101:
176:
824:
1186:
939:
595:
386:, adopted from botany. It is used to describe arrangements of leaves of a plant, pine cones, or pineapples, but also planar patterns of florets in a sunflower head. While the arrangement in the former are cylindrical, the spirals in the latter are arranged on a disk. For columnar structures phyllotaxis in the context of cylindrical structures is adopted.
1071:
They calculated the packing fraction for all these structures as a function of the diameter ratio. At the peaks of this curve lie the uniform structures. In-between these discrete diameter ratios are the line slips at a lower packing density. Their packing fraction is significantly smaller than that
187:
structures confined inside a glass tube. They can be realised experimentally with equal-sized soap bubbles inside a glass tube, produced by blowing air of constant gas flow through a needle dipped in a surfactant solution. By putting the resulting foam column under forced drainage (feeding it with
842:
All spheres in a uniform structure have the same number of contacts, but the number of contacts for spheres in a line slip may differ from sphere to sphere. For the example line slip in the image on the right side, some spheres count five and others six contacts. Thus a line slip structure is
76:
The book "Columnar
Structures of Spheres: Fundamentals and Applications" serves as a notable contributions to this field of study. Authored by Winkelmann and Chan, the book reviews theoretical foundations and practical applications of densely packed spheres within cylindrical confinements.
894:
By shearing the row of spheres below the loss of contact against a row above the loss of contact, one can regenerate two uniform structures related to this line slip. Thus, each line slip is related to two adjacent uniform structures, one at a higher and one at a lower diameter ratio
531:
describes a family of spirals in the 3-dimensional packing. They count the number of spirals in each direction until the spiral repeats. This notation, however, only applies to triangular lattices and is therefore restricted to the ordered structures without internal spheres.
159:
form a columnar structure in autumn. Its berries are similar to that of the corpse flower, since the titan arum is its larger relative. However, the cuckoo-pint is much smaller in height (height ≈ 20 cm). The berry arrangement varies with the stem to berry size.
1120:
also discovered that such structures can be related to disk packings on a surface of a cylinder. The contact network of both packings are identical. For both packing types, it was found that different uniform structures are connected with each other by line slips.
268:
built rods of the size of several microns. These microrods are created by densely packing silica colloidal particles inside cylindrical pores. By solidifying the assembled structures the microrods were imaged and examined using scanning electron microscopy (SEM).
838:
The differences between uniform and line-slip structures are marginal and difficult to spot from images of the sphere packings. However, by comparing their rolled-out contact networks, one can spot that certain lines (which represent contacts) are missing.
606:
A uniform structure is identified by each sphere having the same number of contacting neighbours. This gives each sphere an identical neighbourhood. In the example image on the side each sphere has six neighbouring contacts.
195:
Due to this simple experimental set-up, many columnar structures have been discovered and investigated in the context of foams with experiments as well as simulation. Many simulations have been carried out using the
233:
in the centre of the tube. For many more arrangements of this type, it was observed that the outside bubble layer is ordered, with each internal layer resembling a different, simpler columnar structure by using
1283:
207:
In the zigzag structure the bubbles are stacked on top of each other in a continuous w-shape. For this particular structure a moving interface with increasing liquid fraction was reported by
Hutzler
261:
examined the morphologies of virus capsid proteins self-assembled around metal nanorods. Drug particles were coated as densely as possible on a spherocylinder to provide the best medical treatment.
225:
Further discovered structures include complex structures with internal spheres/foam cells. Some dry foam structures with interior cells were found to consist of a chain of pentagonal
364:
85:
Columnar structures appear in various research fields on a broad range of length scales from metres down to the nanoscale. On the largest scale, such structures can be found in
755:
1079:
The rich variety of such ordered structures can also be obtained by sequential depositioning the spheres into the cylinder. Chan reproduced all dense sphere packings up to
437:
1111:
689:
469:
1726:
Norman, James; Sorrell, Emma L.; Hu, Yi; Siripurapu, Vaishnavi; Garcia, Jamie; Bagwell, Jennifer; Charbonneau, Patrick; Lubkin, Sharon R.; Bagnat, Michel (2018-11-05).
1475:
1160:
813:
727:
1026:
2044:
Troche, Karla S.; Coluci, Vitor R.; Braga, Scheila F.; Chinellato, David D.; Sato, Fernando; Legoas, Sergio B.; Rurali, Riccardo; Galvão, Douglas S. (2005-02-01).
1343:
889:
648:
584:
1370:
250:. Their physical or chemical properties can be altered by trapping identical particles inside them. These are usually done by self-assembling fullerenes such as
3098:
980:
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509:
322:
921:
1323:
1303:
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628:
529:
489:
302:
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Such a structure is termed line slip because the losses of contacts occur along a line in the rolled-out contact network. It was first identified by Picket
691:, the regular hexagonal lattice is its characterising feature since this lattice type has the maximum number of contacts. For different uniform structures
20:
3044:
2099:
Sanwaria, Sunita; Horechyy, Andriy; Wolf, Daniel; Chu, Che-Yi; Chen, Hsin-Lung; Formanek, Petr; Stamm, Manfred; Srivastava, Rajiv; Nandan, Bhanu (2014).
