Mysteries in Packing Regular Tetrahedra

Lagarias, Jeffrey C.; Zong, Chuanming

doi:10.1090/noti918

Cited by 34 publications

(27 citation statements)

References 30 publications

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“…The kissing problem has been generalized to tetrahedra [22], but we are not aware of results for ellipsoidal particles. Motivated by recent work on the packing properties of disordered granular ellipsoids [12][13][14][15], we consider a modified kissing problem for uniaxial ellipsoids 1∶1∶α, also known as spheroids, or ellipsoids of rotation.…”

Section: Introductionmentioning

confidence: 99%

Densest Local Structures of Uniaxial Ellipsoids

2016

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Connecting the collective behavior of disordered systems with local structure on the particle scale is an important challenge, for example, in granular and glassy systems. Compounding complexity, in many scientific and industrial applications, particles are polydisperse, aspherical, or even of varying shape. Here, we investigate a generalization of the classical kissing problem in order to understand the local building blocks of packings of aspherical grains. We numerically determine the densest local structures of uniaxial ellipsoids by minimizing the Set Voronoi cell volume around a given particle. Depending on the particle aspect ratio, different local structures are observed and classified by symmetry and Voronoi coordination number. In extended disordered packings of frictionless particles, knowledge of the densest structures allows us to rescale the Voronoi volume distributions onto the single-parameter family of k-Gamma distributions. Moreover, we find that approximate icosahedral clusters are found in random packings, while the optimal local structures for more aspherical particles are not formed.

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Section: Introductionmentioning

confidence: 99%

Densest Local Structures of Uniaxial Ellipsoids

2016

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“…Then, by (13), one can deduce that ρ rC + X is a covering of E n , ρC + X is a packing in E n , and therefore, by (14),…”

Section: Remarkmentioning

confidence: 96%

“…On the other hand, both δ c (K) and θ c (K) are no longer invariants under affine linear transformations (see [13]). Therefore, to study δ c (K) and θ c (K), we have to work in {K n , · H }.…”

Section: Definition 2 We Say a Family Of Ballsmentioning

confidence: 99%

Functionals on the spaces of convex bodies

Zong

2014

Acta. Math. Sin.-English Ser.

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In geometry, there are several challenging problems studying numbers associated to convex bodies. For example, the packing density problem, the kissing number problem, the covering density problem, the packing-covering constant problem, Hadwiger's covering conjecture and Borsuk's partition conjecture. They are fundamental and fascinating problems about the same objects. However, up to now, both the methodology and the technique applied to them are essentially different. Therefore, a common foundation for them has been much expected. By treating problems of these types as functionals defined on the spaces of n-dimensional convex bodies, this paper tries to create such a foundation. In particular, supderivatives for these functionals will be studied.

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“…Several long-standing conjectures on the mathematics of space-tiling and packing, such as the 3D Kepler conjecture, were solved only in the last decade, while many important questions remain open. 13,33,38 Similarly, for optimal simplicial meshing, while the two-dimensional optimal triangle is often associated with the regular triangle, the question becomes subtle in three dimension as regular tetrahedra fail to be space-filling. Studies in Ref.…”

Section: Introductionmentioning

confidence: 99%

“…This is a simple fact, but several renowned scholars have made mistakes, and many questions about it remain unsolved. 38 The earliest and perhaps also the most famous mistake has been attributed to Aristotle 3 which created a controversy for nearly 2000 years. The mystery surrounding this kind of puzzle has also partly contributed to the study of Hilbert's 18th problem.…”

Section: Introductionmentioning

confidence: 99%

Optimization of Subdivision Invariant Tetrahedra

Liu

2015

Int. J. Comput. Geom. Appl.

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While two-dimensional triangles are always subdivision invariant, the same does not always hold for their three-dimensional counterparts. We consider several interesting properties of those three-dimensional tetrahedra which are subdivision invariant and offer them a new classification. Moreover, we study the optimization of these tetrahedra, arguing that the second Sommerville tetrahedra are the closest to being regular and are optimal by many measures. Anisotropic subdivision invariant tetrahedra with high aspect ratios are characterized. Potential implications and applications of our findings are also discussed.

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Mysteries in Packing Regular Tetrahedra

Cited by 34 publications

References 30 publications

Densest Local Structures of Uniaxial Ellipsoids

Densest Local Structures of Uniaxial Ellipsoids

Functionals on the spaces of convex bodies

Optimization of Subdivision Invariant Tetrahedra

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