How do Shapes Fill Space?

形状如何填充空间？

• 批准号: EP/H004866/1
• 负责人: Uwe Grimm
• 金额: $ 2.55万
• 依托单位: The Open University
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2009
• 资助国家: 英国
• 起止时间: 2009 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FH004866%2F1
• 关键词: How; do; Shapes; Fill; Space

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.

从浴室瓷砖到砖墙，我们周围的空间都充满了垃圾。在人类历史的大部分时间里，人们一直在考虑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。从古希腊开始，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和群论中的数学描述起源于对镶嵌的研究，并成为晶体结构分类的基础。镶嵌的现代时代始于20世纪60年代，当时伯杰证明了一组给定的形状是否可以镶嵌平面是不可判定的，这一结果在2007年由马根斯特恩扩展到双曲平面。这直接导致了平铺理论和彭罗斯平铺等迷人例子的新世界的发现。准晶体（具有“禁止”对称性的晶体）的发现为彭罗斯和相关的镶嵌提供了非周期有序结构的模型。从零散的奇怪例子中，我们的理解正在演变成一个连贯的理论。特别是，替代规则的理论为彭罗斯镶嵌提供了一个自然的环境。因此，镶嵌提供了深刻的数学与美丽的图像相结合，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，以及相关谜题类型活动的互动特征，以及与准晶体等神秘材料的当前研究的联系，吸引了所有年龄段的观众。由于瓷砖是熟悉的对象，这个主题避免了符号和方程的数学语言往往造成的障碍，并使我们能够沟通的非平凡的数学概念，以公众audience.This项目将创建在皇家学会夏季展览2009年，预计将吸引超过5000名游客的展览材料。在2009年6月/7月的展览之后，这些材料被改编用于继续在皇家研究所（Ri）支持的英国范围内的数学大师班（10-18岁奥尔兹1至2.5小时的互动课程）中使用，并用于在Ri举办的家庭娱乐日。