1441:
1929:"Corrected Article: Simulation and observation of line-slip structures in columnar structures of soft spheres [Phys. Rev. E 96, 012610 (2017)]"
1379:
Depending on number of spheres and rotational speed, a variety of ordered structures that are comparable to the dense sphere packings were discovered.
3094:
42:
of specified diameter and length. For cylinders with diameters on the same order of magnitude as the spheres, such packings result in what are called
1222:
610:
The number of contacts is best visualised in the rolled-out contact network. It is created by rolling out the contact network into a plane of height
1177:. Such packings include achiral structures with specific spheroid orientations and chiral helical structures with rotating spheroid orientations.
257:
Such structures also assemble when particles are coated on the surface of a spherocylinder as in the context of pharmaceutical research. Lazáro
1856:
1459:
280:
are constructing such a resonator by self-assembling nanospheres on the surface of the cylinder. The nanospheres are suspended in an
1650:
2209:
Chopra, Nasreen G.; Luyken, R. J.; Cherrey, K.; Crespi, Vincent H.; Cohen, Marvin L.; Louie, Steven G.; Zettl, A. (1995-08-18).
891:, since each number represents one of the lattice vectors in the hexagonal lattice. This is usually indicated by a bold number.
3231:
602:
structure and its corresponding rolled-out contact network. The identical vicinity of each sphere defines a uniform structure.
167:. It assembles its seed capsules around a branch of the plant. The structure depends on the seed capsule size to branch size.
3163:
3037:
1404:
2146:
Yamazaki, T; Kuramochi, K; Takagi, D; Homma, Y; Nishimura, F; Hori, N; Watanabe, K; Suzuki, S; Kobayashi, Y (2008-01-30).
1189:
Columnar structures are assembled by using rapid rotations around a central axis to drive the spheres towards this axis.
124:
276:(i.e. materials with a negative refractive index) which find applications in super lenses or optical cloaking. Tanjeem
230:
3262:
3148:
3102:
3090:
89:
where seeds of a plant assemble around the stem. On a smaller scale bubbles of equal size crystallise to columnar
3252:
3138:
3030:
835:
For each uniform structure, there also exists a related but different structure, called a line-slip arrangement.
2101:"Helical Packing of Nanoparticles Confined in Cylindrical Domains of a Self-Assembled Block Copolymer Structure"
930:
were the first to experimentally realise such a structure using soap bubbles in a system of deformable spheres.
654:. Each dot in this pattern represents a sphere of the packing and each line a contact between adjacent spheres.
3267:
3153:
831:
structure and its corresponding rolled-out contact network. A line slip is identified by the loss of contacts.
1166:
and discovered 17 new dense structures with internal spheres that are not in contact with the cylinder wall.
327:
215:
97:
such structures can be found in man-made objects which are on length scales from a micron to the nanoscale.
986:
Columnar structures arise naturally in the context of dense hard sphere packings inside a cylinder. Mughal
3112:
281:
150:. This flower can be up to 3m in height and is natively solely found in western Sumatra and western Java.
147:
2888:"Non-Equilibrium Self-Assembly of Monocomponent and Multicomponent Tubular Structures in Rotating Fluids"
3143:
273:
70:
1786:. A Collection of Papers Presented at the 10th Eufoam Conference, Thessaloniki, Greece,7-10 July 2014.
3010:
1204:
When the lathe is static, the beads float on top of the liquid. With increasing rotational speed, the
1068:. This includes some structures with internal spheres that are not in contact with the cylinder wall.
650:
of each sphere. For a uniform structure such as the one in the example image, this leads to a regular
100:
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2722:
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2593:
2456:
2285:
2222:
2159:
2057:
1999:
1950:
1890:
1779:
1630:
1575:
1502:
1169:
A similar variety of dense crystalline structures have also been discovered for columnar packings of
1073:
251:
2765:
2147:
1836:
732:
3205:
3067:
2764:
Fu, Lin; Steinhardt, William; Zhao, Hao; Socolar, Joshua E. S.; Charbonneau, Patrick (2016-02-23).
1373:
991:
2499:
2385:
2988:
2954:
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2887:
2819:
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2746:
2712:
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1940:
1817:
1708:
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1534:
1469:
1174:
1163:
392:
128:
111:
660:
540:
Ordered columnar structures without internal spheres are categorised into two separate classes:
131:" (1917). But they are also of interest in other biological areas, including bacteria, viruses,
1680:
1082:
853:
The direction, in which the loss of contacts occur can be denoted in the phyllotactic notation
442:
211:
in 1997. This included an unexpected 180° twist interface, whose explanation is still lacking.
188:
surfactant solution from the top), the foam can be adjusted to either a dry (bubbles shaped as
175:
3210:
3107:
2980:
2915:
2907:
2868:
2811:
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2738:
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2301:
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2081:
2073:
2015:
1968:
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1809:
1757:
1700:
1656:
1646:
1611:
1593:
1526:
1518:
1455:
1217:
1205:
1113:
using an algorithm, in which the spheres are placed sequentially dropped inside the cylinder.
651:
58:
2433:
Cai, Wenshan; Chettiar, Uday K.; Kildishev, Alexander V.; Shalaev, Vladimir M. (April 2007).
