### Architecture Case Study in Transformity Factorization

#### Abstract

This paper studies the Giannantoni factorization of H.T. Odum’s transformity into dissipative and generative components. A dissipative component of architecture was developed in the author’s paper “ ‘Tropical’ Emergy and (Dis-) Order” at the 4th Biennial Emergy Research Conference, and is related to the number of surfaces used up in architectural construction, for example making walls out of bricks. A generative component was developed in the author’s paper “An Algorithm to Measure Symmetry and Positional Emergy of n Points,” presented at the 2007 annual meeting of the American Mathematical Society, New Orleans, LA and included the the ISSS 2007 Bulletin; the generative component is related to the number of equal distances created between different parts of a structure. There is some evidence of ordinality; for example higher-dimensional structures can have orders of magnitude more symmetry. Emergy maximization is analyzed as a constrained calculus problem which for maximization requires middle values of both dissipation and generation. For example a placement of bricks around a yard in a highly symmetric fashion may have high symmetry but if they are not connected , will not lead to a desirable architectural structure. Similarly connectling the bricks into haphazard walls may have high dissipation but without some symmetry of construction into regular structures such as rooms, will be considered a waste of materials. Some other questions such as evolution of biological and animal structure are discussed.