Economou, A., & Hong, T. C. K. (2023). Back to the drawing board: Shape calculations in Shape Machine. In J. S. Gero (Ed.), Design computing and cognition’22 (pp. 477-496). Springer. https://doi.org/10.1007/978-3-031-20418-0_33

Abstract:
Three computer implementations of one of the earliest and most iconic shape grammars are given to showcase different ways to use shape rules in Shape Machine. These three modes of working with shape rules—manual, automatic and conditional—are used interchangeably to produce three skeuomorphic variations of the original checkerboard lattice grammar in different design domains—3D prints, origami and kerfing—and showcase the versatility and applicability of the shape rules for the specification of diverse artifacts for different scales, materials and functions. Some initial remarks on the extension of the shape grammar formalism to include programming constructs including states, loops, jumps, and conditionals are discussed.

Hong, TC.K., Economou, A. Five Criteria for Shape Grammar Interpreters. In: Gero, J.S. (eds) Design Computing and Cognition’20. Springer, Cham. 2022. https://doi.org/10.1007/978-3-030-90625-2_11

Abstract:
Shape grammar interpreters have been studied for more than forty years addressing several areas of design research including architectural, engineering, and product design. At the core of all these implementations, the operation of embedding – the ability of a shape grammar interpreter to search for subshapes in a geometry model even if they are not explicitly encoded in the database of the system – resists a general solution. Here, a detailed account on various constructions of embedding is provided, including determinate and indeterminate ones, to give a sense of the rising complexity of their implementation in a shape grammar interpreter, and to provide a visual map of the work accomplished in the field so far, and the work ahead too.

Economou, A. (2021). Durand redrawn: A formal description of Durand’s "Précis of the Lectures on Architecture". In Conference Proceedings of NEXUS 20/21: Relationships Between Architecture and Mathematics (pp. 101-106). Kaiserslautern, Germany.

Abstract:
Jean-Nicholas-Louis Durand’s Précis des leçons d'architecture données à l’École polytechnique  was published in1802 with no less ambition than to revolutionize architectural pedagogy and production. Emphatically, and obsessively, Durand completed his book with the chapter “Procedure to Be Followed in the Composition of any Project”. Even if the architectural vocabulary pursued in the Précis has changed a lot since the nineteenth century, its insistence on method and procedural logic dominates stronger than ever contemporary architectural discourse. The work here revisits Durand’s theoretical proposition to formally test its claims, conventions and expressiveness and to ultimately reconsider its relation to current theories and practices of architecture. Significantly, the work reworks the complete thesis in the form of drawing instructions pictorially encoded within the Shape Machine ‒ a radically new modeling software that allows for the visual specification of shape rules and the automated production of mechanically generated drawings ‒ to generate the designs illustrated in the Précis (and more too). A sample of mechanically executed plans for various building typologies is given at the end to give a sense of the detail and expressiveness of the work.

Economou, A., Hong, T. C. K., Ligler, H., & Park, J. (2021). Shape Machine: A primer for visual computation. In J. H. Lee (Ed.), A new perspective of cultural DNA (pp. 65-92). Springer. https://doi.org/10.1007/978-981-15-7707-9_6

Abstract:
A brief survey of the expressive power of Shape Machine, a new shape grammar interpreter, is presented. The work is discussed in two parts: A brief presentation of a series of shape computations that have been routinely used as benchmarks for the design tasks a shape grammar interpreter should be able to accomplish; and a brief exploration of design applications in various domains, namely, product modeling, mechanical and architectural design to suggest possibilities for new design workflows and/or new trajectories in new domains too. Some speculations on the future of the technology pertaining to its potential usage in a new paradigm of programming with shapes (programming by drawing) and/or the envision of a new paradigm of a physical computer are given in the end.

Ligler, H., & Economou, A. (2019). From drawing shapes to scripting shapes: Architectural theory mediated by Shape Machine. In Proceedings of the Symposium on Simulation for Architecture + Urban Design (SimAUD).

