## 70 Citations

New dense superball packings in three dimensions

- Mathematics
- 2018

Abstract We construct a new family of lattice packings for superballs in three dimensions (unit balls for the l3p $\begin{array}{} \displaystyle l^p_3 \end{array}$ norm) with p ∈ (1, 1.58]. We…

Lattice packing and covering of convex bodies

- Mathematics
- 2011

The aim of this article is twofold. First, to indicate briefly major problems and developments dealing with lattice packings and coverings of balls and convex bodies. Second, to survey more recent…

On lattice coverings by simplices

- Computer Science
- 2015

This paper presents the first nontrivial lower bound for the lattice covering density by n-dimensional simplices, which is based on the volumes of generalized difference bodies.

A Dense Packing of Regular Tetrahedra

- Mathematics, Materials ScienceDiscret. Comput. Geom.
- 2008

It is shown that a dense packing of regular tetrahedra, with packing density D>.7786157, can be constructed using standard packing principles.

Mathematical modeling of the interaction of non-oriented convex polytopes

- Mathematics
- 2012

An Φ-function for two non-oriented convex polytopes is set up. The Φ-function can be used to construct a mathematical model of packing optimization problem for non-oriented polytopes. An example of…

Dense regular packings of irregular nonconvex particles.

- MathematicsPhysical review letters
- 2011

A new numerical scheme to study systems of nonconvex, irregular, and punctured particles in an efficient manner is presented and it is proved that the densest packing is obtained for both rhombiuboctahedra and rhombic enneacontrahedra.

Packing, tiling, and covering with tetrahedra.

- MathematicsProceedings of the National Academy of Sciences of the United States of America
- 2006

The results suggest that the regular tetrahedron may not be able to pack as densely as the sphere, which would contradict a conjecture of Ulam.

A Note on Lattice Packings via Lattice Refinements

- Mathematics, Computer ScienceExp. Math.
- 2018

A simple o(nn/2) running time algorithm that refines successively the packing lattice Dn (checkboard lattice) of the unit ball Bn and terminates with packing lattices achieving the best-known lattice densities.

Packing Cones and Their Negatives in Space

- MathematicsDiscret. Comput. Geom.
- 2007

An explicit constant c > 0 such that the density of any such C is smaller than 1 - c, answering a question of Wlodek Kuperberg.

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- Mathematics
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- Mathematics
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Packing and Covering the Plane with Translates of a Convex Polygon

- Mathematics, Computer ScienceJ. Algorithms
- 1990

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- MathematicsNature
- 1947

THE meeting of the London Mathematical Society on December 19, 1946, took the form of a symposium on the geometry of numbers, arranged by Prof. H. Davenport. Prof. E. C. Titchmarsh, president of the…

Convex bodies which tile space by translation

- Mathematics
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It is shown that a convex body K tiles E d by translation if, and only if, K is a centrally symmetric d -polytope with centrally symmetric facets, such that every belt of K (consisting of those of…

Lattice Octahedra

- MathematicsCanadian Journal of Mathematics
- 1960

Let Ai, A2 , … , An be n linearly independent points in n-dimensional Euclidean space of a lattice Λ. The points ± A1, ±A2 , . . , ±An define a closed n-dimensional octahedron (or “cross poly tope“)…

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- Mathematics
- 1994

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On the Frustrum of a Sphere

- Mathematics
- 1950

(1) K: f(x,y,z) _ 1, which is symmetrical about 0, defines a convex body K say, in a three-dimensional Euclidean space. Let A denote any lattice of points admissible for K, so that all points of A,…