软胞元、开尔文泡沫与施瓦茨极小曲面
Soft cells, Kelvin's foam and the minimal surfaces of Schwarz
AI总结：
本文研究软胞元与极小曲面的关系，证明施瓦茨P和D曲面可生成非标准软镶嵌并连续互化，同时揭示其与DV-bcc镶嵌的等价类结构及开尔文泡沫的中间角色。
摘要：
最近，我们引入了一类新的形状，称为软胞元（soft cells），它们以软镶嵌（soft tilings）的方式无间隙、无重叠地填充空间，同时最小化尖锐角点的数量。我们引入了边弯曲算法（edge bending algorithm），将多面体镶嵌变形为软镶嵌，并证明了无穷多类多面体镶嵌可以被光滑变形为标准软镶嵌。在本文中，我们证明某些三重周期极小曲面（triply periodic minimal surfaces）自然地产生非标准软镶嵌。通过扩展边弯曲算法，我们进一步证明，由施瓦茨 P 曲面（Schwarz P）和施瓦茨 D 曲面（Schwarz D）导出的软镶嵌可以通过一个单参数族的中间非标准软镶嵌连续地相互转化。值得注意的是，由于保持了组合结构，所得两种镶嵌都属于体心立方 bcc 格子上狄利克雷-沃罗诺伊镶嵌（Dirichlet-Voronoi tiling）的一阶等价类，这凸显了这些极小曲面构型背后深刻的几何联系。通过要求一阶类中边的端切线相同，我们还定义了镶嵌之间的二阶等价类，并证明在共享 DV-bcc 镶嵌完全对称群的软镶嵌中恰好存在两个这样的类。此外，我们构造了一个单参数镶嵌族，连接标准与非标准软镶嵌，并明确地将经典开尔文泡沫（Kelvin foam）结构作为中间构型包含在内。这一构造表明，软胞元本身及其生成所用的几何方法都为理解自然形式背后的结构原理提供了有价值的见解。我们还给出了由螺旋二十四面体（gyroid）结构诱导的软镶嵌。
英文摘要：
Recently, we introduced a new class of shapes, called soft cells which fill space as soft tilings without gaps and overlaps while minimizing the number of sharp corners. We introduced the edge bending algorithm that deforms a polyhedral tiling into a soft tiling and we proved that an infinite class of polyhedral tilings can be smoothly deformed into standard soft tilings. Here, we demonstrate that certain triply periodic minimal surfaces naturally give rise to non-standard soft tilings. By extending the edge-bending algorithm, we further establish that the soft tilings derived from the Schwarz P and Schwarz D surfaces can be continuously transformed into one another through a one-parameter family of intermediate non-standard soft tilings. Notably, by carrying its combinatorial structure, both resulting tilings belong to the first order equivalence class of the Dirichlet-Voronoi tiling on the body-centered cubic bcc lattice, highlighting a deep geometric connection underlying these minimal surface configurations. By requiring identical end-tangents for edges in a first order class, we also define second order equivalence classes among tilings and prove that there exist exactly two such classes among soft tilings which share the full symmetry group of the DV-bcc tiling. Additionally, we construct a one-parameter family of tilings bridging standard and non-standard soft tilings, explicitly including the classic Kelvin foam structure as an intermediate configuration. This construction highlights that both the soft cells themselves and the geometric methods employed in their generation provide valuable insights into the structural principles underlying natural forms. We also present the soft tiling induced by the gyroid structure.
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