1197:. Here, polymeric beads are placed together with a fluid of higher density inside a rotating
780:
694:
123:
Columnar structures were first studied in botany due to their diverse appearances in plants.
3257:
3053:
2972:
2899:
2858:
2795:
2730:
2665:
2601:
2562:
2464:
2405:
2397:
2350:
2309:
2293:
2230:
2167:
2112:
2065:
2007:
1958:
1898:
1844:
1799:
1791:
1747:
1739:
1692:
1638:
1601:
1583:
1510:
1447:
1409:
1131:
2210:
1490:
997:
2337:
Wu, Gaoxiang; Cho, Hyesung; Wood, Derek A.; Dinsmore, Anthony D.; Yang, Shu (2017-04-12).
1387:
367:
197:
146:
One of the largest flowers where the berries arrange in a regular cylindrical form is the
31:
2171:
1328:
856:
633:
551:
2968:
2854:
2791:
2726:
2661:
2597:
2460:
2289:
2226:
2163:
2061:
2046:"Prediction of Ordered Phases of Encapsulated C60, C70, and C78 Inside Carbon Nanotubes"
2003:
1954:
1894:
1579:
1506:
1216:
the central axis. Hence, the beads are essentially confined by a potential given by the
3189:
3168:
3130:
3117:
3082:
2581:
2314:
2273:
1752:
1727:
1606:
1553:
1399:
1348:
965:
945:
494:
307:
155:
106:
94:
62:
2942:
2700:
2635:
1778:
Meagher, A. J.; García-Moreno, F.; Banhart, J.; Mughal, A.; Hutzler, S. (2015-05-20).
1642:
1633:, in Lauffer, Max A.; Bang, Frederik B.; Maramorosch, Karl; Smith, Kenneth M. (eds.),
1386:
It is based on analytic energy calculations using a generic sphere model and predicts
389:
The phyllotactic notation describes such structures by a triplet of positive integers
3246:
3226:
3173:
2525:
2027:
898:
2992:
2927:
2823:
2750:
2685:
2484:
2258:
2195:
1821:
1538:
2941:
Winkelmann, J.; Mughal, A.; Williams, D. B.; Weaire, D.; Hutzler, S. (2019-02-25).
2863:
2838:
2561:(Thesis thesis). Trinity College Dublin. School of Physics. Discipline of Physics.
1795:
1308:
1288:
1051:
1031:
760:
613:
514:
474:
287:
226:
201:
132:
2434:
2234:
1514:
3072:
2605:
1185:
823:
383:
164:
54:
2976:
2734:
2669:
1963:
1928:
1927:
Winkelmann, J.; Haffner, B.; Weaire, D.; Mughal, A.; Hutzler, S. (2017-07-31).
2011:
1902:
1848:
757:
plane. Each uniform structure is thus distinguished by its periodicity vector
594:
189:
140:
2911:
2807:
2476:
2419:
2401:
2362:
2339:"Confined Assemblies of Colloidal Particles with Soft Repulsive Interactions"
2305:
2242:
2179:
2124:
2077:
2019:
1910:
1813:
1704:
1597:
1522:
3015:
2468:
1987:
1878:
1780:"An experimental study of columnar crystals using monodisperse microbubbles"
1588:
247:
163:
Another plant that can be found in many gardens of residential areas is the
136:
2984:
2919:
2903:
2872:
2815:
2742:
2677:
2613:
2556:
2370:
2323:
2250:
2187:
2132:
2116:
2085:
1972:
1761:
1743:
1637:, Advances in Virus Research, vol. 20, Academic Press, pp. 1–32,
1615:
1193:
A further dynamic method to assemble such structures was introduced by Lee
183:
A further occurrence of ordered columnar arrangement on the macroscale are
19:
1530:
1451:
1076:, bcc, or hcp due to the free volume left by the cylindrical confinement.
938:
2354:
1660:
1372:
proportionality, the confining potential resembles that of a cylindrical
1170:
246:
Columnar structures have also been studied intensively in the context of
39:
2701:"Densest columnar structures of hard spheres from sequential deposition"
2451:
2799:
2410:
2338:
2297:
2272:
Lázaro, Guillermo R.; Dragnea, Bogdan; Hagan, Michael F. (2018-07-18).
1712:
548:
structures. For each structure that can be identified with the triplet
272:
Columnar arrangements are also investigated as a possible candidate of
235:
50:
2839:"Shape-Anisotropy-Induced Ordered Packings in Cylindrical Confinement"
2566:
2069:
1804:
1732:
Philosophical
Transactions of the Royal Society B: Biological Sciences
1382:
A comprehensive theory to this experiment was developed by
Winkelmann
254:, C70, or C78 into carbon nanotubes, but also boron nitride nanotubes
127:
analysed such arrangement of plant parts around the stem in his book "
61:, and so forth due to the analogous assembly of small particles (like
86:
35:
2148:"Ordered fullerene nanocylinders in large-diameter carbon nanotubes"
2045:
1696:
3022:
2959:
2782:
1945:
1728:"Tissue self-organization underlies morphogenesis of the notochord"
366:). The nanospheres then stick to the surface of the cylinders by a
2717:
2652:
1570:
1198:
174:
66:
18:
2274:"Self-assembly of convex particles on spherocylindrical surfaces"
1986:
Saadatfar, M.; Barry, J.; Weaire, D.; Hutzler, S. (2008-09-01).
