A new counter-example to Kelvin's conjecture on minimal surfaces

Gabbrielli, Ruggero

doi:10.1080/09500830903022651

Cited by 37 publications

(26 citation statements)

References 21 publications

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“…The observation that the truncated octahedron is a "large" local maximum for the isoperimetric ratio in 3D for space-filling tessellations suggests a weak re-formulation of the Kelvin conjecture of global optimality of the truncated octahedron, which has been proved false [55,56]. indicates that large cells preferentially feature large isoperimetric quotients (the bigger, the bulkier).…”

Section: Discussionmentioning

confidence: 99%

“…In both cases, the mean area increases quadratically (with very similar coefficient) for small values of  , which shows that the Voronoi tessellations of the BCC and FCC cubic crystals are local minima for the mean surface in the set of space-filling tessellations. We know that neither the truncated octahedron nor the rhombic dodecahedron are global minima, since (at least) the WeairePhelan [55] and the Gabbrielli [56] structures have a smaller surface. It is reasonable to expect that a similar quadratic increase of the average surface should be observed when perturbing with spatial gaussian noise the crystalline structure corresponding to the Weaire-Phelan and Gabbrielli cells.…”

Section:  mentioning

confidence: 99%

“…The FCC (together with the Hexagonal Close Packed structure) features the largest possible packing fraction: the related 1611 Kepler's conjecture has been recently proved by [54] The Voronoi cell of BCC has been conjectured by Kelvin in 1887 as being the space-filling cell with the smallest surface to volume ratio, and only recently two counter-examples have been given in [55] and [56]. Interestingly, Gabbrielli [56] found a novel space-filling cell by imaginatively looking at the patterns generated as stationary solution of the 3D Swift-Hohenberg equation, thus methodologically mimicking the more usual 2D case. Finally, the truncated octahedron is conjectured to have the lowest cost among all 3D spacefilling cells [51].…”

Section: Introductionmentioning

confidence: 99%

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Symmetry-Break in Voronoi Tessellations

Lucarini

2009

Symmetry

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Abstract:We analyse in a common framework the properties of the Voronoi tessellations resulting from regular 2D and 3D crystals and those of tessellations generated by Poisson distributions of points, thus joining on symmetry breaking processes and the approach to uniform random distributions of seeds. We perturb crystalline structures in 2D and 3D with a spatial Gaussian noise whose adimensional strength is α and analyse the statistical properties of the cells of the resulting Voronoi tessellations using an ensemble approach. In 2D we consider triangular, square and hexagonal regular lattices, resulting into hexagonal, square and triangular tessellations, respectively. In 3D we consider the simple cubic (SC), body-centred cubic (BCC), and face-centred cubic (FCC) crystals, whose corresponding Voronoi cells are the cube, the truncated octahedron, and the rhombic dodecahedron, respectively. In 2D, for all values α>0, hexagons constitute the most common class of cells. Noise destroys the triangular and square tessellations, which are structurally unstable, as their topological properties are discontinuous in α=0. On the contrary, the honeycomb hexagonal tessellation is topologically stable and, experimentally, all Voronoi cells are hexagonal for small but finite noise with α<0.12. Basically, the same happens in the 3D case, where only the tessellation of the BCC crystal is topologically stable even against noise of small but finite intensity. In both 2D and 3D cases, already for a moderate amount of Gaussian noise (α>0.5), memory of the specific initial unperturbed state is lost, because the statistical properties of the three perturbed regular tessellations are indistinguishable. When α>2, results converge to those of Poisson-Voronoi tessellations. In 2D, while the isoperimetric ratio increases with noise for the perturbed hexagonal tessellation, for the OPEN ACCESSSymmetry 2009, 1 22 perturbed triangular and square tessellations it is optimised for specific value of noise intensity. The same applies in 3D, where noise degrades the isoperimetric ratio for perturbed FCC and BCC lattices, whereas the opposite holds for perturbed SCC lattices. This allows for formulating a weaker form of the Kelvin conjecture. By analysing jointly the statistical properties of the area and of the volume of the cells, we discover that also the cells shape heavily fluctuates when noise is introduced in the system. In 2D, the geometrical properties of n-sided cells change with α until the Poisson-Voronoi limit is reached for α>2; in this limit the Desch law for perimeters is shown to be not valid and a square root dependence on n is established, which agrees with exact asymptotic results. Anomalous scaling relations are observed between the perimeter and the area in the 2D and between the areas and the volumes of the cells in 3D: except for the hexagonal (2D) and FCC structure (3D), this applies also for infinitesimal noise. In the Poisson-Voronoi limit, the anomalous exponent is about 0.17 in both the 2D and 3D case. A positive anomal...

