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Four Algebraic Structures in Design
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Reinventing the Discourse: Proceedings of the Twenty First Annual Conference of the Association for Computer-Aided Design in Architecture (ACADIA)
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Parametric model, Group theory, Symmetry, Configuration, Lattices, Prismatic groups, Shape studies
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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.
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