112:
187:
835:
1197:
950:
606:
397:, 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.
1082:
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
198:
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
853:
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
87:
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.
905:
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
542:
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.
170:
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.
1131:
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.
279:
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).
849:
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.
617:
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.
206:
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
244:
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
1294:
218:
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
272:
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.
236:
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
375:
96:
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
766:
1090:
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
448:
1122:
700:
480:
1737:
Norman, James; Sorrell, Emma L.; Hu, Yi; Siripurapu, Vaishnavi; Garcia, Jamie; Bagwell, Jennifer; Charbonneau, Patrick; Lubkin, Sharon R.; Bagnat, Michel (2018-11-05).
1486:
1171:
824:
738:
1037:
2055:
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).
1354:
900:
659:
595:
1381:
261:. Their physical or chemical properties can be altered by trapping identical particles inside them. These are usually done by self-assembling fullerenes such as
3109:
991:
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520:
333:
932:
1334:
1314:
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1057:
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639:
540:
500:
313:
857:
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
702:, the regular hexagonal lattice is its characterising feature since this lattice type has the maximum number of contacts. For different uniform structures
31:
3055:
2110:
Sanwaria, Sunita; Horechyy, Andriy; Wolf, Daniel; Chu, Che-Yi; Chen, Hsin-Lung; Formanek, Petr; Stamm, Manfred; Srivastava, Rajiv; Nandan, Bhanu (2014).
1452:
1940:"Corrected Article: Simulation and observation of line-slip structures in columnar structures of soft spheres [Phys. Rev. E 96, 012610 (2017)]"
1390:
Depending on number of spheres and rotational speed, a variety of ordered structures that are comparable to the dense sphere packings were discovered.
3105:
53:
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
1233:
621:
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
1188:. Such packings include achiral structures with specific spheroid orientations and chiral helical structures with rotating spheroid orientations.
268:
Such structures also assemble when particles are coated on the surface of a spherocylinder as in the context of pharmaceutical research. Lazáro
1867:
1470:
291:
are constructing such a resonator by self-assembling nanospheres on the surface of the cylinder. The nanospheres are suspended in an
1661:
2220:
Chopra, Nasreen G.; Luyken, R. J.; Cherrey, K.; Crespi, Vincent H.; Cohen, Marvin L.; Louie, Steven G.; Zettl, A. (1995-08-18).
902:, since each number represents one of the lattice vectors in the hexagonal lattice. This is usually indicated by a bold number.
3242:
613:
structure and its corresponding rolled-out contact network. The identical vicinity of each sphere defines a uniform structure.
178:. It assembles its seed capsules around a branch of the plant. The structure depends on the seed capsule size to branch size.
3174:
3048:
1415:
2157:
Yamazaki, T; Kuramochi, K; Takagi, D; Homma, Y; Nishimura, F; Hori, N; Watanabe, K; Suzuki, S; Kobayashi, Y (2008-01-30).
1200:
Columnar structures are assembled by using rapid rotations around a central axis to drive the spheres towards this axis.
135:
287:(i.e. materials with a negative refractive index) which find applications in super lenses or optical cloaking. Tanjeem
241:
3273:
3159:
3113:
3101:
100:
where seeds of a plant assemble around the stem. On a smaller scale bubbles of equal size crystallise to columnar
3263:
3149:
3041:
846:
For each uniform structure, there also exists a related but different structure, called a line-slip arrangement.
2112:"Helical Packing of Nanoparticles Confined in Cylindrical Domains of a Self-Assembled Block Copolymer Structure"
941:
were the first to experimentally realise such a structure using soap bubbles in a system of deformable spheres.
665:. Each dot in this pattern represents a sphere of the packing and each line a contact between adjacent spheres.
3278:
3164:
842:
structure and its corresponding rolled-out contact network. A line slip is identified by the loss of contacts.
1177:
and discovered 17 new dense structures with internal spheres that are not in contact with the cylinder wall.
338:
226:
108:
such structures can be found in man-made objects which are on length scales from a micron to the nanoscale.
997:
Columnar structures arise naturally in the context of dense hard sphere packings inside a cylinder. Mughal
3123:
292:
161:. This flower can be up to 3m in height and is natively solely found in western Sumatra and western Java.
158:
2899:"Non-Equilibrium Self-Assembly of Monocomponent and Multicomponent Tubular Structures in Rotating Fluids"
3154:
284:
81:
1797:. A Collection of Papers Presented at the 10th Eufoam Conference, Thessaloniki, Greece,7-10 July 2014.
3021:
1215:
When the lathe is static, the beads float on top of the liquid. With increasing rotational speed, the
1079:. This includes some structures with internal spheres that are not in contact with the cylinder wall.
