:
2877:
2873:
2809:{\displaystyle n}
2747:{\displaystyle n}
2589:Hermann Minkowski
2545:
2336:of sets, so that
2325:{\displaystyle K}
2251:{\displaystyle n}
2196:
2105:{\displaystyle K}
2087:Steiner's formula
2034:{\displaystyle d}
2014:{\displaystyle d}
1995:László Fejes Tóth
1974:{\displaystyle n}
1954:{\displaystyle x}
1934:{\displaystyle V}
1807:
1749:
1719:
1698:{\displaystyle n}
1669:
1650:
1629:
1574:{\displaystyle V}
1534:
1483:{\displaystyle x}
1463:{\displaystyle a}
1422:
1399:
1395:
1317:
1271:
1267:
1238:{\displaystyle a}
1218:{\displaystyle r}
1186:
1135:
1015:{\displaystyle n}
995:{\displaystyle x}
941:
891:
815:
775:
762:
742:
723:
651:{\displaystyle r}
580:{\displaystyle r}
560:{\displaystyle n}
262:{\displaystyle n}
202:{\displaystyle d}
181:{\displaystyle d}
36:Kepler conjecture
29:László Fejes Tóth
3909:
3770:
3710:Abstract packing
3703:Packing problems
3696:
3689:
3682:
3673:
3667:
3666:
3648:
3624:
3615:
3614:
3604:
3580:
3574:
3573:
3533:
3527:
3526:
3525:
3524:
3491:
3485:
3484:
3474:
3465:
3459:
3458:
3430:
3421:
3420:
3384:
3356:
3354:
3353:
3348:
3343:
3342:
3320:
3318:
3317:
3312:
3301:
3300:
3284:
3282:
3281:
3276:
3265:
3264:
3243:
3242:
3226:
3224:
3223:
3218:
3200:
3198:
3197:
3192:
3187:
3186:
3164:
3162:
3161:
3156:
3144:
3142:
3141:
3136:
3134:
3133:
3114:
3112:
3111:
3106:
3104:
3103:
3092:
3082:
3081:
3070:
3068:
3064:
3063:
3038:
3034:
3033:
3017:
3009:
3008:
2990:
2989:
2978:
2968:
2967:
2956:
2954:
2950:
2949:
2924:
2920:
2919:
2903:
2888:
2886:
2885:
2880:
2878:
2869:
2868:
2854:
2853:
2832:
2831:
2815:
2813:
2812:
2807:
2795:
2793:
2792:
2787:
2776:
2775:
2753:
2751:
2750:
2745:
2733:
2731:
2730:
2725:
2714:
2713:
2691:
2689:
2688:
2683:
2672:
2671:
2655:
2653:
2652:
2647:
2636:
2635:
2619:
2617:
2616:
2611:
2578:
2576:
2575:
2570:
2556:
2554:
2553:
2548:
2546:
2544:
2531:
2530:
2514:
2497:
2489:
2488:
2457:
2455:
2454:
2449:
2437:
2435:
2434:
2429:
2411:
2409:
2408:
2403:
2377:
2375:
2374:
2369:
2358:
2357:
2331:
2329:
2328:
2323:
2311:
2309:
2308:
2303:
2301:
2300:
2284:
2282:
2281:
2276:
2274:
2273:
2257:
2255:
2254:
2249:
2237:
2235:
2234:
2229:
2227:
2226:
2207:
2205:
2204:
2199:
2197:
2195:
2185:
2184:
2168:
2151:
2143:
2142:
2111:
2109:
2108:
2103:
2066:
2064:
2063:
2058:
2040:
2038:
2037:
2032:
2021:dimensions is a
2020:
2018:
2017:
2012:
1980:
1978:
1977:
1972:
1960:
1958:
1957:
1952:
1940:
1938:
1937:
1932:
1917:
1915:
1914:
1909:
1907:
1906:
1890:
1888:
1887:
1882:
1864:
1862:
1861:
1856:
1834:
1832:
1831:
1826:
1824:
