This paper examines a range of geometric concepts of importance to the further understanding of two- and three-dimensional design. A brief explanation is given of symmetry in patterns and tilings, and attention is focused on a particular set of polyhedra, known as the Platonic solids. The difficulties encountered in attempting to apply two-dimensional repeating designs to regular polyhedra, avoiding gap and overlap and ensuring precise registration, are recognized. The results of ongoing research at the University of Leeds are presented. The symmetry characteristics of importance to the process are identified, and the patterning of each of the five solids is explained and illustrated. Avenues for further research are suggested.