Niels Bohr Institute, University of Copenhagen, Denmark.
School of Chemistry, University of Sydney, Australia.
Soft Matter. 2023 Feb 22;19(8):1586-1595. doi: 10.1039/d2sm01403c.
Nature employs an impressive range of topologically complex ordered nanostructures that occur in various forms in both natural and synthetic materials. A particular class of these exhibits negative curvature and forms periodic saddle-shaped surfaces in three dimensions. Unlike pattern formation on flat or positively curved surfaces like spherical systems, the understanding of patterning on such surfaces is highly complicated due to the structures being intrinsically intertwined in three dimensions. We present a new method for visualisation and analysis of patterns on triply periodic negatively curved surfaces by mapping to two-dimensional hyperbolic space analogous to spherical projections in cartography thus effectively creating a more accessible "hyperbolic map" of the pattern. Specifically, we exemplify the method the simplest triply periodic minimal surfaces: the Primitive, Diamond, and Gyroid in their universal cover along with decorations from a soft materials, whose structures involve decorations of soft matter on negatively curved surfaces, not necessarily minimal.
大自然采用了令人印象深刻的多种拓扑复杂有序纳米结构，这些结构以各种形式存在于天然和合成材料中。这些结构中有一类具有负曲率，并在三维空间中形成周期性的鞍形表面。与在平面或正曲率表面（如球形系统）上的图案形成不同，由于这些结构在三维空间中本质上是交织在一起的，因此理解在这种表面上的图案形成非常复杂。我们提出了一种新的方法，通过映射到二维双曲空间来可视化和分析三重周期性负曲率表面上的图案，类似于制图学中的球形投影，从而有效地创建了更易于访问的图案“双曲地图”。具体来说，我们举例说明了该方法在最简单的三重周期性最小曲面中的应用：原始、钻石和胞状结构及其通用覆盖，以及来自软物质的装饰，其结构涉及负曲率表面上的软物质装饰，不一定是最小的。