1823:
1808:
1803:
1756:
1751:
1750:
1747:
1731:
1729:
1728:
1723:
1721:
1720:
1717:
1704:
1702:
1701:
1696:
1680:
1678:
1677:
1672:
1670:
1665:
1664:
1663:
1651:
1646:
1641:
1640:
1635:
1631:
1630:
1625:
1598:
1580:
1578:
1577:
1572:
1556:
1554:
1553:
1548:
1540:
1536:
1535:
1530:
1489:
1487:
1486:
1481:
1469:
1467:
1466:
1461:
1446:
1444:
1443:
1438:
1436:
1435:
1423:
1418:
1413:
1412:
1400:
1391:
1390:
1385:
1384:
1365:
1363:
1362:
1357:
1355:
1354:
1334:
1332:
1331:
1326:
1318:
1313:
1288:
1286:
1285:
1280:
1272:
1263:
1262:
1244:
1242:
1241:
1236:
1224:
1222:
1221:
1216:
1197:
1195:
1194:
1189:
1187:
1182:
1141:
1136:
1131:
1108:
1105:
1100:
1078:
1073:
1057:
1052:
1021:
1019:
1018:
1013:
1001:
999:
998:
993:
974:
972:
971:
966:
964:
960:
959:
947:
943:
942:
934:
909:
905:
904:
892:
884:
879:
878:
833:
829:
828:
816:
808:
803:
802:
781:
777:
776:
773:
764:
763:
760:
748:
744:
743:
740:
725:
724:
721:
708:
707:
684:
682:
681:
676:
674:
673:
657:
655:
654:
649:
637:
635:
634:
629:
586:
584:
583:
578:
566:
564:
563:
558:
535:
533:
532:
527:
509:
507:
506:
501:
480:
478:
477:
472:
434:
432:
431:
426:
408:
406:
405:
400:
382:
380:
379:
374:
332:
330:
329:
324:
288:
286:
285:
280:
268:
266:
265:
260:
233:which assigns a
208:
206:
205:
200:
187:
185:
184:
179:
119:
107:
95:
3917:
3916:
3912:
3911:
3910:
3908:
3907:
3906:
3892:
3891:
3890:
3885:
3864:
3843:
3827:
3774:
3768:
3767:Tammes problem
3726:
3705:
3700:
3670:
3626:
3625:
3618:
3582:
3581:
3577:
3535:
3534:
3530:
3522:
3520:
3518:
3493:
3492:
3488:
3472:
3467:
3466:
3462:
3455:
3432:
3431:
3424:
3386:
3385:
3370:
3366:
3360:
3334:
3323:
3322:
3292:
3287:
3286:
3256:
3234:
3229:
3228:
3203:
3202:
3178:
3167:
3166:
3147:
3146:
3125:
3120:
3119:
3073:
3071:
3049:
3039:
3025:
3018:
3000:
2959:
2957:
2935:
2925:
2911:
2904:
2897:
2896:
2845:
2823:
2818:
2817:
2798:
2797:
2767:
2756:
2755:
2736:
2735:
2705:
2694:
2693:
2663:
2658:
2657:
2627:
2622:
2621:
2596:
2595:
2561:
2560:
2522:
2515:
2498:
2480:
2463:
2462:
2440:
2439:
2414:
2413:
2388:
2387:
2349:
2338:
2337:
2314:
2313:
2292:
2287:
2286:
2285:of the spheres
2265:
2260:
2259:
2240:
2239:
2218:
2213:
2212:
2176:
2169:
2152:
2134:
2117:
2116:
2094:
2093:
2079:convex geometry
2074:
2043:
2042:
2023:
2022:
2003:
2002:
1987:
1963:
1962:
1943:
1942:
1923:
1922:
1898:
1893:
1892:
1867:
1866:
1841:
1840:
1815:
1757:
1742:
1737:
1736:
1712:
1707:
1706:
1687:
1686:
1655:
1611:
1607:
1606:
1599:
1586:
1585:
1563:
1562:
1516:
1512:
1495:
1494:
1472:
1471:
1452:
1451:
1427:
1404:
