Economou, A., Hong, T. C. K., & Newton, R. (2024). Shape meets Euclid: Integrating shape computation with ruler and compass procedures. Automation in Construction, 165, 105562. https://doi.org/10.1016/j.autcon.2024.105562

Abstract:
The recent success of a class of mechanical applications of shape rules in CAD shape grammar interpreters has produced a renewed interest in the design and support of shape embedding and shape rewrite technologies. Despite this, some fundamental constructions in rule-based modeling, namely, any of Euclid's straightedge and compass constructions, appear awkward to simulate within the fuse-embed cycle of a shape grammar interpreter. It is suggested here that this problem of modeling Euclid's straightedge and compass constructions can be efficiently addressed with the introduction of a new schema rule leveraging symbolic representations of shape. Two applications based on the usage of three instances of this schema rule are given to show the simulation of straightedge and compass constructions in a general purpose 2D CAD shape grammar interpreter, the Shape Machine for Rhino. A reflection upon the usage of rule schemas in shape grammar interpreters is given in the end.   

Hong, T. C. K., & Economou, A. (2022). What shape grammars do that CAD should: The 14 cases of shape embedding. Artificial Intelligence for Engineering Design, Analysis and Manufacturing, 36(E4), 1-20. https://doi.org/10.1017/S0890060421000263

Abstract:
The process of shape embedding, that is, the inquiry whether for two shapes u and w there is a transformation f that embeds the shape f(u) in w is the most critical – and elusive – process for all shape grammar interpreters implemented within CAD systems. This paper identifies three major challenges underlying the implementation of shape embedding within a CAD environment and proposes three corresponding solutions. The focus is given here in the calculation of shape embedding for shapes made up of lines in a 2D CAD environment to provide the testbed for the mechanisms proposed. The integration of these three sets of algorithms into one coherent framework produces the blueprint of Shape Machine, a general-purpose shape grammar interpreter implemented in Rhinoceros, a NURBS 2D/3D CAD software. Some preliminary thoughts regarding the impact of the proposed shape-rewrite modeler in CAD industry and more broadly, in design education are given in the end.

Hong, T. C. K., & Economou, A. (2022). What shape grammars do that CAD should: The 14 cases of shape embedding. Artificial Intelligence for Engineering Design, Analysis and Manufacturing, 36(E4), 1-20. https://doi.org/10.1017/S0890060421000263

Abstract:
The process of shape embedding, that is, the inquiry whether for two shapes u and w there is a transformation f that embeds the shape f(u) in w is the most critical – and elusive – process for all shape grammar interpreters implemented within CAD systems. This paper identifies three major challenges underlying the implementation of shape embedding within a CAD environment and proposes three corresponding solutions. The focus is given here in the calculation of shape embedding for shapes made up of lines in a 2D CAD environment to provide the testbed for the mechanisms proposed. The integration of these three sets of algorithms into one coherent framework produces the blueprint of Shape Machine, a general-purpose shape grammar interpreter implemented in Rhinoceros, a NURBS 2D/3D CAD software. Some preliminary thoughts regarding the impact of the proposed shape-rewrite modeler in CAD industry and more broadly, in design education are given in the end.

Okhoya, V. W., Bernal, M., Economou, A., Saha, N., Hong, T. C. K., & Haymaker, J. (2022). Generative workplace and space planning in architectural practice. International Journal of Architectural Computing, 20(3), 645-672. https://doi.org/10.1177/14780771221120580

Abstract:
Generative design is emerging as an important approach for design exploration and design analysis in architectural practice. At the interior design scale, although many approaches exist, they do not meet many requirements for implementing generative design in practice. These requirements include the need for end-user accessible tools and skills, rapid execution, the use of standard inputs and outputs, and being scalable and reusable. In this paper, we describe a hybrid process that uses both space allocation and shape grammar algorithms to solve workplace and space planning interior design problems. Space allocation algorithms partition spaces according to program requirements while shape grammar automates the placement of inventory and the production of high-resolution drawings. We evaluate using three real world example projects how this hybrid approach meets the identified requirements of generative space planning in architectural practice.

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 presented 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 presented in the end.  

Yu, Y., Hong, T. K., Economou, A., & Paulino, G. (2021). Rethinking origami: On the generation of origami patterns in Shape Machine. Computer-Aided Design, 137, 103029. https://doi.org/10.1016/j.cad.2021.103029

Abstract:
A generative description of Mies van der Rohe’s courthouse design language is presented in the form of a three-dimensional parametric shape grammar and its significance in the discourse of courthouse building type is discussed. The implementation of the grammar automates the generations of existing and newly emerged Miesian courthouse design variations in the forms of three-dimensional models. These design variations are systematically examined in terms of their formal and functional properties to evaluate the consistency, flexibility and potential of the grammar towards a generative theory of the courthouse design at large.