1784:
Colloids and
Surfaces A: Physicochemical and Engineering Aspects
729:
the rolled-out contact pattern only varies by a rotation in the
184:
90:
3026:
2886:
Lee, Taehoon; Gizynski, Konrad; Grzybowski, Bartosz A. (2017).
2634:
Mughal, A.; Chan, H. K.; Weaire, D.; Hutzler, S. (2012-05-11).
1278:{\displaystyle E_{\text{rot}}={\frac {1}{2}}mR^{2}\omega ^{2},}
2580:
Pickett, Galen T.; Gross, Mark; Okuyama, Hiroko (2000-10-23).
586:, there exist a uniform structure and at least one line slip.
1443:
Columnar
Structures of Spheres: Fundamentals and Applications
536:
Types of ordered columnar structures without internal spheres
23:
Illustration of a columnar structure assembled by golf balls.
179:
Spherical soap bubbles confined in a cylindrical glass tube.
1988:"Ordered cylindrical foam structures with internal bubbles"
2943:"Theory of rotational columnar structures of soft spheres"
2558:
Structures of columnar packings with soft and hard spheres
1879:"Moving boundaries in ordered cylindrical foam structures"
34:
with the objective of packing a given number of identical
1835:
Weaire, D.; Hutzler, S.; Verbist, G.; Peters, E. (2007),
1491:"Tubular Packing of Spheres in Biological Fine Structure"
49:
These problems are studied extensively in the context of
2837:
Jin, Weiwei; Chan, Ho-Kei; Zhong, Zheng (2020-06-16).
1351:
1331:
1311:
1291:
1134:
1085:
1054:
1034:
1000:
942:
Optimal packing fraction for hard spheres of diameter
901:
859:
763:
735:
636:
616:
554:
517:
477:
445:
330:
290:
2636:"Dense packings of spheres in cylinders: Simulations"
1554:"Dislocation-mediated growth of bacterial cell walls"
1225:
968:
948:
783:
697:
663:
657:
For all uniform structures above a diameter ratio of
497:
395:
310:
1877:
Hutzler, S.; Weaire, D.; Crawford, R. (1997-06-01).
3219:
3198:
3182:
3129:
3081:
3060:
304:, much larger than the diameter of the nanospheres
204:for the wet limit where the bubbles are spherical.
1364:
1337:
1317:
1297:
1277:
1154:
1105:
1060:
1040:
1020:
974:
954:
915:
883:
807:
769:
749:
721:
683:
642:
622:
578:
523:
503:
483:
463:
431:
358:
316:
296:
843:characterised by these gaps or loss of contacts.
1843:, John Wiley & Sons, Ltd, pp. 315–374,
1558:Proceedings of the National Academy of Sciences
777:, which is defined by the phyllotactic triplet
1181:Columnar structures created by rapid rotations
284:solution together with a cylinder of diameter
3038:
2386:"The Magical World of Photonic Metamaterials"
1128:extended this work to higher diameter ratios
153:On smaller length scales, the berries of the
93:structures when confined in a glass tube. In
8:
1474:: CS1 maint: multiple names: authors list (
1440:Chan, Jens Winkelmann, Ho-Kei (2023-03-31).
3045:
3031:
3023:
374:Classification using phyllotactic notation
2958:
2862:
2781:
2716:
2651:
2582:"Spontaneous Chirality in Simple Systems"
2450:
2409:
2313:
1962:
1944:
1803:
1751:
1605:
1587:
1569:
1356:
1350:
1330:
1310:
1290:
1266:
1256:
1239:
1230:
1224:
1138:
1133:
1089:
1084:
1072:of an unconfined lattice packing such as
1053:
1033:
1004:
999:
967:
947:
905:
900:
858:
782:
762:
739:
734:
696:
667:
662:
635:
615:
553:
516:
496:
476:
444:
394:
348:
334:
329:
309:
289:
2343:Journal of the American Chemical Society
1325:the distance from the central axis, and
1184:
937:
822:
593:
214:The first experimental observation of a
192:) or wet (spherical bubbles) structure.
99:
2766:"Hard sphere packings within cylinders"
2105:Angewandte Chemie International Edition
1446:. New York: Jenny Stanford Publishing.
1421:
359:{\textstyle D/d\approx 3{\text{ to }}5}
3011:"Packing spheres into a Thin Cylinder"
1552:Amir, A.; Nelson, D. R. (2012-06-19).
1467:
2629:
2627:
2625:
2623:
2550:
2548:
2546:
2544:
2542:
2435:"Optical cloaking with metamaterials"
2039:
2037:
1922:
1920:
7:
1773:
1771:
1635:Advances in Virus Research Volume 20
1435:
1433:
1431:
1429:
1427:
1425:
200:to investigate dry structure or the
378:The most common way of classifying
1631:"The Structure of Tubular Viruses"
934:Dense sphere packings in cylinders
14:
1345:the rotational speed. Due to the
16:Three-dimensional packing problem
3009:Becker, Aaron T. and Huang, L.