1376:
1371:
1370:
1346:
1341:
1340:
1298:
1297:
1250:
1249:
1227:
1226:
1207:
1206:
1142:
1109:
1027:
1026:
1004:
1003:
984:
983:
962:
961:
951:
926:
922:
907:
906:
896:
870:
831:
830:
820:
794:
779:
778:
768:
755:
746:
745:
735:
716:
709:
699:
690:
689:
665:
660:
659:
640:
639:
593:
592:
569:
568:
549:
548:
542:
512:
511:
486:
485:
484:For dimensions
457:
456:
411:
410:
385:
384:
335:
334:
291:
290:
271:
270:
251:
250:
247:
223:packing density
219:
217:Optimal packing
191:
190:
170:
169:
166:
158:cluster packing
154:
152:Cluster packing
142:
134:sausage packing
130:
128:Sausage packing
123:
122:Cluster packing
120:
111:
108:
99:
98:Sausage packing
96:
84:
55:
17:
12:
11:
5:
3915:
3913:
3905:
3904:
3894:
3893:
3887:
3886:
3884:
3883:
3878:
3872:
3870:
3866:
3865:
3863:
3862:
3857:
3851:
3849:
3845:
3844:
3842:
3841:
3839:Square packing
3835:
3833:
3829:
3828:
3826:
3825:
3820:
3818:Kissing number
3815:
3810:
3805:
3800:
3795:
3790:
3784:
3782:
3780:Sphere packing
3776:
3775:
3773:
3772:
3764:
3759:
3754:
3736:
3734:
3732:Circle packing
3728:
3727:
3725:
3724:
3719:
3713:
3711:
3707:
3706:
3701:
3699:
3698:
3691:
3684:
3676:
3669:
3668:
3616:
3595:(2): 197–227.
3575:
3548:(2): 197–199.
3528:
3516:
3486:
3460:
3453:
3422:
3367:
3365:
3362:
3346:
3341:
3337:
3333:
3330:
3310:
3307:
3304:
3299:
3295:
3274:
3271:
3268:
3263:
3259:
3255:
3252:
3249:
3246:
3241:
3237:
3216:
3213:
3210:
3190:
3185:
3181:
3177:
3174:
3165:dimensions is
3154:
3132:
3128:
3116:
3115:
3102:
3099:
3096:
3091:
3088:
3085:
3080:
3076:
3067:
3062:
3059:
3056:
3052:
3048:
3045:
3042:
3037:
3032:
3028:
3024:
3021:
3015:
3012:
3007:
3003:
2999:
2996:
2993:
2988:
2985:
2982:
2977:
2974:
2971:
2966:
2962:
2953:
2948:
2945:
2942:
2938:
2934:
2931:
2928:
2923:
2918:
2914:
2910:
2907:
2876:
2872:
2866:
2863:
2860:
2857:
2852:
2848:
2844:
2841:
2838:
2835:
2830:
2826:
2805:
2785:
2782:
2779:
2774:
2770:
2766:
2763:
2743:
2723:
2720:
2717:
2712:
2708:
2704:
2701:
2692:such that for
2681:
2678:
2675:
2670:
2666:
2645:
2642:
2639:
2634:
2630:
2609:
2606:
2603:
2568:
2558:
2557:
2543:
2540:
2537:
2534:
2529:
2525:
2521:
2518:
2513:
2510:
2507:
2504:
2501:
2495:
2492:
2487:
2483:
2479:
2476:
2473:
2470:
2447:
2427:
2424:
2421:
2401:
2398:
2395:
2367:
2364:
2361:
2356:
2352:
2348:
2345:
2321:
2299:
2295:
2272:
2268:
2247:
2225:
2221:
2209:
2208:
2194:
2191:
2188:
2183:
2179:
2175:
2172:
2167:
2164:
2161:
2158:
2155:
2149:
2146:
2141:
2137:
2133:
2130:
2127:
2124:
2101:
2081:, such as the
2073:
2070:
2068:42 and above.