Abstract:
The asymptotic relation between the three Vitruvian prerequisites of design (firmness, utility, and delight) and the six principles of design (order, arrangement, eurythmy, symmetry, propriety and economy) has been one of the most striking and unresolved characteristics of the foundations of architectural theory from Vitruvius’ text onwards. The conundrum is even more exacerbated because of the convoluted and/or obscure nature of the six principles of design. This work here builds on an attractive proposition that recasts the six principles of design in three dyads, but significantly extends it by critically relating it with key ideas in contemporary computational design discourse. It is furthermore suggested that the emergent tripartite scheme of design, including algebras of design, design rules and criticism rules, provides a new light in the Vitruvian model that can disambiguate the esoteric nature of the six principles of design, relate them coherently to the three Vitruvian prerequisites of design, and more significantly, propose an intellectual thread of architectural theory and discourse that can remain useful throughout the ever changing nature of architectural briefs, technologies and aesthetic concerns.

Economou, A. (2018). The six Vitruvian principles of architectural design reframed within contemporary computational design discourse. In K. Williams & M. G. Bevilaqua (Eds.), Nexus 2018 Architecture and Mathematics Conference Book (pp. 117-121). Kim Williams Books, University of Pisa.

Abstract:
The asymptotic relation between the three Vitruvian prerequisites of design (firmness, utility, and delight) and the six principles of design (order, arrangement, eurythmy, symmetry, propriety and economy) has been one of the most striking and unresolved characteristics of the foundations of architectural theory from Vitruvius’ text onwards. The conundrum is even more exacerbated because of the convoluted and/or obscure nature of the six principles of design. This work here builds on an attractive proposition that recasts the six principles of design in three dyads, but significantly extends it by critically relating it with key ideas in contemporary computational design discourse. It is furthermore suggested that the emergent tripartite scheme of design, including algebras of design, design rules and criticism rules, provides a new light in the Vitruvian model that can disambiguate the esoteric nature of the six principles of design, relate them coherently to the three Vitruvian prerequisites of design, and more significantly, propose an intellectual thread of architectural theory and discourse that can remain useful throughout the ever changing nature of architectural briefs, technologies and aesthetic concerns.

Ligler, H., & Economou, A. (2018). On John Portman's atria: Two exercises in hotel composition. In J. S. Gero (Ed.), Conference Proceedings of Design Computing and Cognition DCC’18 (pp. 439-458). Springer.

Abstract:
Two formal exercises in hotel composition are presented. In both, the hos-pitality work of the architect John Portman is the focus. His language of hollow forms is addressed following his unique claim on the organizing principles found in his 1964 house, Entelechy I. The first exercise outlines a generative specification for his atrium hotel language in a parametric shape grammar informed by the logic of the house that generates an atrium hotel prototype. The second exercise speculates with a sketch on how transfor-mation grammars can yield various configurations to explore Portman’s atrium hotel language for a series of initial shapes. The overall goal of the research is to progress an ongoing effort to build a constructive theory on Portman’s architectural language as explored for a variety of scales and con-texts.

Dortdivanlioglu, H., & Economou, A. (2017). Outlining Terragni: A riddle reworked. In G. Çağdaş et al. (Eds.), CAAD Futures 2017, CCIS 724 (pp. 381-394). Springer.

Abstract:
Despite his controversial political background, the leading architect of the Italian Rationalist Movement, Giuseppe Terragni, has attracted the attention of a large group of architectural scholars. He has often been acknowledged as an enigmatic architect whose work oscillated between classicism and modernism. This work takes on two of the most emblematic projects by Terragni, the Mambretti Tomb and the Danteum and provides a formal generation of both projects in the form of two parametric shape grammars to inquire on the possibility of a common compositional strategy underlying both projects. A new interpretation of Terragni’s work as a non-classical/classical computational model is given in the end.

Ligler, H., & Economou, A. (2015). Entelechy I: Towards a formal specification of John Portman's domestic architecture. In B. Martens et al. (Eds.), Real Time: Extending the Reach of Computation, Proceedings of the 33rd eCAADe Conference, Volume 1 (pp. 445-452). Vienna University of Technology.

