How do Shapes Fill Space?

形状充斥着我们周围的空间，从浴室的瓷砖到砖墙。如何将形状组合在一起，使它们填满空间，形成我们所说的瓷砖，这个令人困惑的问题在人类历史的大部分时间里都被考虑过。这个问题可能首先出现在艺术领域，瓷砖被用来制作有趣的图案，比如伊斯兰艺术。自古希腊以来，它也被作为数学的一部分进行研究。例如，我们对对称性的理解及其在群论方面的数学描述起源于对瓷砖的研究，并奠定了晶体结构分类的基础。瓦片的现代时代始于20世纪60年代，当时伯杰证明了一组给定形状能否在平面上瓦片的问题是不可确定的，2007年，马根斯特恩将这一结果推广到了双曲平面上。这直接导致了瓦片理论的新世界的发现和迷人的例子，如彭罗斯瓦片。准晶体（具有“禁止”对称的晶体）的发现提供了额外的动力，因为彭罗斯和相关的瓦片为非周期有序结构提供了模型。从零星的奇怪例子，我们的理解现在正在演变成一个连贯的理论。特别是，替代规则理论为彭罗斯瓷砖提供了一个自然的背景。因此，瓷砖将深奥的数学与美丽的图像结合在一起，这使它们成为公众参与活动的理想主题。视觉吸引力，与艺术和建筑的联系以及相关拼图类型活动的互动特征，以及与当前对准晶体等神秘材料的研究的联系，吸引了所有年龄段的观众。因为瓷砖是人们熟悉的对象，这个主题避免了符号和方程的数学语言经常造成的障碍，使我们能够向公众传达不平凡的数学概念。该项目将为2009年皇家学会夏季展览制作材料，预计将吸引5000多名参观者。在2009年6月/ 7月的展览之后，这些材料将被改编用于由皇家学会（Ri）支持的全英国范围的数学大师班（10-18岁的1至2.5小时互动课程）和Ri举办的家庭欢乐日。

Shapes fill space all around us, from bathroom tilings to brick walls. The puzzling problem of how to fit shapes together so that they fill space to form what we call tilings has been considered throughout much of the history of humanity. The problem probably emerged first in the arts, where tiles were used to produce interesting patterns, like those of Islamic art. It has also been studied as part of mathematics since the ancient Greeks. For example, our understanding of symmetries and their mathematical description in terms of group theory originated from the investigation of tilings, and underlies the classification of crystal structures.The modern era for tilings began in the 1960s when Berger proved that the problem of whether a given set of shapes could tile the plane was undecidable, a result extended to the hyperbolic plane by Margenstern in 2007. This led directly to the discovery of new worlds of tiling theory and fascinating examples such as the Penrose tiling. The discovery of quasicrystals (crystals with 'forbidden' symmetry) gave additional impetus as the Penrose and related tilings provide models for non-periodic ordered structures. From a scattering of strange examples, our understanding is now evolving to coalesce into a coherent theory. In particular, the theory of substitution rules is giving a natural setting for the Penrose tiling.Tilings therefore offer a combination of deep mathematics with beautiful imagery, which makes them an ideal topic for public engagement activities. The visual appeal, the link to arts and architecture and the interactive character of related puzzle-type activities, as well as the link to current research on mysterious materials such as quasicrystals, fascinates audiences across all age groups. Because tilings are familiar objects, this topic avoids the barrier often caused by the mathematical language of symbols and equations, and enables us to communicate non-trivial mathematical concepts to a public audience.This project will create material for an exhibit at the Royal Society Summer Exhibition 2009, which is expected to attract in excess of 5000 visitors. After the exhibition in June/July 2009, the materials are adapted for continued use in UK-wide mathematics masterclasses (1 to 2.5 hour interactive sessions for 10-18 year olds) supported by the Royal Institution (Ri) and for use in Family Fun Days hosted at the Ri.