661:
of each sphere. For a uniform structure such as the one in the example image, this leads to a regular
111:
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2733:
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2233:
2170:
2068:
2010:
1961:
1901:
1790:
1641:
1586:
1513:
1180:
A similar variety of dense crystalline structures have also been discovered for columnar packings of
1084:
262:
2776:
2158:
1847:
743:
3216:
3078:
2775:
Fu, Lin; Steinhardt, William; Zhao, Hao; Socolar, Joshua E. S.; Charbonneau, Patrick (2016-02-23).
1384:
1002:
2510:
2396:
2999:
2965:
2934:
2898:
2830:
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2757:
2723:
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1951:
1828:
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1545:
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1185:
1174:
403:
139:
122:
671:
551:
Ordered columnar structures without internal spheres are categorised into two separate classes:
142:" (1917). But they are also of interest in other biological areas, including bacteria, viruses,
1691:
1093:
864:
The direction, in which the loss of contacts occur can be denoted in the phyllotactic notation
453:
222:
in 1997. This included an unexpected 180° twist interface, whose explanation is still lacking.
199:
surfactant solution from the top), the foam can be adjusted to either a dry (bubbles shaped as
186:
3221:
3118:
2991:
2926:
2918:
2879:
2822:
2814:
2749:
2684:
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1979:
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1711:
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1657:
1622:
1604:
1537:
1529:
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1228:
1216:
1124:
using an algorithm, in which the spheres are placed sequentially dropped inside the cylinder.
662:
69:
2444:
Cai, Wenshan; Chettiar, Uday K.; Kildishev, Alexander V.; Shalaev, Vladimir M. (April 2007).
1208:. Here, polymeric beads are placed together with a fluid of higher density inside a rotating
791:
705:
134:
Columnar structures were first studied in botany due to their diverse appearances in plants.
3268:
3064:
2983:
2910:
2869:
2806:
2741:
2676:
2612:
2573:
2475:
2416:
2408:
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2320:
2304:
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2178:
2123:
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2018:
1969:
1909:
1855:
1810:
1802:
1758:
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1703:
1649:
1612:
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1521:
1458:
1420:
1142:
2221:
1501:
1008:
2348:
Wu, Gaoxiang; Cho, Hyesung; Wood, Derek A.; Dinsmore, Anthony D.; Yang, Shu (2017-04-12).
1398:
378:
208:
157:
One of the largest flowers where the berries arrange in a regular cylindrical form is the
42:
2182:
1339:
867:
644:
562:
2979:
2865:
2802:
2737:
2672:
2608:
2471:
2300:
2237:
2174:
2072:
2057:"Prediction of Ordered Phases of Encapsulated C60, C70, and C78 Inside Carbon Nanotubes"
2014:
1965:
1905:
1590:
1517:
1227:
the central axis. Hence, the beads are essentially confined by a potential given by the
3200:
3179:
3141:
3128:
3093:
2592:
2325:
2284:
1763:
1738:
1617:
1564:
1410:
1359:
976:
956:
505:
318:
166:
117:
105:
73:
17:
2953:
2711:
2646:
1789:
Meagher, A. J.; García-Moreno, F.; Banhart, J.; Mughal, A.; Hutzler, S. (2015-05-20).
1653:
1644:, in Lauffer, Max A.; Bang, Frederik B.; Maramorosch, Karl; Smith, Kenneth M. (eds.),
1397:
It is based on analytic energy calculations using a generic sphere model and predicts
400:
The phyllotactic notation describes such structures by a triplet of positive integers
3257:
3237:
3184:
2536:
2038:
909:
3003:
2938:
2834:
2761:
2696:
2495:
2269:
2206:
1832:
1549:
2952:
Winkelmann, J.; Mughal, A.; Williams, D. B.; Weaire, D.; Hutzler, S. (2019-02-25).
2874:
2849:
2572:(Thesis thesis). Trinity College Dublin. School of Physics. Discipline of Physics.
1806:
1319:
1299:
1062:
1042:
771:
624:
525:
485:
298:
237:
212:
143:
2445:
2245:
1525:
3083:
2616:
1196:
834:
394:
175:
65:
2987:
2745:
2680:
1974:
1939:
1938:
Winkelmann, J.; Haffner, B.; Weaire, D.; Mughal, A.; Hutzler, S. (2017-07-31).