750:{\textstyle z{\text{-}}\theta }
3174:Sphere-packing (Hamming) bound
2864:10.1103/PhysRevLett.124.248002
2172:10.1088/0957-4484/19/04/045702
1992:Philosophical Magazine Letters
1796:10.1016/j.colsurfa.2014.12.020
1489:Erickson, R. O. (1973-08-24).
1405:Close-packing of equal spheres
962:inside a cylinder of diameter
878:
860:
802:
784:
716:
698:
573:
555:
426:
396:
1:
2500:"Self-assembled metamaterial"
1643:10.1016/s0065-3527(08)60500-x
382:columnar structures uses the
218:was discovered by Winkelmann
2235:10.1126/science.269.5226.966
1841:Advances in Chemical Physics
1679:Bryan, Joseph (1974-12-01).
1515:10.1126/science.181.4101.705
994:up to the diameter ratio of
990:studied such packings using
850:, but not termed line slip.
28:Sphere packing in a cylinder
2699:Chan, Ho-Kei (2011-11-14).
2606:10.1103/PhysRevLett.85.3652
2384:Ozbay, Ekmel (2008-11-01).
1837:"A Review of Foam Drainage"
432:{\displaystyle (l=m+n,m,n)}
110:form a columnar structure (
3284:
2977:10.1103/PhysRevE.99.020602
2735:10.1103/PhysRevE.84.050302
2670:10.1103/PhysRevE.85.051305
1964:10.1103/PhysRevE.97.059902
1629:Hull, Roger (1976-01-01),
1305:is the mass of the beads,
1106:{\textstyle D/d<2.7013}
684:{\displaystyle D/d>2.0}
464:{\textstyle l\geq m\geq n}
2555:Winkelmann, Jens (2020).
2390:Optics and Photonics News
2211:"Boron Nitride Nanotubes"
2012:10.1080/09500830802307658
1903:10.1080/13642819708205711
1849:10.1002/9780470141618.ch5
3099:isosceles right triangle
2402:10.1364/OPN.19.11.000022
1883:Philosophical Magazine B
222:in a system of bubbles.
2843:Physical Review Letters
2586:Physical Review Letters
2469:10.1038/nphoton.2007.28
1589:10.1073/pnas.1207105109
1390:structure transitions.
1175:Monte Carlo simulations
1155:{\textstyle D/d<4.0}
808:{\displaystyle (l,m,n)}
722:{\displaystyle (l,m,n)}
30:is a three-dimensional
3113:Circle packing theorem
2904:10.1002/adma.201704274
2117:10.1002/anie.201403565
1744:10.1098/rstb.2017.0320
1366:
1339:
1319:
1299:
1279:
1208:then pushes the fluid
1190:
1156:
1107:
1062:
1042:
1028:for cylinder diameter
1022:
1021:{\textstyle D/d=2.873}
983:
976:
956:
917:
885:
832:
809:
771:
751:
723:
685:
644:
624:
603:
580:
525:
505:
485:
465:
433:
360:
318:
298:
180:
165:Australian bottlebrush
115:
71:crystalline structures
24:
2498:Manoharan, Vinothan.
1452:10.1201/9780429092114
1367:
1340:
1320:
1300:
1280:
1188:
1157:
1108:
1063:
1043:
1023:
977:
957:
941:
918:
886:
826:
810:
772:
752:
724:
686:
645:
625:
597:
581:
526:
506:
486:
466:
434:
384:phyllotactic notation
361:
319:
299:
274:optical metamaterials
178:
103:
81:Appearance in science
22:
3095:equilateral triangle
2355:10.1021/jacs.6b12975
1349:
1338:{\textstyle \omega }
1329:
1309:
1289:
1223:
1132:
1083:
1052:
1032:
998:
966:
946:
899:
884:{\textstyle (l,m,n)}
857:
781:
761:
733:
695:
661:
643:{\textstyle \theta }
634:
630:and azimuthal angle
614:
579:{\textstyle (l,m,n)}
552:
515:
495:
475:
443:
393:
328:
308:
288:
3232:Slothouber–Graatsma
2969:2019PhRvE..99b0602W
2855:2020PhRvL.124x8002J
2792:2016SMat...12.2505F
2727:2011PhRvE..84e0302C
2662:2012PhRvE..85e1305M
2598:2000PhRvL..85.3652P
2526:"Tanjeem's website"
2461:2007NaPho...1..224C
2290:2018SMat...14.5728L
2227:1995Sci...269..966C
2164:2008Nanot..19d5702Y
2062:2005NanoL...5..349T
2004:2008PMagL..88..661S
1955:2017PhRvE..97e9902W
1895:1997PMagB..75..845H
1580:2012PNAS..109.9833A
1507:1973Sci...181..705E
1374:harmonic oscillator
1048:to sphere diameter
992:simulated annealing
819:Line-slip structure
216:line-slip structure
104:The berries of the
69:) into cylindrical
44:columnar structures
2892:Advanced Materials
2800:10.1039/C5SM02875B
2298:10.1039/C8SM00129D
1738:(1759): 20170320.
