One of the most interesting aspects of symmetry theory and pattern analysis and composition is the part relation (<) of symmetry groups; the elaborate and complex hierarchies of symmetry groups and subgroups, all nested one within the other, point to their direct correspondence with complex compositional structures of spatial patterns and suggest a precise methodology for formal analysis and composition in architectural design. This work looks closely at a specific set of symmetry groups - the two infinite types of the planar groups, the cyclic and the dihedral ones - and provides an automated environment to enumerate and represent the subgroups and their relationships with lattices in a graph theoretic manner. The complexity of these structures can be astonishing and it is suggested here that its graph theoretical representation can con­tribute to a better understanding of problems of spatial complexity in architectural design. The computational approach outlined in this work may indeed be used in formal analysis to identify all spatial repetitions and spatial correspondences that can be observed in a design. Alterna­tively, the approach may be used in formal composition to structure the design choices and bring to the foreground the whole range of spatial relationships available to the designer at any level of the design inquiry.

The computation and manual illustration of all possible subgroups for a given group n is not a trivial task; the theory for the computation here relies on a sorting based on two theorems proved by Lagrange and Sylow respectively: Lagrange theorem identifies a very precise numeri­cal relationship between subgroups and groups, namely that the order of a subgroup always divides the order of a group; Sylow's theorem proposes that if a number h is a power of a prime and divides the order of a group, then the group has a subgroup of order h. Here the automation relies to a routine to generate the complete list of all prime factors for a given number n. The simplest, albeit not most efficient, algorithm to generate the primes is the sieve of Eratosthenes. Since computing time is not a key feature for the relatively small magnitudes dealt in this project, algorithmic simplicity has been chosen over efficiency. Once the primes are extracted all possible distinct products are computed and tested to produce the possible lists of factors and the corresponding cyclic and dihedral subgroups. The generation of the graphs is then a straightforward task of iterating through the factors and generating the n/f nodes noting each time which set of operations the node represents.

The completion of the illustration of the structure of the graph is done with the pictorial representation of the edges of the graph deduced from each label and iterating over the nodes. The pictorial representation of the graphs is given in two ways: The first representation shows all possible subgroups and maps all possible relationships between them; this representation offers the most detailed view in the inner structure of any design and illustrates nicely the stun­ning complexity found even in the simplest of structures. The second representation fore­grounds the qualitative difference between rotational and reflective symmetries and classifies all subgroups and their relations in terms of two distinct classes, the cyclic and the dihedral groups respectively; this representation relies on a specific construct from group theory, the construal of conjugacy or equivalence relationships for elements of the group and essentially produces graphs where all identical instances of a dihedral group of some order n are collapsed to a single node; this preserves the relations between the various subgroups and produces a leaner graph. The application uses the Fruchterman-Reingold algorithm for the collapsed repre­sentation of the graphs.