2022:
1913:
1859:
768:
plane. Each uniform structure is thus distinguished by its periodicity vector
605:
200:
151:
2922:
2818:
2487:
2430:
2412:
2373:
2350:"Confined Assemblies of Colloidal Particles with Soft Repulsive Interactions"
2316:
2253:
2190:
2135:
2088:
2030:
1921:
1824:
1715:
1608:
1533:
3026:
2479:
1998:
1889:
1791:"An experimental study of columnar crystals using monodisperse microbubbles"
1599:
258:
174:
Another plant that can be found in many gardens of residential areas is the
147:
2995:
2930:
2914:
2883:
2826:
2753:
2688:
2624:
2567:
2381:
2334:
2261:
2198:
2143:
2127:
2096:
1983:
1772:
1754:
1648:, Advances in Virus Research, vol. 20, Academic Press, pp. 1–32,
1626:
1204:
A further dynamic method to assemble such structures was introduced by Lee
194:
A further occurrence of ordered columnar arrangement on the macroscale are
30:
1541:
1462:
1087:, bcc, or hcp due to the free volume left by the cylindrical confinement.
949:
2365:
1671:
1383:
proportionality, the confining potential resembles that of a cylindrical
1181:
257:
Columnar structures have also been studied intensively in the context of
50:
2712:"Densest columnar structures of hard spheres from sequential deposition"
2462:
2810:
2421:
2349:
2308:
2283:
Lázaro, Guillermo R.; Dragnea, Bogdan; Hagan, Michael F. (2018-07-18).
1723:
559:
structures. For each structure that can be identified with the triplet
283:
Columnar arrangements are also investigated as a possible candidate of
246:
61:
2850:"Shape-Anisotropy-Induced Ordered Packings in Cylindrical Confinement"
2577:
2080:
1815:
1743:
Philosophical
Transactions of the Royal Society B: Biological Sciences
1393:
A comprehensive theory to this experiment was developed by
Winkelmann
265:, C70, or C78 into carbon nanotubes, but also boron nitride nanotubes
138:
analysed such arrangement of plant parts around the stem in his book "
72:, and so forth due to the analogous assembly of small particles (like
97:
46:
2159:"Ordered fullerene nanocylinders in large-diameter carbon nanotubes"
2056:
1707:
3033:
2970:
2793:
1956:
1739:"Tissue self-organization underlies morphogenesis of the notochord"
377:). The nanospheres then stick to the surface of the cylinders by a
2728:
2663:
1581:
1209:
185:
77:
29:
2285:"Self-assembly of convex particles on spherocylindrical surfaces"
1997:
Saadatfar, M.; Barry, J.; Weaire, D.; Hutzler, S. (2008-09-01).
1795:
Colloids and
Surfaces A: Physicochemical and Engineering Aspects
740:
the rolled-out contact pattern only varies by a rotation in the
195:
101:
3037:
2897:
Lee, Taehoon; Gizynski, Konrad; Grzybowski, Bartosz A. (2017).
2645:
Mughal, A.; Chan, H. K.; Weaire, D.; Hutzler, S. (2012-05-11).
1289:{\displaystyle E_{\text{rot}}={\frac {1}{2}}mR^{2}\omega ^{2},}
2591:
Pickett, Galen T.; Gross, Mark; Okuyama, Hiroko (2000-10-23).
597:, there exist a uniform structure and at least one line slip.
1454:
Columnar
Structures of Spheres: Fundamentals and Applications
547:
Types of ordered columnar structures without internal spheres
34:
Illustration of a columnar structure assembled by golf balls.
190:
Spherical soap bubbles confined in a cylindrical glass tube.
1999:"Ordered cylindrical foam structures with internal bubbles"
2954:"Theory of rotational columnar structures of soft spheres"
2569:
Structures of columnar packings with soft and hard spheres
1890:"Moving boundaries in ordered cylindrical foam structures"
45:
with the objective of packing a given number of identical
1846:
Weaire, D.; Hutzler, S.; Verbist, G.; Peters, E. (2007),
1502:"Tubular Packing of Spheres in Biological Fine Structure"
60:
These problems are studied extensively in the context of
2848:
Jin, Weiwei; Chan, Ho-Kei; Zhong, Zheng (2020-06-16).
1362:
1342:
1322:
1302:
1145:
1096:
1065:
1045:
1011:
953:
Optimal packing fraction for hard spheres of diameter
912:
870:
774:
746:
647:
627:
565:
528:
488:
456:
341:
301:
2647:"Dense packings of spheres in cylinders: Simulations"
1565:"Dislocation-mediated growth of bacterial cell walls"
1236:
979:
959:
794:
708:
674:
668:
For all uniform structures above a diameter ratio of
508:
406:
321:
1888:
Hutzler, S.; Weaire, D.; Crawford, R. (1997-06-01).
3230:
3209:
3193:
3140:
3092:
3071:
315:, much larger than the diameter of the nanospheres
215:for the wet limit where the bubbles are spherical.
