Lattice Packing in R^2 Pubblico
Li, Shuo (2017)
Abstract
The sphere packing problem has a long history. A sphere packing problem refers to the problem of finding arrangements of equal-sized nonoverlapping spheres that can fill a given space with maximized density. In the 17th century, German mathematician, Johannes Kepler proposed the so-called Kepler's Conjecture which is about the sphere packing problem in three-dimensional Euclidean space. This mathematical conjecture bothered mathematicians for more than 400 years.
Moreover, since every dimension has its own version of sphere packing problem, this classic problem will continue to be a hot topic for mathematicians. In this paper, we will first briefly introduce the sphere packing problem in dimension 2 and 3. Since we can visualize these packings, our intuition can help us to understand the problem better. Then we will discuss the recent breakthrough in some higher dimensions such as dimension 8 and 24. Then we will focus on sphere packings in dimension 2 and we will give an exposition of the theorem that the best lattice packing in dimension 2 is given by hexagonal lattice. This theorem is due to Axel Thue.
To achieve our main goal in this paper, we will use the definitions of well-rounded lattice and successive minima. Moreover, we will break down the proof of main theorem into several parts. In other words, we will prove some preliminary lemmas before proceeding to the main proof.
Table of Contents
1 Introduction ................................. 1
1.1 Introduction ................................. 1
1.2 Solution for the R2 Lattice Packing Problem ................................. 6
2 Background and Strategy ................................. 7
2.1 Definition ................................. 7
2.2 Some Sample Packings in R2 ........................ 15
2.3 Three-Step Strategy for Proving Theorem 1 .................................19
3 Confirming the Three Steps ................................. 20
3.1 Every Lattice Has a Minimal Basis .................... 20
3.2 Well Rounded Lattices ........................... 23
3.3 Well Rounded Similarity Class....................... 26
3.4 Proof of Theorem 1............................. 30
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