2056:
2053:
2050:
2030:
2010:
1986:
1983:
1970:
1961:and therefore
1950:
1930:
1905:
1901:
1880:
1877:
1874:
1854:
1851:
1848:
1837:
1836:
1822:
1818:
1814:
1811:
1806:
1802:
1799:
1796:
1793:
1790:
1787:
1784:
1781:
1778:
1775:
1772:
1769:
1766:
1763:
1760:
1754:
1745:
1715:
1694:
1683:
1682:
1668:
1662:
1658:
1654:
1649:
1644:
1639:
1634:
1628:
1623:
1620:
1617:
1614:
1610:
1605:
1602:
1596:
1593:
1570:
1559:
1558:
1546:
1543:
1539:
1533:
1528:
1525:
1522:
1519:
1515:
1511:
1508:
1505:
1502:
1479:
1459:
1448:
1447:
1434:
1430:
1426:
1421:
1416:
1411:
1407:
1403:
1398:
1394:
1388:
1383:
1379:
1353:
1349:
1337:
1336:
1324:
1321:
1316:
1311:
1308:
1305:
1291:
1290:
1278:
1275:
1270:
1266:
1260:
1257:
1234:
1214:
1200:
1199:
1185:
1181:
1178:
1175:
1172:
1169:
1166:
1163:
1160:
1157:
1154:
1151:
1148:
1145:
1139:
1134:
1130:
1127:
1124:
1121:
1118:
1115:
1112:
1104:
1099:
1096:
1093:
1089:
1085:
1082:
1077:
1072:
1069:
1066:
1062:
1056:
1051:
1048:
1045:
1041:
1037:
1034:
1011:
991:
976:
975:
958:
954:
950:
946:
940:
937:
932:
929:
925:
921:
918:
915:
912:
910:
908:
903:
899:
895:
890:
887:
882:
877:
873:
869:
866:
863:
860:
857:
854:
851:
848:
845:
842:
839:
836:
834:
832:
827:
823:
819:
814:
811:
806:
801:
797:
793:
790:
787:
784:
782:
780:
771:
767:
758:
754:
751:
749:
747:
738:
734:
731:
728:
719:
715:
712:
710:
706:
702:
698:
697:
672:
668:
647:
627:
624:
621:
618:
615:
612:
609:
606:
603:
600:
576:
556:
541:
538:
525:
522:
519:
499:
496:
493:
470:
467:
464:
424:
421:
418:
398:
395:
392:
372:
369:
366:
363:
360:
357:
354:
351:
348:
345:
342:
322:
319:
316:
313:
310:
307:
304:
301:
298:
278:
258:
246:
243:
218:
215:
198:
177:
165:
162:
153:
150:
141:
138:
129:
126:
125:
124:
121:
114:
112:
109:
102:
100:
97:
90:
83:
82:Packing shapes
80:
65:In general, a
54:
51:
15:
13:
10:
9:
6:
4:
3:
2:
3914:
3903:
3900:
3899:
3897:
3882:
3879:
3877:
3874:
3873:
3871:
3867:
3861:
3858:
3856:
3853:
3852:
3850:
3846:
3840:
3837:
3836:
3834:
3830:
3824:
3821:
3819:
3816:
3814:
3813:Close-packing
3811:
3809:
3808:In a cylinder
3806:
3804:
3801:
3799:
3796:
3794:
3791:
3789:
3786:
3785:
3783:
3781:
3777:
3771:
3765:
3763:
3760:
3758:
3755:
3753:
3749:
3745:
3741:
3738:
3737:
3735:
3733:
3729:
3723:
3720:
3718:
3715:
3714:
3712:
3708:
3704:
3697:
3692:
3690:
3685:
3683:
3678:
3677:
3674:
3664:
3660:
3656:
3652:
3647:
3642:
3638:
3634:
3630:
3623:
3621:
3617:
3612:
3608:
3603:
3598:
3594:
3590:
3586:
3579:
3576:
3571:
3567:
3563:
3559:
3555:
3551:
3547:
3543:
3539:
3532:
3529:
3519:
3513:
3509:
3505:
3501:
3497:
3490:
3487:
3482:
3478:
3471:
3464:
3461:
3456:
3450:
3446:
3442:
3438:
3437:
3429:
3427:
3423:
3418:
3414:
3410:
3406:
3402:
3398:
3394:
3390:
3383:
3381:
3379:
3377:
3375:
3373:
3369:
3363:
3361:
3358:
3339:
3335:
3328:
3305:
3297:
3293:
3269:
3261:
3257:
3253:
3247:
3239:
3235:
3214:
3211:
3208:
3183:
3179:
3172:
3152:
3130:
3126:
3100:
3097:
3094:
3086:
3078:
3074:
3060:
3057:
3054:
3050:
3043:
3040:
3030:
3026:
3019:
3013:
3005:
3001:
2994:
2991:
2986:
2983:
2980:
2972:
2964:
2960:
2946:
2943:
2940:
2936:
2929:
2926:
2916:
2912:
2905:
2895:
2894:
2893:
2890:
2874:
2870:
2864:
2858:
2850:
2846:
2842:
2836:
2828:
2824:
2803:
2780:
2772:
2768:
2764:
2761:
2754:), while for
2741:
2718:
2710:
2706:
2702:
2699:
2676:
2668:
2664:
2640:
2632:
2628:
2607:
2604:
2601:
2592:
2590:
2586:
2582:
2581:mixed volumes
2566:
2538:
2535:
2532:
2527:
2523:
2516:
2508:
2502:
2499:
2493:
2485:
2481:
2477:
2474:
2468:
2461:
2460:
2459:
2445:
2425:
2422:
2419:
2399:
2396:
2393:
2384:
2383:Claude Rogers
2379:
2362:
2359:
2354:
2350:
2343:
2335:
2319:
2297:
2293:
2270:
2266:
2245:
2223:
2219:
2189:
2186:
2181:
2177:
2170:
2162:
2156:
2153:
2147:
2139:
2135:
2131:
2128:
2122:
2115:
2114:
2113:
2099:
2090:
2088:
2084:
2080:
2071:
2069:
2054:
2051:
2048:
2028:
2008:
2000:
1996:
1992:
1984:
1982:
1968:
1948:
1928:
1919:
1903:
1899:
1878:
1875:
1872:
1852:
1849:
1846:
1820:
1816:
1812:
1809:
1804:
1800:
1797:
1791:
1788:
1785:
1779:
1773:
1770:
1767:
1761:
1758:
1752:
1743:
1735:
1734:
1733:
1713:
1692:
1666:
1660:
1656:
1652:
1647:
1642:
1637:
1632:
1626:
1621:
1618:
1615:
1612:
1608:
1603:
1600:
1594:
1591:
1584:
1583:
1582:
1568:
1544:
1541:
1537:
1531:
1526:
1523:
1520:
1517:
1513:
1509:
1506:
1503:
1500:
1493:
1492:
1491:
1477:
1457:
1432:
1428:
1424:
1419:
1414:
1409:
1405:
1401:
1396:
1392:
1386:
1381:
1377:
1369:
1368:
1367:
1351:
1347:
1322:
1319:
1314:
1309:
1306:
1303:
1296:
1295:
1294:
1276:
1273:
1268:
1264:
1258:
1255:
1248:
1247:
1246:
1232:
1212:
1205:
1183:
1176:
1173:
1170:
1164:
1158:
1155:
1152:
1146:
1143:
1137:
1132:
1125:
1122:
1119:
1113:
1110:
1102:
1097:
1094:
1091:
1087:
1083:
1080:
1075:
1070:
1067:
1064:
1060:
1054:
1049:
1046:
1043:
1039:
1035:
1032:
1025:
1024:
1023:
1009:
989:
981:
956:
952:
948:
944:
938:
935:
930:
927:
923:
919:
916:
913:
911:
901:
897:
893:
888:
885:
880:
875:
871:
867:
861:
858:
855:
849:
846:
843:
840:
837:
835:
825:
821:
817:
812:
809:
804:
799:
795:
791:
788:
785:
783:
769:
765:
756:
752:
750:
736:
732:
729:
726:
717:
713:
711:
704:
700:
688:
687:
686:
670:
666:
645:
622:
619:
616:
610:
607:
604:
601:
598:
590:
574:
554:
545:
539:
537:
523:
520:
517:
497:
494:
491:
482:
468:
465:
462:
453:
449:
444:
442:
438:
422:
419:
416:
396:
393:
390:
370:
367:
364:
361:
358:
355:
352:
349:
346:
343:
340:
320:
317:
314:
311:
308:
305:
302:
299:
296:
276:
256:
244:
242:
240:
236:
232:
226:
224:
216:
214:
211:
209:
196:
175:
163:
161:
159:
151:
149:
147:
146:pizza packing
140:Pizza packing
139:
137:
135:
127:
118:
113:
110:Pizza packing
106:
101:
94:
89:
87:
81:
79:
77:
73:
68:
59:
52:
50:
47:
45:
41:
37:
32:
30:
26:
22:
3792:
3750: /
3746: /
3742: /
3636:
3632:
3592:
3588:
3578:
3545:
3541:
3531:
3521:, retrieved
3499:
3489:
3480:
3476:
3463:
3435:
3395:(1): 16–21.