1365:{\textstyle R^{2}}
1362:
1335:
1315:
1295:
1275:
1191:
1164:linear programming
1152:
1103:
1058:
1038:
1018:
984:
972:
952:
913:
881:
833:
805:
767:
747:
719:
681:
640:
620:
604:
576:
521:
501:
481:
461:
429:
356:
314:
294:
181:
129:On Growth and Form
116:
25:
3263:Discrete geometry
3240:
3239:
3199:Other 3-D packing
3183:Other 2-D packing
3108:Apollonian gasket
2947:Physical Review E
2705:Physical Review E
2640:Physical Review E
2592:(17): 3652–3655.
2524:Tanjeem, Nabila.
2349:(14): 5095–5101.
2284:(28): 5728–5740.
2221:(5226): 966–967.
2111:(34): 9090–9093.
2070:10.1021/nl047930r
1998:(9–10): 661–668.
1933:Physical Review E
1858:978-0-470-14161-8
1564:(25): 9833–9838.
1501:(4101): 705–716.
1461:978-0-429-09211-4
1247:
1233:
1218:rotational energy
1206:centripetal force
975:{\displaystyle D}
955:{\displaystyle d}
742:
652:hexagonal lattice
590:Uniform structure
504:{\displaystyle m}
351:
317:{\displaystyle d}
202:hard sphere model
59:materials science
3275:
3253:Packing problems
3121:
3061:Abstract packing
3054:Packing problems
3047:
3040:
3033:
3024:
2997:
2996:
2962:
2938:
2932:
2931:
2883:
2877:
2876:
2866:
2834:
2828:
2827:
2785:
2776:(9): 2505–2514.
2761:
2755:
2754:
2720:
2696:
2690:
2689:
2655:
2631:
2618:
2617:
2577:
2571:
2570:
2552:
2537:
2536:
2534:
2532:
2521:
2515:
2514:
2512:
2510:
2495:
2489:
2488:
2454:
2439:Nature Photonics
2430:
2424:
2423:
2413:
2381:
2375:
2374:
2334:
2328:
2327:
2317:
2269:
2263:
2262:
2206:
2200:
2199:
2143:
2137:
2136:
2096:
2090:
2089:
2041:
2032:
2031:
1983:
1977:
1976:
1966:
1948:
1924:
1915:
1914:
1874:
1868:
1867:
1866:
1865:
1832:
1826:
1825:
1807:
1775:
1766:
1765:
1755:
1723:
1717:
1716:
1676:
1670:
1669:
1668:
1667:
1626:
1620:
1619:
1609:
1591:
1573:
1549:
1543:
1542:
1486:
1480:
1479:
1473:
1465:
1437:
1410:Packing problems
1371:
1369:
1368:
1363:
1361:
1360:
1344:
1342:
1341:
1336:
1324:
1322:
1321:
1316:
1304:
1302:
1301:
1296:
1284:
1282:
1281:
1276:
1271:
1270:
1261:
1260:
1248:
1240:
1235:
1234:
1231:
1161:
1159:
1158:
1153:
1142:
1112:
1110:
1109:
1104:
1093:
1067:
1065:
1064:
1059:
1047:
1045:
1044:
1039:
1027:
1025:
1024:
1019:
1008:
981:
979:
978:
973:
961:
959:
958:
953:
922:
920:
919:
916:{\textstyle D/d}
914:
909:
890:
888:
887:
882:
814:
812:
811:
806:
776:
774:
773:
768:
756:
754:
753:
748:
743:
740:
728:
726:
725:
720:
690:
688:
687:
682:
671:
649:
647:
646:
641:
629:
627:
626:
621:
585:
583:
582:
577:
530:
528:
527:
522:
510:
508:
507:
502:
490:
488:
487:
482:
470:
468:
467:
462:
438:
436:
435:
430:
365:
363:
362:
357:
352:
349:
338:
323:
321:
320:
315:
303:
301:
300:
295:
236:X-ray tomography
3283:
3282:
3278:
3277:
3276:
3274:
3273:
3272:
3268:Crystallography
3243:
3242:
3241:
3236:
3215:
3194:
3178:
3125:
3119:
3118:Tammes problem
3077:
3056:
3051:
3006:
3001:
3000:
2940:
2939:
2935:
2898:(47): 1704274.
2885:
2884:
2880:
2836:
2835:
2831:
2763:
2762:
2758:
2698:
2697:
2693:
2633:
2632:
2621:
2579:
2578:
2574:
2554:
2553:
2540:
2530:
2528:
2523:
2522:
2518:
2508:
2506:
2497:
2496:
2492:
2452:physics/0611242
2432:
2431:
2427:
2383:
2382:
2378:
2336:
2335:
2331:
2271:
2270:
2266:
2208:
2207:
2203:
2145:
2144:
2140:
2098:
2097:
2093:
2043:
2042:
2035:
1985:
1984:
1980:
1926:
1925:
1918:
1876:
1875:
1871:
1863:
1861:
1859:
1834:
1833:
1829:
1777:
1776:
1769:
1725:
1724:
1720:
1697:10.2307/1297089
1691:(12): 701–711.
