Densest lattice packings of 3-polytopes

  title={Densest lattice packings of 3-polytopes},
  author={Ulrich Betke and Martin Henk},
  journal={Comput. Geom.},
New dense superball packings in three dimensions
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
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
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
  • E. R. Chen
  • Mathematics, Materials Science
    Discret. 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
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.
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.
  • J. Conway, S. Torquato
  • Mathematics
    Proceedings 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
  • M. Henk
  • Mathematics, Computer Science
    Exp. 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
An explicit constant c > 0 such that the density of any such C is smaller than 1 - c, answering a question of Wlodek Kuperberg.


An optimal convex hull algorithm in any fixed dimension
A deterministic algorithm for computing the convex hull of n points inEd in optimalO(n logn+n⌞d/2⌟) time and a by-product of this result is an algorithm for Computing the Voronoi diagram ofn points ind-space in optimal O(nLogn+ n⌜d/ 2⌝) time.
On the densest packing of sections of a cube (1)
Sunto.Determinazione dei reticolati critici relativi alla porzione di un cubo compresa fra due piani perpendicolari ad una diagonale e simmetrici rispetto al centro.
The problem of finding the densest packing of tetrahedra was first suggested by Hubert [3, p. 319]. Minkowski [4] attempted to find the densest lattice packing of tetrahedra, but his result is
The Geometry of Numbers
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
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
  • L. Mordell
  • Mathematics
    Canadian 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“)
Lectures on Polytopes
Based on a graduate course given at the Technische Universitat, Berlin, these lectures present a wealth of material on the modern theory of convex polytopes. The clear and straightforward
On the Frustrum of a Sphere
(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,