Symmetry is an implicit objective in structural form-finding that often reconciles efficiency and aesthetics. This paper identifies the symmetry of polyhedral diagrams in three-dimensional graphic statics (3DGS) as point groups and formulates them as constraints, enabling the optimization and manipulation of polyhedral diagrams that preserve such symmetry. 3DGS has been an efficient and effective tool for the form-finding of funicular structures. However, when modifying complex diagrams for design exploration or optimization, one can easily break the symmetry of the reciprocal design input, rendering the result undesirable for practical use. To address this problem, this paper investigates symmetry transformations and introduces point groups, an abstract algebra tool commonly used in crystallography to represent the symmetry and equivalence between a network of atoms (points with labels). It then discusses the hierarchy of symmetry in the geometry types of a polyhedral diagram, and proposes the constraint of symmetry through edge lengths. Based on the crystal symmetry search algorithm by spglib and pymatgen, a fast fingerprinting algorithm is developed to identify the point group of a polyhedral diagram and sort equivalent edges into sets. Finally, the paper shows that the necessary and sufficient condition for preserving the point group symmetry is that each set of edges has the same length. This constraint is compatible with the algebraic formulation of 3DGS and effectively preserves symmetry while reducing the dimension of the solution space. The method is implemented in the PolyFrame 2 plug-in for Rhino and Grasshopper.

翻译：对称性是结构形态优化中隐含的目标，往往能兼顾效率与美学。本文确定三维图解静力学（3DGS）中多面体图的对称性为点群，并将其形式化为约束条件，从而能够优化和操控保持此类对称性的多面体图。3DGS一直是索结构形态优化的高效工具，然而在对复杂图形进行设计探索或优化修改时，很容易破坏互易设计输入的对称性，导致结果不适用于实际应用。为解决此问题，本文研究对称变换并引入点群——晶体学中常用的抽象代数工具，用于表示原子（带标签的点）网络的对称性与等价关系。进而讨论多面体图几何类型中的对称层级，并提出通过边长度施加对称约束。基于spglib和pymatgen的晶体对称性搜索算法，本文开发了快速指纹识别算法以识别多面体图的点群，并将等价边归入集合。最后证明保持点群对称性的充要条件是每个边集具有相同长度。该约束与3DGS代数表达式兼容，可在降低解空间维度的同时有效保持对称性。方法已集成至Rhino与Grasshopper平台的PolyFrame 2插件中。