1678:
1677:
1673:
1665:
1663:
1653:
1628:
1627:
1623:
1551:
1550:
1546:
1488:
1487:
1483:
1466:
1462:
1439:
1438:
1423:
1418:
1396:
1352:
1347:
1346:
1327:
1326:
1307:
1306:
1287:
1286:
1262:
1252:
1226:
1221:
1220:
1183:
1130:
1129:
1081:
1080:
1050:
1049:
1030:
1029:
996:
995:
964:
963:
944:
943:
936:
897:
896:
855:
854:
821:
779:
778:
759:
758:
731:
730:
693:
692:
659:
658:
632:
631:
612:
611:
592:
550:
549:
538:
513:
512:
493:
492:
473:
472:
441:
440:
391:
390:
376:
368:depletion force
326:
325:
306:
305:
286:
285:
244:
198:Surface Evolver
173:
125:D'Arcy Thompson
121:
83:
32:packing problem
17:
12:
11:
5:
3281:
3279:
3271:
3270:
3265:
3260:
3255:
3245:
3244:
3238:
3237:
3235:
3234:
3229:
3223:
3221:
3217:
3216:
3214:
3213:
3208:
3202:
3200:
3196:
3195:
3193:
3192:
3190:Square packing
3186:
3184:
3180:
3179:
3177:
3176:
3171:
3169:Kissing number
3166:
3161:
3156:
3151:
3146:
3141:
3135:
3133:
3131:Sphere packing
3127:
3126:
3124:
3123:
3115:
3110:
3105:
3087:
3085:
3083:Circle packing
3079:
3078:
3076:
3075:
3070:
3064:
3062:
3058:
3057:
3052:
3050:
3049:
3042:
3035:
3027:
3021:
3020:
3005:
3004:External links
3002:
2999:
2998:
2933:
2878:
2849:(24): 248002.
2829:
2756:
2691:
2619:
2572:
2538:
2516:
2490:
2445:(4): 224–227.
2425:
2376:
2329:
2264:
2201:
2152:Nanotechnology
2138:
2091:
2056:(2): 349–355.
2033:
1978:
1916:
1889:(6): 845–857.
1869:
1857:
1827:
1767:
1718:
1681:"Microtubules"
1671:
1651:
1621:
1544:
1481:
1460:
1420:
1419:
1417:
1414:
1413:
1412:
1407:
1402:
1400:Sphere packing
1395:
1392:
1359:
1355:
1334:
1318:{\textstyle R}
1314:
1298:{\textstyle m}
1294:
1274:
1269:
1265:
1259:
1255:
1251:
1246:
1243:
1238:
1229:
1212:and the beads
1182:
1179:
1151:
1148:
1145:
1141:
1137:
1102:
1099:
1096:
1092:
1088:
1061:{\textstyle d}
1057:
1041:{\textstyle D}
1037:
1017:
1014:
1011:
1007:
1003:
971:
951:
935:
932:
912:
908:
904:
880:
877:
874:
871:
868:
865:
862:
820:
817:
804:
801:
798:
795:
792:
789:
786:
770:{\textstyle V}
766:
746:
738:
718:
715:
712:
709:
706:
703:
700:
680:
677:
674:
670:
666:
639:
623:{\textstyle z}
619:
591:
588:
575:
572:
569:
566:
563:
560:
557:
537:
534:
524:{\textstyle n}
520:
500:
484:{\textstyle l}
480:
471:. Each number
460:
457:
454:
451:
448:
428:
425:
422:
419:
416:
413:
410:
407:
404:
401:
398:
375:
372:
355:
350: to
347:
344:
341:
337:
333:
313:
297:{\textstyle D}
293:
243:
240:
172:
169:
156:Arum maculatum
120:
117:
107:Arum maculatum
82:
79:
15:
13:
10:
9:
6:
4:
3:
2:
3280:
3269:
3266:
3264:
3261:
3259:
3256:
3254:
3251:
3250:
3248:
3233:
3230:
3228:
3225:
3224:
3222:
3218:
3212:
3209:
3207:
3204:
3203:
3201:
3197:
3191:
3188:
3187:
3185:
3181:
3175:
3172:
3170:
3167:
3165:
3164:Close-packing
3162:
3160:
3159:In a cylinder
3157:
3155:
3152:
3150:
3147:
3145:
3142:
3140:
3137:
3136:
3134:
3132:
3128:
3122:
3116:
3114:
3111:
3109:
3106:
3104:
3100:
3096:
3092:
3089:
3088:
3086:
3084:
3080:
3074:
3071:
3069:
3066:
3065:
3063:
3059:
3055:
3048:
3043:
3041:
3036:
3034:
3029:
3028:
3025:
3018:
3017:
3012:
3008:
3007:
3003:
2994:
2990:
2986:
2982:
2978:
2974:
2970:
2966:
2961:
2956:
2953:(2): 020602.
2952:
2948:
2944:
2937:
2934:
2929:
2925:
2921:
2917:
2913:
2909:
2905:
2901:
2897:
2893:
2889:
2882:
2879:
2874:
2870:
2865:
2860:
2856:
2852:
2848:
2844:
2840:
2833:
2830:
2825:
2821:
2817:
2813:
2809:
2805:
2801:
2797:
2793:
2789:
2784:
2779:
2775:
2771:
2767:
2760:
2757:
2752:
2748:
2744:
2740:
2736:
2732:
2728:
2724:
2719:
2714:
2711:(5): 050302.
