1 INTRODUCTION

The symmetry group of a two-dimensional design is either finite or infinite. The finite group of plane symmetry is called the point group. Spatial transformations take place in a fixed point or line. Within this category, bilateral symmetry, by which structures, building elements, and rooms are laid out in a mirrored balance along a central vertical axis, prevails over the building façade in most classical and modern buildings. The infinite groups are the symmetries of infinite patterns such as one directional translation of the frieze or two directional translations of wallpaper patterns. Wallpaper patterns on building facade are found abundant in the housing site plans, and facades of high-rise buildings in which translation of identical windows line up in a coordinate plane.

Using symmetry, complex designs as a whole can be decomposed into parts for analysis. Conversely, the same principles can be used to recombine the parts into multiple sequential designs. This study further develops a parallel procedure to study the potential application to building façades with regard to the seven frieze groups.
 
 

2 SEVEN FREEZE GROUP

Simply put, a frieze pattern consists of two-dimensional motifs repeating in one direction only. There are seven distinct groups that can be the symmetry group of a frieze in an infinite sequence of equidistant points arranged in a linear series on the Euclidian plane (March and Steadman, 1971; Martin, 1982; Park, 2004). Thus, every frieze pattern belongs to one of only seven mathematically possible patterns. Five types of isometries occur in frieze groups: translation (T), half-turn (S: rotation through 180 degrees), vertical reflection (R_v), horizontal reflection (R_h), and glide reflection (G). Different possible combinations of above isometries form seven distinct symmetry groups together.

3 BUILDING FAÇADE ANALYSIS: SOME EXAMPLES

A building façade is characterized by the way building elements are arranged. Using symmetry, one can better understand its composition through pattern recognition (Park, 2000, 2001). While numerous symmetric facade examples exist, most are based on bilateral symmetry. The following three designs provide different uses of symmetry in floor plan and façade, summarizing a shift from the dominating role of symmetry in classical architecture to the modern. The first is Andrea Palladio’s Villa Emo at Fanzolo designed and built during the decade 1555-1565. The second is early 20^th century design, the Villa Moller designed by Adolf Loos. The last is the Mackey Duplex designed by Rudolph M. Schindler in 1939.

The use of symmetry as a design principle is especially evident in the villas of Palladio. Symmetry at Palladio’s time was viewed as "ideal" for both the design of the building and the grounds, characterized by a central axis with mirror image components on either side. Villa Emo for instance was set along a vertical axis in the floor plan and façade. On the other hand, symmetry plays a double game in the work of Loos and Schindler, marking a decisive break with classical historicism. In Loos’ Moller house of 1928, in spite of freeing the façade from the shackles of historical ornament, the front façade retains a clear set of symmetric guidelines (Gravagnuolo, 1982). Nevertheless, the unusually strong façade symmetry is not carried through to the interior design; the order that organizes the exterior of the building and that of its interiors are heterogeneous characters. While masked with simple but elegant symmetric façade, there is no order on the inside floor plan. The application was reversed in Schindler’s Mackey Duplex (Sheine, 2001). The floor plan is set along an axis, while the façade displays no hint of symmetry. Basically, the symmetry of the floor plan is unexpected from such a facade. It is typically difficult in Schindler’s design to predict the space forms from floor plans.
 

4. FAÇADE SYNTHESIS

Let us consider a finite series of design motifs including doors, windows, balconies, recesses, stairs, etc. to create a row house façade. These elements in multiple layers will be aligned and superimposed on top of each other with respect to the seven frieze groups, creating a global asymmetric facade. The façade is constructed in such a way that the elements associated with respect to the symmetry coincide respectively. Let’s imagine that we design rows of houses on a regularly divided lot along a street façade. Then can we array those motifs on the façade with respect to the frieze symmetry? In this paper, we provide a constructive example of such a design. 
 
 

5. CONCLUSION

This research has sought to highlight the potential for the seven frieze groups of symmetry to serve as guiding principles for creating unique building facades. The mathematical notion of symmetry serves for the analytic tool of frieze patterns of various symmetries and also for the construction of a new facade design of building. The constructed building façade leads to the conclusion that with the seven frieze groups of symmetry, a new possibility in creating a variety of dynamic building façades is shown. By combining a set of different design elements with the principle, a variety of designs can be further generated. When a series of row houses are designed and combined as such, the street facade of the pedestrian level can become more elaborate, providing dynamic townscape.
 
 

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