Abstract:
John Portman's work attracts much interest, although little scholarship exists that directly engages his contribution in formal composition. Most of the discussion of Portman's architecture tends to focus on his commercial work and hotels, although a key to understanding his work is found in his personal domestic projects where he has had the freedom to explore his architectural ideas. This study focuses on his first residence, Entelechy I, to begin outlining his design principles formally. The ambition is to open up the whole question of his architectural contribution in the United States and at large.

Park, J., & Economou, A. (2015). The Dirksen variations: Towards a generative description of Mies's courthouse language. In B. Martens et al. (Eds.), Real Time: Extending the Reach of Computation, Proceedings of the 33rd eCAADe Conference, Volume 1 (pp. 453-462). Vienna University of Technology.

Abstract:
A generative description of Mies van der Rohe's courthouse language is presented in the form of a shape grammar. The grounding of the work is based on a set of 135 sketches produced by the office of Mies during the design process of the Everett McKinley Dirksen United States Courthouse in Chicago, and documented in the Mies van der Rohe Archive at the Museum of Modern Art. The work here postulates a set of 39 unique courthouse designs all showcasing distinct variations of the courtroom type in the Miesian language and re-casts them in two-dimensional diagrams to make their differences and similarities transparent. A series of spatial relations between five types of spaces are extracted, including courtrooms, circulation networks, vertical cores, office spaces, and support spaces, and are deployed to specify the shape rules of the grammar. A set of conventions to specify how the two-dimensional diagrams represent three-dimensional models is briefly outlined to prepare the ground for the implementation of the grammar in a three-dimensional shape grammar interpreter.

Economou, A., & Kotsopoulos, S. (2014). From shape rules to rule schemata and back. In J. S. Gero & S. Hanna (Eds.), Design Computing and Cognition DCC’14 (pp. 419-438). Springer.

Abstract:
Shape rules and rule schemata are compared in terms of their expressive and productive features in design inquiry. Two kinds of formal processes are discussed to facilitate the comparison. The first proceeds from shape rule instances and infers rule schemata that the shape rules can be defined in. The second proceeds from rule schemata and postulates shape rule instances that can be defined within the schemata. These two parallel processes mirror our intuition in design: the conceptual need to frame explicit actions within general frameworks of principles, and the productive need to supply general principles with an explicit system of actions.

Grasl, T., & Economou, A. (2014). Towards controlled grammars: Approaches to automating rule selection for shape grammars. In E. Thompson (Ed.), Fusion: Proceedings of the 32nd eCAADe Conference, Volume 2 (pp. 357-363). Department of Architecture and Built Environment, Newcastle upon Tyne, England.

Abstract:
An approach to automating the rule selection process of shape grammars is introduced. The methods presented are divided into two approaches: Extensive enumeration and goal directed generation. Complete design catalogue enumeration is interesting only when the results are small. Bigger numbers are useful only to the extent they give a sense of the upper boundary of the set of possibilities specified by the grammar. Smaller numbers are useful because they allow an actual, constructive and meaningful visual encounter with these possibilities. Goal directed design generation is essentially an attempt to reduce the grammars solution space and filter out only those solutions that are of interest and may be given for any desired subset of the complete set of designs specified by the grammar. A shape grammar interpreter is extended here by a computational framework to allow the rule selection to be executed by agents. Hereby each agent is based on a different paradigm taken from the field of artificial intelligence. The results are compared.

Quan, J. S., Economou, A., Grasl, T., & Pei-Ju Yang, P. (2014). Computing energy performance of building density, shape, and typology in urban context. In Proceedings of the 6th International Conference on Applied Energy – ICAE2014 (pp. 1602-1605). Elsevier.

