The densest packing of equal circles on a sphere

Kottwitz, D.A.

doi:10.1107/s0108767390011370

Cited by 67 publications

(58 citation statements)

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“…The densest packings found tend be concentrated at low P, for n < 6. The large-n results suggest a convergence to a limiting packing fraction of Φ ≃ 0.8960, not unlike the large-n convergence of area fraction observed in numerical studies of disc packing on spheres (30). The maximal core density is found for a ð5; 5Þ packing, for which Φ ð5;5Þ ¼ 0.89615.…”

Section: Asymptotically Close-packed Filament Bundlesmentioning

confidence: 57%

Non-Euclidean geometry of twisted filament bundle packing

Bruss

Grason

2012

Proc. Natl. Acad. Sci. U.S.A.

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Densely packed and twisted assemblies of filaments are crucial structural motifs in macroscopic materials (cables, ropes, and textiles) as well as synthetic and biological nanomaterials (fibrous proteins). We study the unique and nontrivial packing geometry of this universal material design from two perspectives. First, we show that the problem of twisted bundle packing can be mapped exactly onto the problem of disc packing on a curved surface, the geometry of which has a positive, spherical curvature close to the center of rotation and approaches the intrinsically flat geometry of a cylinder far from the bundle center. From this mapping, we find the packing of any twisted bundle is geometrically frustrated, as it makes the sixfold geometry of filament close packing impossible at the core of the fiber. This geometrical equivalence leads to a spectrum of close-packed fiber geometries, whose low symmetry (five-, four-, three-, and twofold) reflect non-Euclidean packing constraints at the bundle core. Second, we explore the ground-state structure of twisted filament assemblies formed under the influence of adhesive interactions by a computational model. Here, we find that the underlying non-Euclidean geometry of twisted fiber packing disrupts the regular lattice packing of filaments above a critical radius, proportional to the helical pitch. Above this critical radius, the ground-state packing includes the presence of between one and six excess fivefold disclinations in the cross-sectional order.self-assembly | topological defects | geometric frustration P acking problems arise naturally in a multitude of contexts, from models of crystalline and amorphous solids to structure formation in tissues of living organisms. Though often easy to state, packing problems are notoriously difficult to solve. The cannonball stacking of spheres, for example, conjectured by Kepler to be the densest packing in three-dimensional space (1), was only proven so in 1999, and even then with the aid of a computer algorithm (2). In some cases, like the densest packing of spheres, the face-centered cubic lattice-or its two-dimensional analogue, the densest packing of discs, the hexagonal lattice-the optimal structure is a regular, periodic partition of space. In many problems, however, perfect lattice packings are prohibited, and optimal structures necessarily lack the translational and rotational symmetry of periodic order. A well-known example is the problem of finding the densest packing of N discs on a sphere of a given radius, known alternately as the Tammes or the generalized-Thomson problem (3). In this problem, spherical topology requires a variation in the local packing symmetry: All states possess at minimum 12 discs whose nearest-neighbor geometry is fivefold coordinated (4). Beyond its relevance to structural studies of such diverse materials as spherical viral capsids (5) and colloid-stabilized emulsions (6, 7), the problems involving point or disc packing on spheres serve as an important example of systems where topological de...

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Section: Asymptotically Close-packed Filament Bundlesmentioning

confidence: 57%

Non-Euclidean geometry of twisted filament bundle packing

Bruss

Grason

2012

Proc. Natl. Acad. Sci. U.S.A.

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“…The set of conjectured optimal spherical codes displays a wide variety of point groups (95,96), and it is thus unsurprising that the layers of the sphere, icosahedron, and dodecahedron clusters also have a variety of symmetries as N varies, at least when they match optimal spherical codes. Within the set of sphere, icosahedron, and dodecahedron clusters, a majority of clusters have nontriclinic point groups, and these point groups are spread widely across 10 crystal systems.…”

Section: Resultsmentioning

confidence: 99%

Clusters of polyhedra in spherical confinement

Teich

Anders

Klotsa

et al. 2016

Proc. Natl. Acad. Sci. U.S.A.

