非周期性、旋转镶嵌空间与拓扑空间群
Aperiodicity, rotational tiling spaces and topological space groups
摘要：
利用拓扑方法研究了任意维欧几里得空间中非周期镶嵌的旋转结构。经典的拓扑方法来研究非周期性模式主要集中在平移结构，研究相关的空间，连续壳，这里表示为$Ω_t$。在这篇文章中，我们考虑两个进一步的空间$Ω_r$和$Ω_G$（旋转壳），它捕获了底层模式的完全刚性运动特性。旋转壳$Ω_r$被示出为包含作为子匹配箱流形的$Ω_t$的匹配箱流形。我们开发了新的S-MLD不变量来自同伦和上同调性质，这些空间展示了它们的计算以及理论效用。我们计算这些不变量的各种例子，包括一类3维非周期性图案，以及为空间的周期性镶嵌的$\mathbb{R}^3$单位立方体。我们证明了周期图案对称性的经典空间群可以恢复为我们空间$Ω_G$的基本群。类似地，对于与准晶相关的那些图案，晶体学家的非周期性空间群可以恢复为我们的基本不变量的商。
英文摘要：
We study the rotational structures of aperiodic tilings in Euclidean space of arbitrary dimension using topological methods. Classical topological approaches to the study of aperiodic patterns have largely concentrated just on translational structures, studying an associated space, the continuous hull, here denoted $Ω_t$. In this article we consider two further spaces $Ω_r$ and $Ω_G$ (the rotational hulls) which capture the full rigid motion properties of the underlying patterns. The rotational hull $Ω_r$ is shown to be a matchbox manifold which contains $Ω_t$ as a sub-matchbox manifold. We develop new S-MLD invariants derived from the homotopical and cohomological properties of these spaces demonstrating their computational as well as theoretical utility. We compute these invariants for a variety of examples, including a class of 3-dimensional aperiodic patterns, as well as for the space of periodic tessellations of $\mathbb{R}^3$ by unit cubes. We show that the classical space group of symmetries of a periodic pattern may be recovered as the fundamental group of our space $Ω_G$. Similarly, for those patterns associated to quasicrystals, the crystallographers' aperiodic space group may be recovered as a quotient of our fundamental invariant.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