Abstract:
This paper aims to better understand the impact of urban context on building energy consumption. The factors of external shading, shapes generated from zoning ordinances, and local climate are examined concerning three main questions: (1) how density influences building energy consumption generally, (2) how a given density generates alternative building shapes that have different impacts on energy performance, and (3) how different typologies affect the energy-density relationship. To answer them, a series of parametric simulation experiments are conducted based on Martin and March's urban block structure. For more than 14,000 hypothetical models located at the Portland urban grid, the energy consumptions for the purposes of cooling and heating are simulated using AutoCAD script, MATLAB and Energy Plus 8. The results suggest that, different from the common perceptions, building energy consumptions for cooling and heating purposes do not always have a negative relationship with density. Instead, the energy consumption has a negative relationship with density before a turning point, and then the relationship changes to be positive. Also with the same FAR, different building cover ratio and typologies can lead to large variations in energy consumption. By the experiments on different building shapes generated by urban frit, it was found that even with the same typology, the building energy consumption can still vary significantly. Finally, the exploration of climate factors indicates that in both Portland and Atlanta, the findings are similar except that the energy-density relationship is weaker in Atlanta than in Portland.

Economou, A. (2013). f(x): Explicit and recursive definitions of architectural typology. In M. Bernal et al. (Eds.), Knowledge-Based Design: Proceedings of the 11th Iberoamerican Society of Digital Graphics (SIGRADI) Conference (pp. 211-216). Universidad Tecnica Federico Santa María.

Abstract:
The notion of a generative description of architectural typology is discussed. Two different approaches are identified and contrasted in terms of their expressive power in design. The building type of the courthouse is selected to showcase and test the approach. One case study is briefly presented to pictorially illustrate the findings.

Grasl, T., & Economou, A. (2013). Unambiguity: Difficulties in communicating shape grammar rules to a digital interpreter. In R. Stouffs & S. Sariyildiz (Eds.), Computation and Performance: Proceedings of the 31st eCAADe Conference – Volume 2 (pp. 617-620). Delft University of Technology.

Abstract:
Shape grammar interpreters have received increasing attention over the last years. Each software application among all of them offers some solutions to the general problem but few implement or describe rule editing capabilities. A complete shape grammar interpreter should support emergence, parametric rules and rule editing via a graphical editor. Most of the conventions characterizing the design of the interface of a shape grammar interface have been given by Tapia (1999) however the editor in the corresponding shape grammar interpreter did not support parametric rules. The work here discusses the conventions characterizing a parametric shape grammar interpreter and the problems that arise in the implementation of a parametric graphical editor. The approach presented here builds on an underlying graph representation of shapes and it is suggested that most of the findings should be applicable to other shape grammar implementations as well.

Grasl, T., & Economou, A. (2011). Grape: Using graph grammars to implement shape grammars. In SimAUD: Simulation for Architecture and Urban Design (pp. 45-52). Boston, MA.

Abstract:
An implementation of a shape grammar interpreter is described. The underlying graph-theoretic framework is briefly discussed to show how alternative representations from graph theory including graphs, overcomplete graphs and hyperedge graphs can support some of the intuitions handled in shape grammars by direct visual computations with shapes. The resulting plugin implemented in Rhino, code-named GRAPE, is briefly described in the end.

Grasl, T., & Economou, A. (2010). Palladian graphs: Using a graph grammar to automate the Palladian grammar. In G. Schmitt et al. (Eds.), Future Cities, 28th eCAADe Conference Proceedings (pp. 275-284). ETH Zurich.

Abstract:
An implementation of the Palladian grammar using a graph grammar and a graph to shape mapping is presented. The application is embedded in a parametric CAD environment and allows the exploration of Palladian villas by hand or by using a random generator.
 
Abstract:
An implementation of a shape grammar interpreter is described. The underlying graph-theoretic framework is briefly discussed to show how alternative representations from graph theory including graphs, overcomplete graphs and hyperedge graphs can support some of the intuitions handled in shape grammars by direct visual computations with shapes. The resulting plugin implemented in Rhino, code-named GRAPE, is briefly described in the end.

Din, E., & Economou, A. (2009). Emergent symmetries and visual computations: A group theoretical analysis of The Smith House by Richard Meier. In T. Tidafi & T. Dorta (Eds.), Joining Languages, Cultures and Visions: CAADFutures 2009 (pp. 173-187). PUM.