Quan, S. J., Park, J., Economou, A., & Lee, S. (2019). Artificial intelligence-aided design: Smart design for sustainable city development. Environment and Planning B: Urban Analytics and City Science, 46(8), 1581-1599.

Abstract:
Current planning and design decision support systems show limitations in the integration of design, science, and computation. Planning support systems with manual design and post-design evaluations impose major challenges in exploring huge design spaces. Generative design systems largely neglect the wicked nature of design problems and lack appropriate representation methods and simulation tools at the urban scale. To tackle those challenges, this research developed a Smart Design framework featuring urban design decision-making reinforced by artificial intelligence-aided design (AIAD). The Smart Design framework treats urban design as an emergent pattern formation processes with contextualized and dynamic objectives. The framework integrates design thinking, advanced artificial intelligence search techniques (e.g. genetic algorithms), urban scale performance simulations, and participation to better inform decision-making. Through four major stages, the framework combines the ideas of Science for Design and Design in Science. The significance and potential of the Smart Design framework are demonstrated in an urban design study of Gangnam superblocks in Seoul, South Korea. The study explores sustainable urban forms in the high-density, super-complex, and hyper-consumptive environment of Gangnam, which can also be found in many other Asian contexts. The case study illustrates how the framework identifies design solutions for sustainable city development in the process of participatory decision-making through the co-evolution of design problems and solutions.

Economou, A., & Grasl, T. (2018). Paperless grammars. In J. H. Lee (Ed.), Computational studies on cultural variation and heredity (pp. 139-160). Springer.

Abstract:
A workshop in formal composition using machine-based specifications of parametric shape rules is presented. The workshop is structured along two different trajectories: one starting from existing grammars and one starting from scratch, and both in a rising complexity in the specification of the rules and the ways they affect design. Rules, productions and designs in corresponding languages illustrate the findings. The speculation on a new design workflow whereas the designers seamlessly design and test their rules within their design processes is briefly discussed in the end.

Grasl, T., & Economou, A. (2018). From shapes to topologies and back: An introduction to a general parametric shape grammar interpreter. Advances in Implemented Shape Grammars: Solutions and Applications, Artificial Intelligence for Engineering Design, Analysis and Manufacturing, 32(2), 208-224.

Abstract:
The shape grammar formalism has been discussed theoretically extensively. Recently there has been increased activity in implementing shape grammar interpreters, yet there is a lack of implementations that support parametric rules and emergence. Here the structure of a general parametric shape grammar interpreter is discussed in detail. The interpreter is based on graph grammars. It supports emergence, parametric rules, and numerous types of geometric objects. The shape grammar engine, an agent-based rule selection system and several implementations based on the engine are discussed.

Eloy, S., Pauwels, P., & Economou, A. (2018). Promises of shape grammars. Advances in Implemented Shape Grammars: Solutions and Applications, Artificial Intelligence for Engineering Design, Analysis and Manufacturing, 32(2), 131-137.

Abstract:
This paper introduces the special issue “Advances in Implemented Shape Grammars: Solutions and Applications” and frames the topic of computer implementations of shape grammars, both with a theoretical and an applied focus. This special issue focuses on the current state of the art regarding computer implementations of shape grammars and brings a discussion about how those systems can evolve in the coming years so that they can be used in real life design scenarios. This paper presents a brief state of the art of shape grammars implementation and an overview of the papers included in the current special issue categorized under technical design, interpreters and interface design, and uses cases. The paper ends with a comprehensive outlook into the future of shape grammars implementations.

Ligler, H., & Economou, A. (2016). Entelechy revisited: On the generative specification of John Portman's architectural language. Environment and Planning B: Urban Analytics and City Science. https://doi.org/10.1177/0265813516676489

Abstract:
John Portman’s work attracts significant commentary, although the focus is typically on the commercial and social aspects of his work as opposed to the actual designs and their related architectural implications. The obvious place to start unpacking his contribution is in his widely recognized and published commercial portfolio, yet he maintains that his design principles are found in his personal domestic work. Here, his 1964 residence Entelechy I is analyzed to inform the development of a parametric shape grammar that generates the original design as well as a series of variations. The goal of this research is to engage Portman’s architectural philosophy and constructively assess his claims of its implicit relationship to his work to date. Key rules suggesting his principles and anticipating his ongoing architectural contribution are outlined. The structure provided by shape computation, involving both shape rules and rule schemas, is positioned as the theoretical basis for an ongoing study of transformations within Portman’s language.