1375:
1348:
1328:
1308:
1288:
1165:
1116:
1071:
1051:
1031:
985:
965:
926:
894:
818:
780:
760:
732:
694:
653:
633:
589:
534:
514:
494:
474:
442:
369:
327:
307:
854:characterised by these gaps or loss of contacts.
1854:, John Wiley & Sons, Ltd, pp. 315–374,
1569:Proceedings of the National Academy of Sciences
788:, which is defined by the phyllotactic triplet
1192:Columnar structures created by rapid rotations
295:solution together with a cylinder of diameter
3049:
2397:"The Magical World of Photonic Metamaterials"
1139:extended this work to higher diameter ratios
164:On smaller length scales, the berries of the
104:structures when confined in a glass tube. In
8:
1485:: CS1 maint: multiple names: authors list (
1451:Chan, Jens Winkelmann, Ho-Kei (2023-03-31).
3056:
3042:
3034:
385:Classification using phyllotactic notation
2969:
2873:
2792:
2727:
2662:
2593:"Spontaneous Chirality in Simple Systems"
2461:
2420:
2324:
1973:
1955:
1814:
1762:
1616:
1598:
1580:
1367:
1361:
1341:
1321:
1301:
1277:
1267:
1250:
1241:
1235:
1149:
1144:
1100:
1095:
1083:of an unconfined lattice packing such as
1064:
1044:
1015:
1010:
978:
958:
916:
911:
869:
793:
773:
750:
745:
707:
678:
673:
646:
626:
564:
527:
507:
487:
455:
405:
359:
345:
340:
320:
300:
2354:Journal of the American Chemical Society
1336:the distance from the central axis, and
1195:
948:
833:
604:
225:The first experimental observation of a
203:) or wet (spherical bubbles) structure.
110:
2777:"Hard sphere packings within cylinders"
2116:Angewandte Chemie International Edition
1457:. New York: Jenny Stanford Publishing.
1432:
370:{\textstyle D/d\approx 3{\text{ to }}5}
3022:"Packing spheres into a Thin Cylinder"
1563:Amir, A.; Nelson, D. R. (2012-06-19).
1478:
2640:
2638:
2636:
2634:
2561:
2559:
2557:
2555:
2553:
2446:"Optical cloaking with metamaterials"
2050:
2048:
1933:
1931:
7:
1784:
1782:
1646:Advances in Virus Research Volume 20
1446:
1444:
1442:
1440:
1438:
1436:
211:to investigate dry structure or the
389:The most common way of classifying
1642:"The Structure of Tubular Viruses"
945:Dense sphere packings in cylinders
25:
1356:the rotational speed. Due to the
27:Three-dimensional packing problem
3020:Becker, Aaron T. and Huang, L.
761:{\textstyle z{\text{-}}\theta }
3185:Sphere-packing (Hamming) bound
2875:10.1103/PhysRevLett.124.248002
2183:10.1088/0957-4484/19/04/045702
2003:Philosophical Magazine Letters
1807:10.1016/j.colsurfa.2014.12.020
1500:Erickson, R. O. (1973-08-24).
1416:Close-packing of equal spheres
973:inside a cylinder of diameter
889:
871:
813:
795:
727:
709:
584:
566:
437:
407:
1:
2511:"Self-assembled metamaterial"
1654:10.1016/s0065-3527(08)60500-x
393:columnar structures uses the
229:was discovered by Winkelmann
2246:10.1126/science.269.5226.966
1852:Advances in Chemical Physics
1690:Bryan, Joseph (1974-12-01).
1526:10.1126/science.181.4101.705
1005:up to the diameter ratio of
1001:studied such packings using
861:, but not termed line slip.
39:Sphere packing in a cylinder
2710:Chan, Ho-Kei (2011-11-14).
2617:10.1103/PhysRevLett.85.3652
2395:Ozbay, Ekmel (2008-11-01).
1848:"A Review of Foam Drainage"
443:{\displaystyle (l=m+n,m,n)}
121:form a columnar structure (
3295:
2988:10.1103/PhysRevE.99.020602
2746:10.1103/PhysRevE.84.050302
2681:10.1103/PhysRevE.85.051305
1975:10.1103/PhysRevE.97.059902
1640:Hull, Roger (1976-01-01),
1316:is the mass of the beads,
1117:{\textstyle D/d<2.7013}
695:{\displaystyle D/d>2.0}
475:{\textstyle l\geq m\geq n}
2566:Winkelmann, Jens (2020).
2401:Optics and Photonics News
2222:"Boron Nitride Nanotubes"
2023:10.1080/09500830802307658
1914:10.1080/13642819708205711
1860:10.1002/9780470141618.ch5
3110:isosceles right triangle
2413:10.1364/OPN.19.11.000022
1894:Philosophical Magazine B
233:in a system of bubbles.
