123ACADIA 2000: Eternity, Infinity and VirtualityIntuitive and Effective Design of PeriodicSymmetric TilesErgun Akleman, Texas A&M University, USAJianer Chen, Texas A&M University, USABurak Meric, Knowledge Based Information Systems, Inc., USAAbstractThis paper presents a new approach for intuitive and effective design of periodic symmetric tiles. Weobserve that planar graphs can effectively represent symmetric tiles and graph drawing provides anintuitive paradigm for designing symmetric tiles. Moreover, based on our theoretical work to representhexagonal symmetry by rectangular symmetry, we are able to present all symmetric tiles as graphsembedded on a torus and based on simple modulo operations. This approach enables us to developa simple and efficient algorithm, which has been implemented in Java. By using this software, design-ers, architects and artists can create interesting symmetric tiles directly on the web. We also havedesigned a few examples of symmetric tiles to show the effectiveness of the approach.1 IntroductionThe symmetric patterns in Alhambra,Granada are probably the most wellknown architectural usage of symmet-ric patterns. In fact, symmetric patternshave been a part of the architecturalworld throughout the history and fre-quently used by almost every civiliza-tion in wallpapers and wall decorations,ceilings, floor tiles, street pavementsand even facades of the buildings asshown in Figures 1.Although, there has been a great inter-est in art and architecture, the theoreti-cal classification of periodic symmet-ric patterns did not began until theearly twentieth century when Russiancrystallographer E. S. Fedorov enu-merated the seventeen two-dimen-sional periodic symmetry groups.These groups today are also known aswallpaper groups, periodic groups or (plane) crystallographic groups (Grunbaum and Shephard1987). Fedorov’s result shows that, mathematically, there are only seventeen distinct types of patternsthat have different symmetries. Since the paper of Fedorov was written in Russian the classification ofthe 17 symmetry groups were not known until the work of Niggli and Polya in 1924.From the designer’s perspective, the most important implication of Fedorov, Niggli and Polya’s workis the identification of the symmetry groups as a set of distinct symmetry operations. This identifica-tion of symmetry operations encouraged artists such as M. C. Escher, F. Briss and K. Mehmedov toFigure 1. Examples ofarchitectural usage ofsymmetric tile(Hargittai andHargittai 1994).124 Intuitive and Effective Design of Periodic Symmetric Tilesdiscover new and interesting patterns. The most famous drawings of symmetric patterns were createdby M. C. Escher [1] and his works are still extremely popular.Although, the knowledge of 17 symmetry groups helps the design of symmetric patterns, it is stilldifficult to find interesting tiles with paper and pen. In other words, even with the knowledge of 17symmetry groups when using only paper and pen to design symmetric patterns the artistic talent isstill important. The idea of using interesting symmetric tiles has a great use in architecture, art,science and education. Therefore, it is important to find interactive computational approaches tosimplify the design of interesting tiles.With the development of computer graphics, many interactive systems to design symmetric pat-terns have been developed. Most of existing symmetric pattern design systems are based on paint-ing paradigm. For designing tiles, these systems are not fundamentally different than paper andpen. In a painting system, even if the users want to make small changes, they must erase existingimages and redraw new ones.One alternative approach is to use drawing paradigm. Kali developed by N. Amenta at University ofMinnesota Geometry Center is an example of a system that uses drawing paradigm (Kali). Sincedrawing is based on objects such as points, lines and polygons that can be translated, scaled orrotated interactively, it is easy to change the shapes of the times. At the first look, this approachseems to be the appropriate choice. Unfortunately, we observe that even drawing paradigm is notappropriate for symmetric tile design. Tiles are polygons that come together without gaps andoverlaps, on the other hand, drawing paradigm supports gaps and overlaps. The user has to beextremely careful not to include any gap or overlap when designing symmetric tiles.We observed that symmetric tiles are inherently graphs embedded on surfaces without edge cross-ings, where lines are edges and line junctions are vertices. In other words, for symmetric tiles theinternal representation must support not only points, lines and polygons but also graphs. We, there-fore, propose that graph drawing paradigm is the most appropriate approach to develop a graphbased algorithmic approach to design symmetric tiles.In this work we have developed a graph-based approach for designing symmetric tiles. In ourapproach, users construct the graphs that represent symmetric tiles by drawing line segments. Eachline segment corresponds to an edge of the underlying graph and endpoints of the lines give thevertices of the graph. The graph representation is constructed by attaching endpoints. In otherwords, when two vertices come close, they will snap together and become one vertex. We alsoprovide intersection prevention to ensure the graph edges never intersect. We also provide an optionthat removes this constraint to enhance flexibility and improve usability.Based on this approach, we have developed a system. By using our system one can produce draw-ings composed of graphs. These graphs can be modified interactively by moving the vertices, byadding new edges, and by dividing edges. In order to create better drawings collisions need to beavoided. For implementation we have chosen the Java programming language. That allows us tomake use of the object-oriented aspects and to make our work available on the Internet.2 Seventeen Planar SymmetriesAs we have mentioned in introduction, there exist seventeen distinct symmetries. In literature(Grunbaum and Shephard 1987), the periodic symmetry groups are called as p1, p2, p4, pm, pmm,p4m, p4m, cm, cmm, pg, pmg, pgg, p4g, p3, p6, p3m1, p31m and p6m. Each one of these symmetrygroups is a collection of symmetry operations: translation, rotation, reflection and glide reflection.The rotations can be either period 2, 3, 4 or 6. These operations are known as isometries,