Grasl, T., & Economou, A. (2013). From topologies to shapes: Parametric shape grammars implemented by graphs. Environment and Planning B: Planning and Design, 40(5), 905-922.

Abstract:
A method for implementing parametric shape grammars is presented. Subgraph detection is used to find subshapes. Parametric shapes are described by restricting topologies. Shape grammars provide a powerful generative approach for the description, interpretation, and evaluation of existing designs as well as for new designs. Still, while the shape grammar formalism has produced a long and excellent series of formal studies, the corresponding efforts for automation of the tasks that shape grammars accomplish have not met with the same success. Two computational challenges of implementing shape grammars are detecting emergent shapes and enabling parametric rules. A general solution to both is presented here in terms of a graph theoretic model. Typically graphs are seen as multidimensional data structures that, with an appropriate embedding, can formally structure and visually resemble the shapes they represent. Still, the encoding of shapes in terms of these alternative graph representations is not straightforward. Various approaches to representing shapes as graphs are described here and one of them is selected as the most useful for a shape grammar interpreter. The resulting formalism uses two parallel representations, a graph-theoretic one that is used for the subshape detection, and a visual one that illustrates the shape and which can be tested for intersections. The implementation presented here also maps the shape rules into graph rules. Once a rule has been selected, matching subgraphs are mapped back to shapes for further user interaction, such as picking a shape to which to apply the rule. The shape grammar library has been implemented in C# and it is code-named GRAPE to playfully recall the parallel representations of ‘GRAphs’ and ‘shAPEs’.

Economou, A., & Grasl, T. (2012). Spatial sieves. In S. Kanach (Ed.), Xenakis matters: Contexts, processes, applications (pp. 287-313). Pendragon Press.

Abstract:

The sieve is a powerful formal tool developed by Xenakis to create integer-sequence generators that can be used for the generation of various numerical patterns to represent pitch scales, rhythm sequences, as well as patterns of loudness, density, timber and so forth. The key idea in the design of sieves is the notion of a vocabulary of elementary modules that can be combined one with another under operations of union and intersection in various ways. Sieves produce a series of patterns that rise from the simplest ones possible to highly expressive structures, simulating almost random-like linear distributions of points on a line. Significantly, Xenakis asserted that his algebra of sieves revealed insights into fundamental aspects of composition by “giv[ing], when it exists, a more hidden symmetry derived from the decomposition of a modulus”. It is this implicit power of this formalism to reveal hidden symmetries and emergent structures that is revisited and reworked here. The formal tools for the inquiry on sieves are taken from La-grange’s theorem of subgroup numerical relation and Pólya’s theorem of counting non-equivalent configurations with respect to a given permutation group. Here the theorems are applied within an automated computational framework to produce all possible non-equivalent configurations that can be extracted from a regular division of a linear module into an n-number of aliquot parts. The software automatically generates the cyclic index of any symmetry group of any finite linear shape, allows for the computation of a figure inventory of variables upon the cycle index, and visualizes the result of all non-equivalent configurations with correspond-ing isomorphic two-dimensional subshapes of regular n-gons. Significantly, the software illustrates all such possible configurations in partial order lattices as well, in order to show the nested orders and symmetries of the spatial sieves.

Grasl, T., & Economou, A. (2012). Sort machines. In J. Halatsch (Ed.), Digital urban modelling and simulation (pp. 123-143). Springer-Verlag.

Abstract:
A graph grammar for the generation of topologies for the U.S. federal courthouse typology is introduced. The goal is to test and implement potential modelling techniques to account for a constructive description of courtroom organizations. The work reports on a specific graph-theoretic model that can generate and enumerate all possible topologies for the courtroom organization typology given a specific methodological framework. The model includes: a) an initial graph representing an ensemble of two or three zones grouped around one, two or more courtrooms; b) a graph theoretic rule-set covering all legal combinations and modifications of the initial basic element; c) a rewriting process to apply a variable number of rules and generate rudimentary descriptions of possible courthouse designs in terms of accessibility graphs; and d) an expansion of the nodes of the graphs into detailed spatial descriptions. A catalogue of possible configurations and a nomenclature for these configurations is given in the end.

Din, E., & Economou, A. (2010). Surface symmetries: The Smith House revisited. International Journal of Architectural Computing, 8(4), 485-506.