3392:
3388:
3359:
3117:
2891:
2593:
2559:
2380:
2210:
2091:
2075:
1998:
1990:
1988:
1920:
1838:
1684:
1560:
1449:
1338:
1292:
1201:
1022:is given by
977:
591:with length
546:
543:
483:
451:
447:
445:
248:
227:
220:
212:
189:
167:
157:
155:
145:
143:
133:
131:
85:
66:
64:
48:
33:
20:
18:
3855:Tetrahedron
3798:In a sphere
3769:(on sphere)
3740:In a circle
3483:(1): 19–29.
1339:The volume
980:tetrahedral
741:half-sphere
452:catastrophe
437:tetrahedral
289:along with
72:convex hull
3788:Apollonian
3523:2022-04-17
3364:References
3227:, we have
2258:midpoints
441:octahedral
76:convex set
3860:Ellipsoid
3803:In a cube
3663:179051027
3655:0942-5977
3611:0179-5376
3570:189833485
3562:0031-5303
3417:122751296
3409:0343-6993
3329:δ
3294:ρ
3258:ρ
3236:ρ
3098:−
3075:ρ
3058:−
3014:≤
2995:δ
2992:≤
2984:−
2961:ρ
2944:−
2847:ρ
2825:ρ
2769:ρ
2765:≥
2762:ρ
2707:ρ
2703:≤
2700:ρ
2665:ρ
2629:ρ
2605:≥
2567:ρ
2536:ρ
2469:δ
2446:ρ
2423:≈
2420:δ
2394:δ
2123:δ
1989:The term
1813:π
1810:⋅
1798:−
1780:⋅
1762:⋅
1653:⋅
1643:⋅
1616:−
1604:⋅
1542:⋅
1521:−
1510:⋅
1425:⋅
1402:⋅
1320:⋅
1274:⋅
1165:⋅
1147:⋅
1114:⋅
1088:∑
1061:∑
1040:∑
949:π
931:−
920:⋅
894:π
868:⋅
859:−
850:⋅
841:π
818:π
789:π
733:⋅
620:−
611:⋅
521:≥
495:≤
394:≥
239:connected
3896:Category
1204:inradius
1202:Now the
761:cylinder
722:cylinder
589:cylinder
3869:Puzzles
1991:sausage
443:shape.
67:packing
25:spheres
3876:Conway
3793:Finite
3752:square
3661:
3653:
3609:
3568:
3560:
3514:
3451:
3415:
3407:
3201:. For
2211:where
774:sphere
333:. For
269:up to
235:vertex
3659:S2CID
3639:(4).
3566:S2CID
3473:(PDF)
3413:S2CID
231:graph
3651:ISSN
3607:ISSN
3558:ISSN
3512:ISBN
3449:ISBN
3405:ISSN
2656:and
2583:and
1839:For
1595:<
383:and
3722:Set
3717:Bin
3641:doi
3597:doi
3550:doi
3504:doi
3441:doi
3397:doi
3145:in
2587:by
2426:0.9
1879:455
1420:192
1245:is
3898::
3657:.
3649:.
3635:.
3631:.
3619:^
3605:.
3593:19
3591:.
3587:.
3564:.
3556:.
3544:.
3540:.
3510:,
3498:,
3479:.
3475:.
3447:.
3425:^
3411:.
3403:.
3393:20
3391:.
3371:^
3357:.
2591:.
2458::
2112::
1981:.
1853:13
1397:12
1269:12
498:10
423:56
397:65
371:62
365:61
359:60
353:59
347:56
321:64
315:63
309:58
303:57
277:55
241:.
31:.
3695:e
3688:t
3681:v
3665:.
3643::
3637:3
3613:.
3599::
3572:.
3552::
3546:6
3506::
3481:3
3457:.
3443::
3419:.
