Applications of space filling curves to substitution tilings

空间填充曲线在替代平铺中的应用

• 批准号: EP/R013691/1
• 负责人: Michael Whittaker
• 金额: $ 12.88万
• 依托单位: University of Glasgow
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2018
• 资助国家: 英国
• 起止时间: 2018 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FR013691%2F1
• 关键词: Applications; space; filling; curves; substitution; Applications; space; filling; curves; substitution

One of the most spectacular scientific discoveries of the late twentieth century was a new material that was neither crystalline nor amorphous. This created a paradigm shift in crystallography, and these alloys are now called quasicrystals. Quasicrystals are modelled mathematically by patterns called aperiodic tilings that lack symmetry in the usual sense, but still exhibit long-range order. The most famous example is due to Sir Roger Penrose, whose tiling exhibited the same `impossible' symmetry as the quasicrystals discovered by Professor Dan Shechtman. The research in this proposal initiates a new method of studying aperiodic tilings through dimension reduction.The digital revolution has made profound advances in sending two- and even three-dimensional images from place to place by encoding them as a sequence of zeros and ones. The research in this proposal draws analogy with this except that the image is infinite and need not have pixels arranged in a locally systematic way. In particular, the research in this proposal initiates a similar type of encoding of an aperiodic tiling through the use of space-filing curves; a type of fractal that was discovered in the late 18th century that helped to reshape our mathematical notions of size, area and volume. In much the same way as a video feed, some information is compressed through the encoding. However, we can still garner a vast amount of information about the original tiling, especially when the space filling curve comes from the underlying method used to define the tiling in the first place. Significantly, in the case of aperiodic tilings, all the geometric information about how tiles fit together is encoded in the digital sequence making it very easy to work with from a mathematical perspective; it is a purely combinatorial object.To each aperiodic tiling we define a dynamical system that consists of a map on a topological space whose individual points are infinite tilings. It has been shown that this topological space is a Cantor set fibre bundle over a torus; that is, it is a donut with an arbitrary number of holes that has fractals emanating from every point on its surface. The bizarre nature of this space makes it extremely difficult to study. For this reason, topological and operator algebraic invariants have been the focus of research on tiling spaces. The programme of research outlined in this proposal gives a new attack on studying this dynamical system by studying the combinatorial space associated with the space filling curve, which is much simpler while retaining most information about the more complicated system.The new approach taken in this proposal will have impact across research in aperiodic tiling theory, and even to the more general study of hyperbolic dynamical systems, operator algebras and fractal geometry.

二十世纪后期最壮观的科学发现之一是一种既不是结晶也不存在的新材料。这产生了晶体学的范式转移，这些合金现在称为Quasicrystals。准晶体是通过数学上的模型来建模的，这些模式在通常的意义上是缺乏对称性的，但仍表现出远距离顺序。最著名的例子是由于罗杰·彭罗斯爵士（Sir Roger Penrose）爵士，他的平铺表现出与丹·谢赫特曼（Dan Shechtman）教授发现的准晶体相同的“不可能”对称性。该提案中的研究启动了一种新的方法，通过缩小尺寸研究了多个斜利。数字革命通过将它们作为一系列零和一个来发送两维图像，从而在将两维的图像从地点发送到位置方面取得了深远的进步。该提案中的研究与此相比进行了类比，只是图像是无限的，并且不需要以当地系统的方式排列像素。特别是，该提案中的研究启动了通过使用太空式曲线的Aperiodic平铺的类似类型的编码。 18世纪后期发现的一种分形有助于重塑我们对大小，面积和体积的数学概念。与视频提要几乎相同，通过编码会压缩一些信息。但是，我们仍然可以获得有关原始瓷砖的大量信息，尤其是当空间填充曲线来自用于定义瓷砖的基础方法时。值得注意的是，在多个斜利的情况下，所有有关瓷砖如何结合在一起的几何信息都在数字序列中编码，从而使其从数学角度很容易使用。它是一个纯粹的组合对象。对于每个Aperiodic瓷砖，我们定义了一个动力系统，该系统由拓扑空间上的地图组成，其各个点是无限的瓷砖。已经表明，这个拓扑空间是圆环上的圆形纤维束。也就是说，它是一个甜甜圈，有一个任意数量的孔，它们的分形从表面的每个点发出。这个空间的奇怪性质使研究变得极为困难。因此，拓扑和操作员代数不变性一直是瓷砖空间研究的重点。该提案中概述的研究计划通过研究与空间填充曲线相关的组合空间进行了新的攻击，对研究这种动态系统进行了研究，这要简单得多，同时保留有关更为复杂系统的大多数信息。该提案中采用的新方法将影响到跨二元理论的研究，甚至与超级动态动态系统的更一般性研究中的研究中，访问型系统的范围更为普遍的研究。