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. The object of analysis is polemically chosen here to be the Smith House by Richard Meier, a design that clearly exemplifies formal qualities of late modernism architecture such as abstraction, layering, complexity, depth, collage and so on, all aesthetic categories appearing impenetrable to a systematic and rigorous analysis using existing group theoretical formal methods. The paper is divided into four sections: In the first section the introduction and motivation of the work are given. In the second section the formal model of notational representation and subsymmetry analysis is described in detail. In the third section an application of this methodology is given for the Smith House including visual computations capturing sub-symmetry relations for all parts of the house and a complete catalogue of recombination of all symmetry parts of the house. Lastly, in the fourth section the discussion is given on the degree to which the symmetry decompositions can support visually the established discourse on the house and the possibility of using such formal tools in CAD applications.

Economou, A., & Swarts, M. (2006). Performing Palladio. International Journal of Architectural Computing, 4(3), 47-61.

Abstract:

The relevance of music theory as an interpretive framework for the understanding of Palladio’s work has been one of the most debated subjects in the realm of architectural theory and criticism. Typically, the debate is quite abstract and it focuses on possible mappings between the ratios found in Palladio’s plans and corresponding ratios used in contemporary musical temperaments. The paper here rather focuses on the actual performance of the ratios found in Palladio’s work and the implications of this performance, melodic and harmonic, for the perception of the space for a situated observer/performer. To that extent the study suggests a model of mapping between space, sound and color and correlates that with polygon partition theory to simulate movement within a space. A brief account of the computer implementation with game engines technology is provided in the end. All examples to test these ideas are based on Palladio’s Villa Capra.

Economou, A. (2006). Tracing axes of growth. In L. Albertazzi (Ed.), Visual thought: The depictive space of perception (pp. 351-365). John Benjamins Publishing Company.

Abstract:
The notion of congruence as a specific mapping that does not change the dimensions of bodies in space is employed here for the discussion and representation of a specific subset of spaces that have one axis of growth. This specific class of designs is ubiquitous in architectural design and in the arts in general and yet a systematic effort of classifying and illustrating these structures for design purposes has yet to be undertaken. This can be partially explained because all classification schemes of symmetry structures require a mathematical sophistication not required or found any more in architectural or art curricula. Furthermore, the complexities of interactions of isometric transformations in three-dimensional space do not lend themselves easily to intuitive approaches to enumeration of classes of designs by simple trial and error efforts. Interactions of isometric transformations in two-dimensional space are relatively easy to comprehend and construct; and several theoreticians, designers and architects have been credited for their attempts to classify and illustrate corresponding symmetries in the plane with zero, one or more axes of growth in the plane. Still, no artists or architects have been credited yet with a systematic construction of designs that take advantage of properties of growth along an axis in three-dimensional space. This work looks closely at a specific class of three-dimensional designs that have one axis of growth and presents all possible algebraic structures that capture the symmetries of these designs. A specific set of designs is discussed, the symmorphic designs, and is used as a framework to derive the non-symmorphic designs and to complete all three-dimensional linear structures; the complete catalogue of all nineteen space structures that may be generated in this manner is presented in the end.

Economou, A. (2006). Tracing axes of growth. In L. Albertazzi (Ed.), Visual thought: The depictive space of perception (pp. 351-365). John Benjamins Publishing Company.

Abstract:
The notion of congruence as a specific mapping that does not change the dimensions of bodies in space is employed here for the discussion and representation of a specific subset of spaces that have one axis of growth. This specific class of designs is ubiquitous in architectural design and in the arts in general and yet a systematic effort of classifying and illustrating these structures for design purposes has yet to be undertaken. This can be partially explained because all classification schemes of symmetry structures require a mathematical sophistication not required or found any more in architectural or art curricula. Furthermore, the complexities of interactions of isometric transformations in three-dimensional space do not lend themselves easily to intuitive approaches to enumeration of classes of designs by simple trial and error efforts. Interactions of isometric transformations in two-dimensional space are relatively easy to comprehend and construct; and several theoreticians, designers and architects have been credited for their attempts to classify and illustrate corresponding symmetries in the plane with zero, one or more axes of growth in the plane. Still, no artists or architects have been credited yet with a systematic construction of designs that take advantage of properties of growth along an axis in three-dimensional space. This work looks closely at a specific class of three-dimensional designs that have one axis of growth and presents all possible algebraic structures that capture the symmetries of these designs. A specific set of designs is discussed, the symmorphic designs, and is used as a framework to derive the non-symmorphic designs and to complete all three-dimensional linear structures; the complete catalogue of all nineteen space structures that may be generated in this manner is presented in the end.