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Chuanming Zong

Publications and source records attributed to Chuanming Zong.

31 records · Page 2Linked to original sources
arXiv2018

Characterization of the Two-Dimensional Five-Fold Lattice Tiles

In 1885, Fedorov discovered that a convex domain can form a lattice tiling of the Euclidean plane if and only if it is a parallelogram or a centrally symmetric hexagon. It is known that there is no other convex domain which can form a two-, three- or four-fold lattice tiling in the Euclidean plane, but there is a centrally symmetric convex decagon which can form a five-fold lattice tiling. This paper characterizes all the convex domains which can form a five-fold lattice tiling of the Euclidean plane.

math.MG↗
arXiv2018

Characterization of the Two-Dimensional Five-Fold Translative Tiles

In 1885, Fedorov discovered that a convex domain can form a lattice tiling of the Euclidean plane if and only if it is a parallelogram or a centrally symmetric hexagon. It is known that there is no other convex domain which can form two-, three- or four-fold translative tiling in the Euclidean plane, but there are centrally symmetric convex octagons and decagons which can form five-fold translative tilings. This paper characterizes all the convex domains which can form five-fold translative tilings of the Euclidean plane.

math.MG↗
arXiv2018

Multiple Lattice Tilings in Euclidean Spaces

This paper proves the following results: Besides parallelograms and centrally symmetric hexagons, there is no other convex domain which can form a two-, three- or four-fold lattice tiling in the Euclidean plane. If a centrally symmetric octagon can form a lattice multiple tiling, then the multiplicity is at least seven. However, there are decagons which can form five-fold $($or six-fold$)$ lattice tilings. Consequently, whenever $n\ge 3$, there are non-parallelohedral polytopes which can form five-fold lattice tilings in the $n$-dimensional Euclidean space.

math.MG↗
arXiv2015

On Lattice Coverings by Simplices

By studying the volume of a generalized difference body, this paper presents the first nontrivial lower bound for the lattice covering density by $n$-dimensional simplices.

math.MG↗
arXiv2014

A Mathematical Theory for Random Solid Packings

Packings of identical objects have fascinated both scientists and laymen alike for centuries, in particular the sphere packings and the packings of identical regular tetrahedra. Mathematicians have tried for centuries to determine the densest packings; Crystallographers and chemists have been fascinated by the lattice packings for centuries as well. On the other hand, physicists, geologists, material scientists and engineers have been challenged by the mysterious random packings for decades. Experiments have shown the existence of the dense random sphere packings and the loose random sphere packings for more than half a century. However, a rigorous definition for them is still missing. The purpose of this paper is to review the random solid packings and to create a mathematical theory for it.

math.MG↗
arXiv2014

Functionals on the Spaces of n-Dimensional Convex Bodies

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. This article suggests an ideal theoretic structure and a couple of research topics such as supderivatives and integral sums of these functionals which seem to be important. In addition, it proves an inequality between the Hausdorff metric and the Banach-Mazur metric and obtains some estimations on the supderivatives.

math.MG↗
arXiv2012

On Arnold's Problem on the Classifications of Convex Lattice Polytopes

In 1980, V.I. Arnold studied the classification problem for convex lattice polygons of given area. Since then this problem and its analogues have been studied by B'ar'any, Pach, Vershik, Liu, Zong and others. Upper bounds for the numbers of non-equivalent ddimensional convex lattice polytopes of given volume or cardinality have been achieved. In this paper, by introducing and studying the unimodular groups acting on convex lattice polytopes, we obtain lower bounds for the number of non-equivalent d-dimensional convex lattice polytopes of bounded volume or given cardinality, which are essentially tight.

math.MG↗