2854:Physical Review Letters
2597:Physical Review Letters
2480:10.1038/nphoton.2007.28
1600:10.1073/pnas.1207105109
1401:structure transitions.
1186:Monte Carlo simulations
1166:{\textstyle D/d<4.0}
819:{\displaystyle (l,m,n)}
733:{\displaystyle (l,m,n)}
41:is a three-dimensional
18:Cylinder sphere packing
3124:Circle packing theorem
2915:10.1002/adma.201704274
2128:10.1002/anie.201403565
1755:10.1098/rstb.2017.0320
1377:
1350:
1330:
1310:
1290:
1219:then pushes the fluid
1201:
1167:
1118:
1073:
1053:
1039:for cylinder diameter
1033:
1032:{\textstyle D/d=2.873}
994:
987:
967:
928:
896:
843:
820:
782:
762:
734:
696:
655:
635:
614:
591:
536:
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496:
476:
444:
371:
329:
309:
191:
176:Australian bottlebrush
126:
82:crystalline structures
35:
2509:Manoharan, Vinothan.
1463:10.1201/9780429092114
1378:
1351:
1331:
1311:
1291:
1199:
1168:
1119:
1074:
1054:
1034:
988:
968:
952:
929:
897:
837:
821:
783:
763:
735:
697:
656:
636:
608:
592:
537:
517:
497:
477:
445:
395:phyllotactic notation
372:
330:
310:
285:optical metamaterials
189:
114:
92:Appearance in science
33:
3106:equilateral triangle
2366:10.1021/jacs.6b12975
1360:
1349:{\textstyle \omega }
1340:
1320:
1300:
1234:
1143:
1094:
1063:
1043:
1009:
977:
957:
910:
895:{\textstyle (l,m,n)}
868:
792:
772:
744:
706:
672:
654:{\textstyle \theta }
645:
641:and azimuthal angle
625:
590:{\textstyle (l,m,n)}
563:
526:
506:
486:
454:
404:
339:
319:
299:
3243:Slothouber–Graatsma
2980:2019PhRvE..99b0602W
2866:2020PhRvL.124x8002J
2803:2016SMat...12.2505F
2738:2011PhRvE..84e0302C
2673:2012PhRvE..85e1305M
2609:2000PhRvL..85.3652P
2537:"Tanjeem's website"
2472:2007NaPho...1..224C
2301:2018SMat...14.5728L
2238:1995Sci...269..966C
2175:2008Nanot..19d5702Y
2073:2005NanoL...5..349T
2015:2008PMagL..88..661S
1966:2017PhRvE..97e9902W
1906:1997PMagB..75..845H
1591:2012PNAS..109.9833A
1518:1973Sci...181..705E
1385:harmonic oscillator
1059:to sphere diameter
1003:simulated annealing
830:Line-slip structure
227:line-slip structure
115:The berries of the
80:) into cylindrical
55:columnar structures
2903:Advanced Materials
2811:10.1039/C5SM02875B
2309:10.1039/C8SM00129D
1749:(1759): 20170320.
1376:{\textstyle R^{2}}
1373:
1346:
1326:
1306:
1286:
1202:
1175:linear programming
1163:
1114:
1069:
1049:
1029:
995:
983:
963:
924:
892:
844:
816:
778:
758:
730:
692:
651:
631:
615:
587:
532:
512:
492:
472:
440:
367:
325:
305:
192:
140:On Growth and Form
127:
36:
3274:Discrete geometry
3251:
3250:
3210:Other 3-D packing
3194:Other 2-D packing
3119:Apollonian gasket
2958:Physical Review E
2716:Physical Review E
2651:Physical Review E
2603:(17): 3652–3655.
2535:Tanjeem, Nabila.
2360:(14): 5095–5101.
2295:(28): 5728–5740.
2232:(5226): 966–967.
2122:(34): 9090–9093.
2081:10.1021/nl047930r
2009:(9–10): 661–668.
1944:Physical Review E
1869:978-0-470-14161-8
1575:(25): 9833–9838.
1512:(4101): 705–716.
1472:978-0-429-09211-4
1258:
1244:
1229:rotational energy
1217:centripetal force
986:{\displaystyle D}
966:{\displaystyle d}
753:
663:hexagonal lattice
601:Uniform structure
515:{\displaystyle m}
362:
328:{\displaystyle d}
213:hard sphere model
70:materials science
16:(Redirected from
3286:
3264:Packing problems
3132:
3072:Abstract packing
3065:Packing problems
3058:
3051:
3044:
3035:
3008:
3007:
2973:
2949:
2943:
2942:
2894:
2888:
2887:
2877:
2845:
2839:
2838:
2796:
2787:(9): 2505–2514.
