Formal systems in architectural design aim at the systematic description, interpretation, and evaluation of existing works of architecture as well the systematic creation of new works of architecture. A basic algorithmic structure is reviewed and various examples of such systems are presented.

The recent emphasis of architecture discourse on issues of pattern making and parametric variation reaffirms the traditional role of symmetry and extends the research in new trajectories. Some basic questions regarding the extent of fitness and value of symmetry in formal composition remain unanswered. Currently all formal analysis using group theoretical tools focus on repetitive designs that show immediately their recursive structure. It is suggested here that highly complex designs can still be described and analyzed with group theoretical manner.

This work builds upon recent methodological approaches in the field (March 1998); (Park 2000); (Economou 2001) and proposes a model that investigates whether the combination of existing group theoretical formalisms with appropriate systems of representation can indeed cast new light in analysis of such works and therefore construct a rigorous body of foundational research in formal composition in architecture design. The broader question that is opened up here is whether a complex architecture object—or part depending on the interest of the researcher, can be interpreted as a layered object whose parts are all related symmetrically; in other words whether an asymmetric shape or configuration can be understood in terms of nested arrangements of some order of symmetry.

The object of analysis has been polemically selected here to be the NY5 architecture, a set of designs that are all clearly exemplifying formal qualities of abstraction, layering, complexity, depth and so on, all appearing impenetrable to a systematic and rigorous analysis using the existing group theoretical formal methods. For example, Richard Meier’s work has been presented here as a hyper-refinement of the modernist imagery that has been inspired not by machines but by other architecture that was inspired by machines and especially Le Corbusier; similarly, the group formalism that can describe Meier’s architecture could constitute a hyperrefined construction that relies on specific representations and mappings that foreground internal complex relationships of the structure itself, i.e. the symmetry subgroups and super-groups of any given spatial configuration. This analogy far as it goes has its limitations too, and the same exist for many other implicit theses herein.

The computation is entirely visual. A reassembly of the layered symmetries explains the structure of the symmetry of the house and provides an illustration of the basic thesis of this research on the foundation of a theory of emergence based on symmetry considerations. All plans of the house are represented in three different levels of abstraction moving successively away from the architectural representation to a purely diagrammatic one that foregrounds divisions of space. All representations are fed into an analysis algorithm to pick up all symmetry relationships and the parts are constructed as instances of a binary composition of a family of rectangular grids. Finally the process is reversed to fully account for the construction of the space of the house as a three dimensional layered composition.

At the end, this research points to a series of other extensions and domains. These extensions generally fall into two categories; (a) on the improvement of the system itself; and (b) on the interpretative capabilities it affords for the construction and evaluation of critical languages of design.