3399::
3345:)
3340:d
3336:B
3332:(
3309:)
3306:d
3303:(
3298:c
3273:)
3270:d
3267:(
3262:c
3254:=
3251:)
3248:d
3245:(
3240:s
3215:2
3212:=
3209:d
3189:)
3184:d
3180:B
3176:(
3173:V
3153:d
3131:d
3127:B
3101:d
3095:1
3090:)
3087:d
3084:(
3079:s
3066:)
3061:1
3055:d
3051:B
3047:(
3044:V
3041:2
3036:)
3031:d
3027:B
3023:(
3020:V
3011:)
3006:d
3002:B
2998:(
2987:d
2981:1
2976:)
2973:d
2970:(
2965:c
2952:)
2947:1
2941:d
2937:B
2933:(
2930:V
2927:2
2922:)
2917:d
2913:B
2909:(
2906:V
2875:2
2871:3
2865:=
2862:)
2859:2
2856:(
2851:s
2843:=
2840:)
2837:2
2834:(
2829:c
2804:n
2784:)
2781:d
2778:(
2773:c
2742:n
2722:)
2719:d
2716:(
2711:s
2680:)
2677:d
2674:(
2669:c
2644:)
2641:d
2638:(
2633:s
2608:2
2602:d
2542:)
2539:K
2533:+
2528:n
2524:C
2520:(
2517:V
2512:)
2509:K
2506:(
2503:V
2500:n
2494:=
2491:)
2486:n
2482:C
2478:,
2475:K
2472:(
2400:1
2397:=
2366:)
2363:K
2360:+
2355:n
2351:C
2347:(
2344:V
2320:K
2298:i
2294:K
2271:i
2267:c
2246:n
2224:n
2220:C
2193:)
2190:K
2187:+
2182:n
2178:C
2174:(
2171:V
2166:)
2163:K
2160:(
2157:V
2154:n
2148:=
2145:)
2140:n
2136:C
2132:,
2129:K
2126:(
2100:K
2055:5
2052:=
2049:d
2029:d
2009:d
1969:n
1949:x
1929:V
1904:3
1900:r
1876:=
1873:n
1850:=
1847:x
1835:.
1821:3
1817:r
1805:3
1801:2
1795:)
1792:2
1789:+
1786:x
1783:(
1777:)
1774:1
1771:+
1768:x
1765:(
1759:x
1753:=
1748:W
1744:V
1718:W
1714:V
1693:n
1681:.
1667:3
1661:3
1657:r
1648:2
1638:3
1633:)
1627:6
1622:+
1619:1
1613:x
1609:(
1601:2
1592:V
1569:V
1557:.
1545:r
1538:)
1532:6
1527:+
1524:1
1518:x
1514:(
1507:2
1504:=
1501:a
1478:x
1458:a
1433:3
1429:r
1415:=
1410:3
1406:a
1393:2
1387:=
1382:T
1378:V
1352:T
1348:V
1335:.
1323:r
1315:6
1310:2
1307:=
1304:a
1289:.
1277:a
1265:6
1259:=
1256:r
1233:a
1213:r
1198:.
1184:6
1180:)
1177:2
1174:+
1171:x
1168:(
1162:)
1159:1
1156:+
1153:x
1150:(
1144:x
1138:=
1133:2
1129:)
1126:1
1123:+
1120:i
1117:(
1111:i
1103:x
1098:1
1095:=
1092:i
1084:=
1081:j
1076:i
1071:1
1068:=
1065:j
1055:x
1050:1
1047:=
1044:i
1036:=
1033:n
1010:n
990:x
957:3
953:r
945:)
939:3
936:1
928:n
924:(
917:2
914:=
902:3
898:r
889:3
886:4
881:+
876:2
872:r
865:)
862:1
856:n
853:(
847:r
844:2
838:=
826:3
822:r
813:3
810:4
805:+
800:2
796:r
792:h
786:=
770:V
766:+
757:V
753:=
737:V
730:2
727:+
718:V
714:=
705:W
701:V
671:W
667:V
646:r
626:)
623:1
617:n
614:(
608:r
605:2
602:=
599:h
575:r
555:n
524:3
518:d
492:d
469:4
466:=
463:d
420:=
417:n
391:n
368:,
362:,
356:,
350:,
344:=
341:n
318:,
312:,
306:,
300:=
297:n
257:n
197:d
176:d
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