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Section: Discussionmentioning

confidence: 99%

Section:  mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Symmetry-Break in Voronoi Tessellations

Lucarini

2009

Symmetry

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“…The latest solution of the minimization problem of unit cell surface energy, where unit cells have planar faces (the bubbles are in contact), was presented by Gabbrielli [19]. Apart from that, and aiming to analyze the foam behavior theoretically, many other idealized structural models were proposed.…”

Section: Model Of the Structurementioning

confidence: 99%

Limit load and failure mechanisms of soda lime glass foam

et al. 2018

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Foamed glass is widely used in the industry as an insulating material. However, its mechanical properties are not well-investigated yet. Foamed glass is produced from glass waste that causes discrepancy in mechanical properties of the final product. This paper shows a way to increase the limit of the load capacity of foamed glass, which is very fragile and sensitive to mechanical and thermal loading conditions. In this paper, three different methods of load application on cellular glass structure (rough contact, resin and flour interfaces) and their influence on failure mechanisms were investigated in detail. The results of numerical analyses, based on finite elements method and compression strength tests using the digital image correlation method, indicate that the overall strength of the material is limited by boundary effects. A careful adjustment of the interface property is the main factor to draw useful conclusions and to extend load limits of cellular glass in engineering applications.

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“…Certain periodic polyhedral tilings are intimately connected to lattices (2-5) and crystal states of matter (6), and can provide efficient meshings of space for numerical applications (e.g., quadrature and discretizing partial differential equations) (7). Polyhedral tilings arise in the structure of foams and Kelvin's problem (3,8,9,10). Remarkably, crystalline forms of DNA can be generated by using specifically constructed mathematical tiling analogs (11).…”

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confidence: 99%

New family of tilings of three-dimensional Euclidean space by tetrahedra and octahedra

Conway

Jiao

Torquato

2011

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

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It is well known that two regular tetrahedra can be combined with a single regular octahedron to tile (complete fill) three-dimensional Euclidean space R 3 . This structure was called the "octet truss" by Buckminster Fuller. It was believed that such a tiling, which is the Delaunay tessellation of the face-centered cubic (fcc) lattice, and its closely related stacking variants, are the only tessellations of R 3 that involve two different regular polyhedra. Here we identify and analyze a unique family comprised of a noncountably infinite number of periodic tilings of R 3 whose smallest repeat tiling unit consists of one regular octahedron and six smaller regular tetrahedra. We first derive an extreme member of this unique tiling family by showing that the "holes" in the optimal lattice packing of octahedra, obtained by Minkowski over a century ago, are congruent tetrahedra. This tiling has 694 distinct concave (i.e., nonconvex) repeat units, 24 of which possess central symmetry, and hence is distinctly different and combinatorically richer than the fcc tetrahedra-octahedra tiling, which only has two distinct tiling units. Then we construct a one-parameter family of octahedron packings that continuously spans from the fcc to the optimal lattice packing of octahedra. We show that the "holes" in these packings, except for the two extreme cases, are tetrahedra of two sizes, leading to a family of periodic tilings with units composed four small tetrahedra and two large tetrahedra that contact an octahedron. These tilings generally possess 2,068 distinct concave tiling units, 62 of which are centrally symmetric.space-filling | nonoverlapping solids | polytopes T ilings have intrigued artists, architects, scientists, and mathematicians for millenia (1). A "tiling" or "tessellation" is a partition of Euclidean space R d into closed regions whose interiors are disjoint. Tilings of space by polyhedra are of particular interest. Certain periodic polyhedral tilings are intimately connected to lattices (2-5) and crystal states of matter (6), and can provide efficient meshings of space for numerical applications (e.g., quadrature and discretizing partial differential equations) (7). Polyhedral tilings arise in the structure of foams and Kelvin's problem (3,8,9,10). Remarkably, crystalline forms of DNA can be generated by using specifically constructed mathematical tiling analogs (11). Some aperiodic tilings of Euclidean space underlie quasicrystals (12, 13), which possess forbidden crystallographic symmetries, and glassy states of matter (14). Tilings of high-dimensional Euclidean space also have important applications in communications, cryptography, information theory, and in the search for gravitational waves (2, 15).Tilings of two-dimensional (2D) Euclidean space R 2 by regular polygons have been widely used since antiquity. Kepler was the first to provide a systematic mathematical treatment of 2D tiling problems in his book entitled Harmonices Mundi (16). Kepler showed that there are just 11 uniform tilings of R 2 with re...

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A new counter-example to Kelvin's conjecture on minimal surfaces

Cited by 37 publications

References 21 publications

Symmetry-Break in Voronoi Tessellations

Symmetry-Break in Voronoi Tessellations

Limit load and failure mechanisms of soda lime glass foam

New family of tilings of three-dimensional Euclidean space by tetrahedra and octahedra

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