Abstract:
This work proposes the use of partial order lattices along with representational schemes to account for patterns of ambiguity and emergence in the description of designs. The complexity of such designs is viewed as an aggregation of spatial layers that can all be decomposed by the subgroup relations of the symmetry of the configuration. At the end, this methodology points to a combinatorial approach that generates visual prototypes for future use in design synthesis. Here, Meier's work is just a case study that validates the group theoretical approach.

Economou, A., & Grasl, T. (2009). Point worlds. In G. Gagdas & B. Colakoglu (Eds.), Computation: The New Realm of Architectural Design, 27th eCAADe Conference Proceedings (pp. 221-228). ITU/YTU.

Abstract:
Group theory has been extensively used in systematic studies in analysis and synthesis of form. A particular analytical method founded upon group theory, the subsymmetry analysis, has been quite successful in showing how asymmetric designs can be seen as aggregates of parts that all possess various degrees of symmetry. This work generalizes these approaches and provides a computational framework for the complete and automated representation of all underlying group structures of finite n-dimensional shapes with a center of symmetry, for n ≤ 3. The application is implemented using GrGen.Net a graph rewriting system written in C#. Some possible applications to illustrate these ideas with shape grammars are discussed in the end.

Grasl, T., Economou, A., & Branum, C. (2009). Combining triples: Using a graph grammar to generate courthouse topologies. In G. Gagdas & B. Colakoglu (Eds.), Computation: The New Realm of Architectural Design, 27th eCAADe Conference Proceedings (pp. 605-612). ITU/YTU.

Abstract:
Group theory has been extensively used in systematic studies in analysis and synthesis of form. A particular analytical method founded upon group theory, the subsymmetry analysis, has been quite successful in showing how asymmetric designs can be seen as aggregates of parts that all possess various degrees of symmetry. This work generalizes these approaches and provides a computational framework for the complete and automated representation of all underlying group structures of finite n-dimensional shapes with a center of symmetry, for n ≤ 3. The application is implemented using GrGen.Net a graph rewriting system written in C#. Some possible applications to illustrate these ideas with shape grammars are discussed in the end.

Economou, A., & Riether, G. (2008). Vitruvian machine. In Proceedings of CAADRIA: Beyond Computer-Aided Design (pp. 522-528). Chiang Mai, Thailand.

Abstract:
The design machine, an algorithmic structure for design, is considered within current trajectories of architecture discourse to suggest a reinterpretation of the Vitruvian discourse and to produce eight systematic studies in formal composition in architectural design. The work here suggests that Vitruvius' account of the three prerequisites of architecture, namely, utilitas, venustas, firmitas and typically translated as commodity, delight, and firmness, directly allude to contemporary interpretative frameworks of architecture works in terms of functional, geometric, and material characteristics respectively and more significantly to the three modules of the design machine, namely the receptor, language of design, and effector respectively. The resulting diagram for design suggests a complete structure for alternative definitions of design processes all encoded in the eight possible combinatorial subsets of the Vitruvian triad including the empty set. A brief presentation of all eight exercises in formal composition in architectural design is given in the end.

Economou, A., & Grasl, T. (2008). Unraveling complexity. In Proceedings of the Third International Conference on Design Computing and Cognition (pp. 361-374). DCC, USA.

Abstract:
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.

Economou, A., & Grasl, T. (2008). Unraveling complexity. In Proceedings of the Third International Conference on Design Computing and Cognition (pp. 361-374). DCC, USA.

Abstract:
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 pro-vides an automated environment to enumerate and represent all their 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 their graph theoretical representation can contribute to a better understanding of problems of spatial complexity in architectural design. The computational approach outlined in this work can be used in formal analysis to identify all spatial repetitions and spatial correspondence observed in a design. Alternatively, the approach can be used in formal synthesis 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 paper here outlines the computational approach for the generation of all the partial order lattices of two-dimensional shapes with an n-fold symmetry axis and illustrates some of these ideas with a preliminary setting of a formal analysis of the symmetry properties of the typology of courthouses.