2772:
2766:
2765:
2731:
2707:
2701:
2700:
2666:
2642:
2629:
2628:
2588:
2582:
2581:
2563:
2548:
2547:
2545:
2543:
2532:
2526:
2525:
2523:
2521:
2506:
2500:
2499:
2465:
2450:Nature Photonics
2441:
2435:
2434:
2424:
2392:
2386:
2385:
2345:
2339:
2338:
2328:
2280:
2274:
2273:
2217:
2211:
2210:
2154:
2148:
2147:
2107:
2101:
2100:
2052:
2043:
2042:
1994:
1988:
1987:
1977:
1959:
1935:
1926:
1925:
1885:
1879:
1878:
1877:
1876:
1843:
1837:
1836:
1818:
1786:
1777:
1776:
1766:
1734:
1728:
1727:
1687:
1681:
1680:
1679:
1678:
1637:
1631:
1630:
1620:
1602:
1584:
1560:
1554:
1553:
1497:
1491:
1490:
1484:
1476:
1448:
1421:Packing problems
1382:
1380:
1379:
1374:
1372:
1371:
1355:
1353:
1352:
1347:
1335:
1333:
1332:
1327:
1315:
1313:
1312:
1307:
1295:
1293:
1292:
1287:
1282:
1281:
1272:
1271:
1259:
1251:
1246:
1245:
1242:
1172:
1170:
1169:
1164:
1153:
1123:
1121:
1120:
1115:
1104:
1078:
1076:
1075:
1070:
1058:
1056:
1055:
1050:
1038:
1036:
1035:
1030:
1019:
992:
990:
989:
984:
972:
970:
969:
964:
933:
931:
930:
927:{\textstyle D/d}
925:
920:
901:
899:
898:
893:
825:
823:
822:
817:
787:
785:
784:
779:
767:
765:
764:
759:
754:
751:
739:
737:
736:
731:
701:
699:
698:
693:
682:
660:
658:
657:
652:
640:
638:
637:
632:
596:
594:
593:
588:
541:
539:
538:
533:
521:
519:
518:
513:
501:
499:
498:
493:
481:
479:
478:
473:
449:
447:
446:
441:
376:
374:
373:
368:
363:
360:
349:
334:
332:
331:
326:
314:
312:
311:
306:
247:X-ray tomography
21:
3294:
3293:
3289:
3288:
3287:
3285:
3284:
3283:
3279:Crystallography
3254:
3253:
3252:
3247:
3226:
3205:
3189:
3136:
3130:
3129:Tammes problem
3088:
3067:
3062:
3017:
3012:
3011:
2951:
2950:
2946:
2909:(47): 1704274.
2896:
2895:
2891:
2847:
2846:
2842:
2774:
2773:
2769:
2709:
2708:
2704:
2644:
2643:
2632:
2590:
2589:
2585:
2565:
2564:
2551:
2541:
2539:
2534:
2533:
2529:
2519:
2517:
2508:
2507:
2503:
2463:physics/0611242
2443:
2442:
2438:
2394:
2393:
2389:
2347:
2346:
2342:
2282:
2281:
2277:
2219:
2218:
2214:
2156:
2155:
2151:
2109:
2108:
2104:
2054:
2053:
2046:
1996:
1995:
1991:
1937:
1936:
1929:
1887:
1886:
1882:
1874:
1872:
1870:
1845:
1844:
1840:
1788:
1787:
1780:
1736:
1735:
1731:
1708:10.2307/1297089
1702:(12): 701–711.
1689:
1688:
1684:
1676:
1674:
1664:
1639:
1638:
1634:
1562:
1561:
1557:
1499:
1498:
1494:
1477:
1473:
1450:
1449:
1434:
1429:
1407:
1363:
1358:
1357:
1338:
1337:
1318:
1317:
1298:
1297:
1273:
1263:
1237:
1232:
1231:
1194:
1141:
1140:
1092:
1091:
1061:
1060:
1041:
1040:
1007:
1006:
975:
974:
955:
954:
947:
908:
907:
866:
865:
832:
790:
789:
770:
769:
742:
741:
704:
703:
670:
669:
643:
642:
623:
622:
603:
561:
560:
549:
524:
523:
504:
503:
484:
483:
452:
451:
402:
401:
387:
379:depletion force
337:
336:
317:
316:
297:
296:
255:
209:Surface Evolver
184:
136:D'Arcy Thompson
132:
94:
43:packing problem
28:
23:
22:
15:
12:
11:
5:
3292:
3290:
3282:
3281:
3276:
3271:
3266:
3256:
3255:
3249:
3248:
3246:
3245:
3240:
3234:
3232:
3228:
3227:
3225:
3224:
3219:
3213:
3211:
3207:
3206:
3204:
3203:
3201:Square packing
3197:
3195:
3191:
3190:
3188:
3187:
3182:
3180:Kissing number
3177:
3172:
3167:
3162:
3157:
3152:
3146:
3144:
3142:Sphere packing
3138:
3137:
3135:
3134:
3126:
3121:
3116:
3098:
3096:
3094:Circle packing
3090:
3089:
3087:
3086:
3081:
3075:
3073:
3069:
3068:
3063:
3061:
3060:
3053:
3046:
3038:
3032:
3031:
3016:
3015:External links
3013:
3010:
3009:
2944:
2889:
2860:(24): 248002.
