1. Introduction

 Architecture and design have always involved a search for general laws of beauty. Is beauty in the eye of the beholder or does it come about through intrinsic properties of space? Three general principles: repetition, harmony, and variety lie at the basis of beautiful designs. Repetition is achieved by using a system that provides a set of proportions that are repeated in a design or building at different scales. Harmony is achieved through a system that provides a small set of lengths or modules with many additive properties which enables the whole to be created as the sum of its parts while remaining entirely within the system. Variety is provided by a system that provides a sufficient degree of versatility in its ability to tile the plane with geometric figures. Any system that provides the means to attain these objectives has a chance to produce designs of interest. In order to achieve these objectives, architects through the ages have used various systems of proportion. A system based on Ö 2 was used to create ancient Roman architecture [Kappraff 1996a,b]. There is evidence that this root 2 system was employed during the Renaissance by Michelangelo in his creation of the Medici Chapel [Williams 1997]. A system of proportion based on the musical scale was used during the Renaissance by the architects Leon Battista Alberti and Andreas Palladio [Wittkover 1971], [Kappraff 1991]. In modern times Le Corbusier created a successful system of proportions referred to as the Modulor based on the golden mean f where f = (1 + Ö 5)/2 [Le Corbusier 1968], [Kappraff 1991], [Kappraff 1996a,b]. In this paper we shall present the mathematics behind a system of proportions based on Ö 3. There is evidence that Andreas Palladio used this system in his architecture [Wassell 1998]. We shall develop the mathematics and geometry of the root 3 system within the context of a general theory of systems of proportion. It will be shown that each system of proportions gives rise to a sequence of 1's and 0's referred to in the study of dynamical systems as symbolic dynamics. Proportional systems based on f , Ö 2, and Ö 3 were the principal systems used to create the buildings and designs of antiquity [Kappraff 1996a,b], [Nicholson 1998]. We shall show that these proportions provide the simplest systems exhibiting the properties of repetition and harmony. Root2 and root3 geometries also have connections to the symmetry groups of the plane [Coxeter 1973].