AI 中文总结
本文将固定对称群下的软铺砌寻找方法扩展至立方格，施加自然条件得到26种软立方胞，提出Python算法对其分类，丰富了软胞铺砌的研究成果。
AI 中文摘要
软胞是指没有尖角、可无间隙且无重叠地填充空间的形状[2]，尖角是指固体表面上无法通过任何光滑曲线的点。在引入软胞概念的文献[2]中，作者证明存在一种算法，可对由凸多面体组成的铺砌进行软化，同时保留原始铺砌的格点和组合结构。尽管该算法（在少量限制条件下）保证存在与凸多面体铺砌组合等价的软铺砌，但该证明未涉及如何找到所有此类铺砌。对于基于截角八面体胞的多面体铺砌，文献[3]展示了如何为固定对称群找到所有软铺砌。本文将该方法扩展并应用于立方格，仅施加自然条件而非对称约束：铺砌的边半切线方向限制为格方向，且铺砌的边为平面。这产生了26种具有不同几何形状的软立方胞，这些胞可创建总共68个基本域，基于其格对称性可分为8组。本文还提出了一种算法流程（对应Python语言的程序），用于对这26种非等价几何胞类型进行分类。
英文摘要
Soft cells are shapes without sharp corners that can fill the space without gaps and overlaps [2]. A sharp corner is a point on the surface of the solid through which no smooth curve passes. In the paper introducing the concept of soft cells [2], the authors proved that there exists an algorithm that can soften tilings consisting of convex polyhedra, preserving the lattice points and combinatorial structure of the original tiling. Although the algorithm guarantees (with a few restrictions) that there exists a soft tiling that is combinatorially equivalent to the convex polyhedral tiling, the proof does not address how to find all such tilings. For a polyhedral tiling based on a truncated octahedral cell, paper [3] shows how to find all soft tilings for a fixed symmetry group. In this paper, we extend this method and apply it to the cubic lattice, imposing only natural conditions, rather than symmetry constraints. The natural conditions being, the directions of edge half-tangents of the tiling are restricted to lattice directions, and the edges of the tiling are planar. This results in 26 soft cubic cells with different geometries. A total of 68 fundamental domains can be created from the cells, which can be classified into 8 groups based on their lattice symmetry. The paper also presents an algorithmic process (with a corresponding program in language Python) for classifying the 26 non-equivalent geometric cell types.
Comments41 pages, 27 figures, 2 tables
