Architectonics of symmetry in twentieth century architectural and music theories.

摘要

Formal methods and techniques are developed for the analysis and synthesis of designs with respect to their symmetry properties; the formal theory is applied mathematics, in particular group theory and combinatorics. The algebraic structure of the symmetry groups is discussed in detail and the emphasis is given in the use of the lattice of symmetries of subshapes of a design, and the use of the cycle index of a permutation group of a set of subshapes of a design.; The advantage of the use of the partial order of sub-symmetries in the analysis of form is that seemingly complex designs can be understood as superimposed layers of simpler designs. Designs are decomposed in subshapes that are described by specific symmetry groups and are ordered in lattices that show the nesting of symmetries within symmetries.; The advantage of the use of the cycle index or a permutation of a set of subshapes of a design is exemplified within Polya's theorem of enumeration of non-equivalent configurations with respect to a given permutation group; this theorem is applied to provide the counting techniques in the enumeration of distinct configurations of subshapes of the Froebel's building gifts and the equal-temperament twelve-tone scale.; The emphasis is given in three-dimensional representations and cross-modal relationships; the formal methods and techniques that are developed here are applied in architectural design research and music research to show one more aspect of the sought after relationship between architecture, music and mathematics. Symmetry is intimately related to compositional development and its multiple manifestations in various Euclidean design worlds, visual and sound alike, provides an inexhaustible source of inspiration for the designer and the researcher alike.