2840:
2767:
2702:
2630:
2583:
2549:
2527:
2501:
2456:(4): 224–227.
2436:
2387:
2340:
2275:
2212:
2163:Nanotechnology
2149:
2102:
2067:(2): 349–355.
2044:
1989:
1927:
1900:(6): 845–857.
1880:
1868:
1838:
1778:
1729:
1692:"Microtubules"
1682:
1662:
1632:
1555:
1492:
1471:
1431:
1430:
1428:
1425:
1424:
1423:
1418:
1413:
1411:Sphere packing
1406:
1403:
1370:
1366:
1345:
1329:{\textstyle R}
1325:
1309:{\textstyle m}
1305:
1285:
1280:
1276:
1270:
1266:
1262:
1257:
1254:
1249:
1240:
1223:and the beads
1193:
1190:
1162:
1159:
1156:
1152:
1148:
1113:
1110:
1107:
1103:
1099:
1072:{\textstyle d}
1068:
1052:{\textstyle D}
1048:
1028:
1025:
1022:
1018:
1014:
982:
962:
946:
943:
923:
919:
915:
891:
888:
885:
882:
879:
876:
873:
831:
828:
815:
812:
809:
806:
803:
800:
797:
781:{\textstyle V}
777:
757:
749:
729:
726:
723:
720:
717:
714:
711:
691:
688:
685:
681:
677:
650:
634:{\textstyle z}
630:
602:
599:
586:
583:
580:
577:
574:
571:
568:
548:
545:
535:{\textstyle n}
531:
511:
495:{\textstyle l}
491:
482:. Each number
471:
468:
465:
462:
459:
439:
436:
433:
430:
427:
424:
421:
418:
415:
412:
409:
386:
383:
366:
361: to
358:
355:
352:
348:
344:
324:
308:{\textstyle D}
304:
254:
251:
183:
180:
167:Arum maculatum
131:
128:
118:Arum maculatum
93:
90:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
3291:
3280:
3277:
3275:
3272:
3270:
3267:
3265:
3262:
3261:
3259:
3244:
3241:
3239:
3236:
3235:
3233:
3229:
3223:
3220:
3218:
3215:
3214:
3212:
3208:
3202:
3199:
3198:
3196:
3192:
3186:
3183:
3181:
3178:
3176:
3175:Close-packing
3173:
3171:
3170:In a cylinder
3168:
3166:
3163:
3161:
3158:
3156:
3153:
3151:
3148:
3147:
3145:
3143:
3139:
3133:
3127:
3125:
3122:
3120:
3117:
3115:
3111:
3107:
3103:
3100:
3099:
3097:
3095:
3091:
3085:
3082:
3080:
3077:
3076:
3074:
3070:
3066:
3059:
3054:
3052:
3047:
3045:
3040:
3039:
3036:
3029:
3028:
3023:
3019:
3018:
3014:
3005:
3001:
2997:
2993:
2989:
2985:
2981:
2977:
2972:
2967:
2964:(2): 020602.
2963:
2959:
2955:
2948:
2945:
2940:
2936:
2932:
2928:
2924:
2920:
2916:
2912:
2908:
2904:
2900:
2893:
2890:
2885:
2881:
2876:
2871:
2867:
2863:
2859:
2855:
2851:
2844:
2841:
2836:
2832:
2828:
2824:
2820:
2816:
2812:
2808:
2804:
2800:
2795:
2790:
2786:
2782:
2778:
2771:
2768:
2763:
2759:
2755:
2751:
2747:
2743:
2739:
2735:
2730:
2725:
2722:(5): 050302.
2721:
2717:
2713:
2706:
2703:
2698:
2694:
2690:
2686:
2682:
2678:
2674:
2670:
2665:
2660:
2657:(5): 051305.
