Point Group Symmetry of Polyhedral Diagrams in Graphic Statics

Abstract

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.