Grasl, T., & Economou, A. (2007). Spatial similarity metrics: Graph theoretic distance measurement and floor plan abstraction. In T. Maver (Ed.), Proceedings of Computer Aided Architectural Design Futures: CAADFutures (pp. 251-263). Sydney, Australia.

Abstract:
Comparing space allocation programs computationally is a resource intensive task. This paper introduces a method which reduces the complexity of floor plan graphs to facilitate the problem. In a first step the nodes are labeled according to a classification scheme. This in its own right reduces the complexity of the comparison by introducing more diversity among the nodes. Subsequently the graph can be coarsened in several steps based on the category information. This does not only further reduce the necessary computation, but can also offer a visualization of and additional insight into the essential structure of the space allocation program at hand.

Economou, A. (2006). Well-tempered mappings of architecture and music. In V. Trova, K. Manolides, & G. Papakonstantinou (Eds.), Representation as Vehicle of Architectural Thought: Conference Proceedings (pp. 85-94). University of Thessaly Press. (In Greek: Οικονόμου, Α. (2006). Συγκερασμένες Απεικονίσεις Αρχιτεκτονικής και Μουσικής. Πρακτικά Συνεδρίου: Η αναπαράσταση ως Όχημα Αρχιτεκτονικής Σκέψης, σελ. 85-94).

Abstract:
A model of correlation of arithmetical ratios in models of space, models of sound and models of light is presented. A computational environment is designed to allow an interactive movement of a virtual agent within the rooms of a given arrangement of spaces using either the body movement or the eye movement of the agent as the basic trigger of sounds and colors.

Park, H. J., Economou, A., & Papalambros, P. (2005). Hermes: A computational tool for proportional studies in design. In B. Martens & A. Brown (Eds.), Conference Proceedings of the Computer-Aided Architecture Design Futures: Learning from the Past a Foundation for the Future (CAADFutures 2005) (pp. 99-108). Kluwer Academic Publishers.

Abstract:
The absence of computational tools for the application of proportional theory in analysis and synthesis in design has been a persistent problem in the field of formal composition in architectural design. Analysis of existing designs still requires an enormous amount of patience and persistence from the researcher to undertake with pencil and paper. Synthesis of new designs with proportional qualities is even more elusive because of the mathematical sophistication it demands from the designers. And both activities require a command of this body of knowledge that very few architects nowadays possess. This work presents a computational tool as a tool for proportional studies based on the theory of means. The analysis component of the application evaluates existing designs and provides statistical measures about the proportional structure of the design. The synthesis component of the application generates new designs from known ones with additional prescribed proportional properties. In both cases design optimization methodologies are employed. The analysis component is written in Autolisp and runs within AutoCAD. The synthesis component is written using the Genetic Algorithm Toolbox of Matlab and an Autolisp application within AutoCAD.

Economou, A. (2001). Four algebraic structures in design. In J. Wasim (Ed.), Proceedings of the Twenty First Annual Conference of the Association for Computer-Aided Design in Architecture: Reinventing the Discourse (pp. 192-201). Association for Computer-Aided Design in Architecture.

Abstract:
Dihedral and cyclic symmetries have been heavily used for the interpretation or the generation of two-dimensional designs in architecture and the arts in general. Several plans, elevations and sections in architectural design can be understood in terms of their relation to one or more underlying cyclic or dihedral group structures. Alternatively, music scores exhibiting canonic or fugal writing in music composition can be understood in terms of their relation to specific underlying cyclic or dihedral group structures. Designs in these spatial and sound systems generally do not necessarily reveal immediately their underlying structures; a complex array of relations and ordering schemes are typically introduced to relate several parts to one another and to the overall configuration to interpret existing designs or produce new designs from scratch. The four algebraic structures that exemplify the cyclic and dihedral symmetries of the Euclidean plane to the Euclidean space are discussed in this paper. Four instances of them, all of order 8, C8, D4, C4xC2, D2xC2, are represented in terms of partial order lattices and correlated with the seven possible geometric structures of prismatic symmetry. It is suggested that the same lattices can be used for the computation of designs with no global symmetry but with a wealth of rich spatial connections among their parts. A simple case study is given in the end to illustrate the notion of generative applications of lattices and nested underlying structures in design.