2656:
2652:
2648:
2641:
2639:
2637:
2635:
2631:
2626:
2622:
2618:
2614:
2610:
2606:
2602:
2598:
2594:
2587:
2584:
2579:
2575:
2571:
2570:
2562:
2560:
2558:
2556:
2554:
2550:
2538:
2531:
2528:
2516:
2515:Manoharan lab
2512:
2505:
2502:
2497:
2493:
2489:
2485:
2481:
2477:
2473:
2469:
2464:
2459:
2455:
2451:
2447:
2440:
2437:
2432:
2428:
2423:
2418:
2414:
2410:
2407:(11): 22–27.
2406:
2402:
2398:
2391:
2388:
2383:
2379:
2375:
2371:
2367:
2363:
2359:
2355:
2351:
2344:
2341:
2336:
2332:
2327:
2322:
2318:
2314:
2310:
2306:
2302:
2298:
2294:
2290:
2286:
2279:
2276:
2271:
2267:
2263:
2259:
2255:
2251:
2247:
2243:
2239:
2235:
2231:
2227:
2223:
2216:
2213:
2208:
2204:
2200:
2196:
2192:
2188:
2184:
2180:
2176:
2172:
2169:(4): 045702.
2168:
2164:
2160:
2153:
2150:
2145:
2141:
2137:
2133:
2129:
2125:
2121:
2117:
2113:
2106:
2103:
2098:
2094:
2090:
2086:
2082:
2078:
2074:
2070:
2066:
2062:
2058:
2051:
2049:
2045:
2040:
2036:
2032:
2028:
2024:
2020:
2016:
2012:
2008:
2004:
2000:
1993:
1990:
1985:
1981:
1976:
1971:
1967:
1963:
1958:
1953:
1950:(5): 059902.
1949:
1945:
1941:
1934:
1932:
1928:
1923:
1919:
1915:
1911:
1907:
1903:
1899:
1895:
1891:
1884:
1881:
1871:
1865:
1861:
1857:
1853:
1849:
1842:
1839:
1834:
1830:
1826:
1822:
1817:
1812:
1808:
1804:
1800:
1796:
1792:
1785:
1783:
1779:
1774:
1770:
1765:
1760:
1756:
1752:
1748:
1744:
1740:
1733:
1730:
1725:
1721:
1717:
1713:
1709:
1705:
1701:
1697:
1693:
1686:
1683:
1673:
1669:
1665:
1663:9780120398201
1659:
1655:
1651:
1647:
1643:
1636:
1633:
1628:
1624:
1619:
1614:
1610:
1606:
1601:
1596:
1592:
1588:
1583:
1578:
1574:
1570:
1566:
1559:
1556:
1551:
1547:
1543:
1539:
1535:
1531:
1527:
1523:
1519:
1515:
1511:
1507:
1503:
1496:
1493:
1488:
1482:
1474:
1468:
1464:
1460:
1456:
1455:
1447:
1445:
1443:
1441:
1439:
1437:
1433:
1426:
1422:
1419:
1417:
1414:
1412:
1409:
1408:
1404:
1402:
1400:
1396:
1391:
1388:
1386:
1368:
1364:
1343:
1323:
1303:
1283:
1278:
1274:
1268:
1264:
1260:
1255:
1252:
1247:
1238:
1230:
1226:
1222:
1218:
1213:
1211:
1207:
1198:
1191:
1189:
1187:
1183:
1178:
1176:
1160:
1157:
1154:
1150:
1146:
1138:
1133:
1130:
1125:
1111:
1108:
1105:
1101:
1097:
1088:
1086:
1080:
1066:
1046:
1026:
1023:
1020:
1016:
1012:
1004:
1000:
980:
960:
951:
944:
942:
940:
935:
921:
917:
913:
903:
886:
883:
880:
877:
874:
862:
860:
855:
851:
847:
841:
836:
829:
827:
810:
807:
804:
801:
798:
775:
755:
747:
724:
721:
718:
715:
712:
689:
686:
683:
679:
675:
666:
664:
648:
628:
619:
612:
607:
600:
598:
581:
578:
575:
572:
569:
558:
554:
546:
544:
529:
509:
489:
469:
466:
463:
460:
457:
434:
431:
428:
425:
422:
419:
416:
413:
410:
398:
396:
392:
384:
382:
380:
364:
356:
353:
350:
346:
342:
322:
302:
294:
290:
286:
281:
278:
273:
271:
266:
264:
260:
252:
250:
248:
243:
239:
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3160:In a sphere
3131:(on sphere)
3102:In a circle
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2422:11693/23249
2289:Soft Matter
1399:peritectoid
937:Winkelmann
838:An example
609:An example
253:Nanoscience
238:dodecahedra
201:polyhedrons
106:nanoscience
66:nanoscience
3258:Categories
3150:Apollonian
2971:1808.02952
2794:1511.08472
2578:2262/91733
1957:1703.00773
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