Geometric Transformations in Design Generation

Abstract We present a method for transforming a simple plane figure or motif into another, which is related to it in a definite way, by applying a group of systematic geometric transformations. The method creates certain classes of star motifs from a basic plane figure. We demonstrate that surface design is an 'atomic' and a 'self-assembly' process, and expansion, figure and motion are the only properties in design which can be directly represented geometrically. By applying a group of fundamental point transformations of the plane, and rearranging the axis of the reflection, we transform a simple figure into a series of complex star motifs. Our ultimate aim is to develop the invariants of design and creative process, and to bring the formal side of art within the purview of mathematics. Self-Organized Motifs In order to gain a quantitative and mechanistic understanding of creative design, it is important to understand the recurrence relations and sequential moves in the transformation process. Although the use of symmetry and geometric transformations to make patterns is well-understood and applied to practical applications at a certain extend Any figure, drawn, painted, printed, embossed, woven, or otherwise executed on a surface, is termed a 'motif', provided there is at least one possible motion which moves the figure, as a whole but not point for point or part for part, back to its initial positio