Economou, A. (2001). C2C2C2: Pythagorean structures in design. In M. Burry, S. Datta, A. Dawson, & J. Rollo (Eds.), Proceedings of Mathematics and Design 2001: The Third International Conference: Digital . Hand . Eye . Ear . Mind (pp. 128-138). Deakin University.

Abstract:
The structure of the direct product group C2C2C2 is examined in detail. The foundations of the group structure in Pythagorean arithmetic are briefly presented, the realization of the group in three-dimensional space is discussed and illustrated, and some notes are suggested for its implication in music notation. A special emphasis is given in four different decompositions of the group and all subsets are enumerated and illustrated. Various applications of this structure are suggested for further systematic studies in the analysis and synthesis of the architecture of form.

Economou, A. (2000). Shape grammars in architectural design studio. In W. Mitchell & J. Fernandez (Eds.), Proceedings of the 2000 ACSA Technology Conference: The Intersection of Design and Technology (pp. 75-81). Association of Collegiate Schools of Architecture.

Abstract:
Shape grammars applications have been developed in various fields including architecture, landscape, engineering, painting, furniture design, ornamental design, and others. Still, despite their success in formal analysis they have not yet managed to establish themselves in the core of architectural discipline and education, the design studio. Several exciting proposals and projects have been generated at UCLA, MIT and Carnegie Mellon during the last decade but it seems that a range of possibilities is still out there for creative usages and systematic applications of grammars in pedagogy and studio. This paper addresses some of these possibilities and raises some questions regarding the applicability of this powerful paradigm in design theory in creative work and particularly in architectural design studio showcasing some recent work at the School of Architecture at the Georgia Institute of Technology.

Economou, A. (2000). Spatial canons and fugues. In Proceedings of Greenwich 2000: Digital Creativity Symposium (pp. 53-60). University of Greenwich.

Abstract:
The foundations of a contrapuntal composition with shapes are considered. Rules in counterpoint in music composition are briefly presented and correlated with rules in formal composition with shapes in architectural design. In this context, spatial canons and fugues are defined as specific spatial studies generated by spatial relations and shape rules defined in terms of spatial relations. Some Java applets and architectural examples are given in the end.

Economou, A. (1999). Counting, coloring, and computing: Lessons from the kindergarten. In J. Barrallo (Ed.), Arts, Mathematics and Architecture: Proceedings of the First International Conference (pp. 147-155). The University of the Basque Country.

Abstract:
A systematic inquiry into some of the geometric possibilities inherent in Frederick Froebel's kindergarten method. A series of studies pertaining to counting, coloring and computing issues is proposed. The Froebel building gifts are used as samples for this approach. The symmetry groups and subgroups of these building blocks are identified and their conjugacy classes are established. The cycle indices of the permutation groups of the blocks are used within Polya's theorem of counting to provide all non-equivalent perfect coloring schemes for the blocks. A catalogue of all n-colorings of the blocks for n≤3 is provided as well and its application within a specific computational framework is suggested in the end.

Economou, A. (1998). The symmetry of the equal temperament scale. In J. Barrallo (Ed.), Mathematics and Design 98: Proceedings of the Second International Conference (pp. 557-566). The University of the Basque Country.

Abstract:
The structure of the dodecagon is postulated as a model for the structure and the symmetry properties of the equal temperament scale. The cycle index of the permutation group of the vertices of the dodecagon is used in Polya's theory of counting configurations non-equivalent with respect to a given permutation group; the numbers of structurally differentiated scales with n = 0,1,2,...,12 notes are specified. The symmetry properties of the regular n-gons for n≤6 are used in the decomposition of the scalar patterns. A complete enumeration of all scales, illustrated by a mapping of the scalar patterns on the plane, is presented in the end.