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Dense particle packing in a confining volume remains a rich, largely unexplored problem, despite applications in blood clotting, plasmonics, industrial packaging and transport, colloidal molecule design, and information storage. Here, we report densest found clusters of the Platonic solids in spherical confinement, for up to N = 60 constituent polyhedral particles. We examine the interplay between anisotropic particle shape and isotropic 3D confinement. Densest clusters exhibit a wide variety of symmetry point groups and form in up to three layers at higher N. For many N values, icosahedra and dodecahedra form clusters that resemble sphere clusters. These common structures are layers of optimal spherical codes in most cases, a surprising fact given the significant faceting of the icosahedron and dodecahedron. We also investigate cluster density as a function of N for each particle shape. We find that, in contrast to what happens in bulk, polyhedra often pack less densely than spheres. We also find especially dense clusters at so-called magic numbers of constituent particles. Our results showcase the structural diversity and experimental utility of families of solutions to the packing in confinement problem.clusters | confinement | packing | colloids | nanoparticles P henomena as diverse as crowding in the cell (1, 2), DNA packaging in cell nuclei and virus capsids (3, 4), the growth of cellular aggregates (5), biological pattern formation (6), blood clotting (7), efficient manufacturing and transport, the planning and design of cellular networks (8), and efficient food and pharmaceutical packaging and transport (9) are related to the optimization problem of packing objects of a specified shape as densely as possible within a confining geometry, or packing in confinement. Packing in confinement is also a laboratory technique used to produce particle aggregates with consistent structure. These aggregates may serve as building blocks (or "colloidal molecules") in hierarchical structures (10, 11), information storage units (12), or drug delivery capsules (13). Experiments concerning cluster formation via spherical droplet confinement (13-20) are of special interest here. Droplets are typically either oil-in-water or water-inoil emulsions, and particle aggregation is induced via the evaporation of the droplet solvent. Clusters may be hollow [in which case they are termed "colloidosomes" (13)] or filled, depending on the formation protocol, and may contain a few (15) to a few billion (14) particles. Clusters of several metallic nanoparticles are especially intriguing given their ability to support surface plasmon modes over a range of frequencies (21). The subwavelength scale of these clusters means that their optical response is highly dependent on their specific geometry (22). Consequently, control over their structure enables control over their optical properties, with implications for cloaking (23), chemical sensing (24), imaging (25), nonlinear optics (26), and the creation of so-called metafluids (27)(28)(29), a...

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“…The packing density varies with the number of circles (pinholes), but as the number of pinholes grows this ratio converges to the best packing density of equal circles on R 2 , which is π/ √ 12 ≈ 0.9069, [4]. Combining Equation (6) and (7) and isolating n leads to…”

Section: Multi-pinhole Geometrymentioning

confidence: 99%

Theoretical bounds and optimal configurations for multi-pinhole SPECT

Nillius

Danielsson

2008

2008 IEEE Nuclear Science Symposium Conference Record

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Abstract-The pinhole geometry in SPECT has an inherent trade-off between resolution and sensitivity. High resolution requires a small aperture which on the other hand directly reduces the rate of detected photons. Recent systems overcome this to some extent by using multiple pinholes spread out around the imaging object, effectively increasing the sensitivity with a factor equal to the number of pinholes. The images of each pinhole must fit on the detector without overlap. This creates another trade-off between resolution, sensitivity and the field-ofview (FOV) of the system.The present work analytically analyzes the properties of the multi-pinhole SPECT geometry. Optimal configurations are identified and characterized. One of the main results is that there exists a theoretical upper bound for the sensitivity given the resolution and the FOV. This upper bound is proportional to the square of the resolution and inversely proportional to the square FOV diameter. This means that if we want to improve the resolution by a factor of ten, the sensitivity will go down a factor 100, unless we decrease the FOV an equal amount. The bound can not be broken even if the detector sphere is infinitely large.One important parameter when designing a system is how close it will be to its theoretical bound. The closer to the bound the more extreme the system will be, in terms of size and number of pinholes. A moderate distance away from the bound the proportions of the system are more realistic.

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The densest packing of equal circles on a sphere

Cited by 67 publications

References 0 publications

Non-Euclidean geometry of twisted filament bundle packing

Non-Euclidean geometry of twisted filament bundle packing

Clusters of polyhedra in spherical confinement

Theoretical bounds and optimal configurations for multi-pinhole SPECT

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