Unraveling Complexity: A Computational Approach for the Generation of all Underlying Structures of Three-Dimensional Shapes with an n-Fold Symmetry Axis

Athanassios Economou and Thomas Grasl

John Gero and Ashok Goel

Proceedings of the Third International Conference on Design Computing and Cognition (DCC)

DCC 08

361-374

Springer, Dordrecht

June 2008

10.1007/978-1-4020-8728-8_19

Shape studies, Configuration, Symmetry, Partial order lattice, Diagram

A computational approach for the generation of all underlying structures of three-dimensional shapes with an n-fold symmetry axis is briefly discussed. More specifically, the work looks at a specific class of designs in three-dimensional space, namely the three-dimensional designs with an n-fold symmetry axis, and provides a computational approach to a) enumerate all their repeated parts; b) depict their relationships in a graph theoretical manner; and c) illustrate all shape correspondences with pictorial visualizations for each distinct class of designs. The specific sets of symmetry groups that are examined here are the four infinite types of the point space groups, namely, the cyclic groups, the dihedral groups and their direct product groups with a cyclic group of order two. These four types of groups can capture the symmetry of any three-dimensional shape or design with an axis of symmetry of an order n. The complexity of these structures can be staggering and it is suggested here that their graph theoretical representation and pictorial representation can contribute to a better understanding of problems of spatial complexity in architectural design. The paper outlines the computational approach for the generation of all partial order lattices of these shapes and illustrates some of these ideas with consistent mappings of these lattices to a language of diagrams to visualize their part to whole relationships.