2710:
2706:
2702:
2695:
2692:
2687:
2683:
2679:
2675:
2671:
2667:
2663:
2659:
2654:
2649:
2646:(5): 051305.
2645:
2641:
2637:
2630:
2628:
2626:
2624:
2620:
2615:
2611:
2607:
2603:
2599:
2595:
2591:
2587:
2583:
2576:
2573:
2568:
2564:
2560:
2559:
2551:
2549:
2547:
2545:
2543:
2539:
2527:
2520:
2517:
2505:
2504:Manoharan lab
2501:
2494:
2491:
2486:
2482:
2478:
2474:
2470:
2466:
2462:
2458:
2453:
2448:
2444:
2440:
2436:
2429:
2426:
2421:
2417:
2412:
2407:
2403:
2399:
2396:(11): 22–27.
2395:
2391:
2387:
2380:
2377:
2372:
2368:
2364:
2360:
2356:
2352:
2348:
2344:
2340:
2333:
2330:
2325:
2321:
2316:
2311:
2307:
2303:
2299:
2295:
2291:
2287:
2283:
2279:
2275:
2268:
2265:
2260:
2256:
2252:
2248:
2244:
2240:
2236:
2232:
2228:
2224:
2220:
2216:
2212:
2205:
2202:
2197:
2193:
2189:
2185:
2181:
2177:
2173:
2169:
2165:
2161:
2158:(4): 045702.
2157:
2153:
2149:
2142:
2139:
2134:
2130:
2126:
2122:
2118:
2114:
2110:
2106:
2102:
2095:
2092:
2087:
2083:
2079:
2075:
2071:
2067:
2063:
2059:
2055:
2051:
2047:
2040:
2038:
2034:
2029:
2025:
2021:
2017:
2013:
2009:
2005:
2001:
1997:
1993:
1989:
1982:
1979:
1974:
1970:
1965:
1960:
1956:
1952:
1947:
1942:
1939:(5): 059902.
1938:
1934:
1930:
1923:
1921:
1917:
1912:
1908:
1904:
1900:
1896:
1892:
1888:
1884:
1880:
1873:
1870:
1860:
1854:
1850:
1846:
1842:
1838:
1831:
1828:
1823:
1819:
1815:
1811:
1806:
1801:
1797:
1793:
1789:
1785:
1781:
1774:
1772:
1768:
1763:
1759:
1754:
1749:
1745:
1741:
1737:
1733:
1729:
1722:
1719:
1714:
1710:
1706:
1702:
1698:
1694:
1690:
1686:
1682:
1675:
1672:
1662:
1658:
1654:
1652:9780120398201
1648:
1644:
1640:
1636:
1632:
1625:
1622:
1617:
1613:
1608:
1603:
1599:
1595:
1590:
1585:
1581:
1577:
1572:
1567:
1563:
1559:
1555:
1548:
1545:
1540:
1536:
1532:
1528:
1524:
1520:
1516:
1512:
1508:
1504:
1500:
1496:
1492:
1485:
1482:
1477:
1471:
1463:
1457:
1453:
1449:
1445:
1444:
1436:
1434:
1432:
1430:
1428:
1426:
1422:
1415:
1411:
1408:
1406:
1403:
1401:
1398:
1397:
1393:
1391:
1389:
1385:
1380:
1377:
1375:
1357:
1353:
1332:
1312:
1292:
1272:
1267:
1263:
1257:
1253:
1249:
1244:
1241:
1236:
1227:
1219:
1215:
1211:
1207:
1202:
1200:
1196:
1187:
1180:
1178:
1176:
1172:
1167:
1165:
1149:
1146:
1143:
1139:
1135:
1127:
1122:
1119:
1114:
1100:
1097:
1094:
1090:
1086:
1077:
1075:
1069:
1055:
1035:
1015:
1012:
1009:
1005:
1001:
993:
989:
969:
949:
940:
933:
931:
929:
924:
910:
906:
902:
892:
875:
872:
869:
866:
863:
851:
849:
844:
840:
836:
830:
825:
818:
816:
799:
796:
793:
790:
787:
764:
744:
736:
713:
710:
707:
704:
701:
678:
675:
672:
668:
664:
655:
653:
637:
617:
608:
601:
596:
589:
587:
570:
567:
564:
561:
558:
547:
543:
535:
533:
518:
498:
478:
458:
455:
452:
449:
446:
423:
420:
417:
414:
411:
408:
405:
402:
399:
387:
385:
381:
373:
371:
369:
353:
345:
342:
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335:
331:
311:
291:
283:
279:
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267:
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177:
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3120:(on sphere)
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2278:Soft Matter
1388:peritectoid
926:Winkelmann
827:An example
598:An example
242:Nanoscience
227:dodecahedra
190:polyhedrons
95:nanoscience
55:nanoscience
3247:Categories
3139:Apollonian
2960:1808.02952
2783:1511.08472
2567:2262/91733
1946:1703.00773
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