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The Relationship Between the Fractal Dimension of Plans And ARCHITECTURE SCIENCE, No. 4, pp.21~44, December 2011 The Relationship between the Fractal Dimension of Plans and Elevations in the Architecture of Frank Lloyd Wright: Comparing the Prairie Style, Textile Block and Usonian Periods Josephine Vaughan1* and Michael J. Ostwald2 1 Research Academic, University of Newcastle, Australia 2 Professor of Architecture, University of Newcastle, Australia *Corresponding author Email: [email protected] (Received Jun. 15, 2010; Accepted Mar. 4, 2011) ABSTRACT Fractal geometry was first used as a quantifiable method for analyzing the visual complexity of a building in the mid 1990’s. Since then the method has been repeated many times – by scholars using both its manual and computational variations – to mathematically analyse architectural elevations. As a result of this practice, a large amount of comparative data describing the visual complexity of the exterior facades of famous buildings has become available. However, very little research has been undertaken into the fractal analysis of architectural plans and even less into the relationship, if any, that exists between visual complexity in plans and elevations. In order to test whether there is any pattern to the relationship between visual complexity in plans and associated elevations, this paper undertakes a comparative fractal analysis of 15 houses by Frank Lloyd Wright. Importantly, these houses are drawn from three distinct stylistic periods in Wright’s work. This suggests that, if the relationship between plan and elevation is significant, then clear trends should be discernable for each period. This paper describes the standard method of fractal analysis along with its computational variation that is used to calculate the present results. A description of each of the houses is then provided, followed by the presentation of the fractal dimension results. Finally, the paper analyses these results in detail and provides a conclusion regarding the proposed relationship between the visual complexity of plans and elevations in Wright’s domestic architecture. KEYWORDS：Frank Lloyd Wright, Fractal Analysis, Computational Analysis, Visual Complexity 21 ARCHITECTURE SCIENCE, No. 4, December 2011 1 Introduction In the 1970’s the scientist and mathematician Benoit Mandelbrot identified and described a new type of non-Euclidean geometry. Known as fractal geometry, this approach accommodated, for the first time, the mathematical measurement of shapes which were “grainy, hydralike, in between, pimply, pocky, ramified, seaweedy, strange, tangled, torturous, wiggly, wispy [and] wrinkled” (Mandelbrot, 1982). The application of this new, non-linear geometry has since been embraced in many fields and in 1996 it was first applied to architectural analysis (Bovill, 1996). The use of fractal geometry in architectural analysis is potentially important for designers, scholars and historians because it provides a numerical indication of the characteristic visual complexity of a building. This previously unavailable quantitative information allows for comparisons to be made between the visual complexity of projects with similar programs or architect’s works from defined periods. The fractal analysis of architecture is still a relatively new approach (Ben Hamouche, 2009) and fractal analysis has only recently been established as a method for the consistent analysis of the characteristic visual complexity of architectural elevations. For example, with the support of new computational methods, data is now available for the first time describing the visual complexity of the exterior facades of a large number of famous 20th and 21st century buildings (Bovill, 1996; Lorenz, 2003; Ostwald, et al., 2008; 2009). However, while the fractal analysis of architectural elevations has become an accepted quantitative method, its capacity to capture the full three-dimensional complexity of a building remains limited. In response to this situation, the present paper tests the idea originally proposed by Bovill (1996) that the fractal analysis of floor plans, in combination to the standard analysis of elevations, will reveal previously unseen patterns in an architect’s approach to three-dimensional form. Bovill (1996) suggests that the comparison between formal complexity in plan and in elevation may be informative. He calculates the fractal dimension of several plans to show how they differ, but he does not develop this idea any further. In response to this suggestion, Ostwald and Vaughan (2010) undertook a fractal analysis of the plans and elevations of Peter Eisenman’s House VI and John Hejduk’s House 7. The value of that research is that it demonstrates the method used to construct a comparison. However, the results of the analysis were too limited to offer any conclusions about patterns or relationships. The first, small-scale attempt to test Bovill’s proposal was produced by Vaughan and Ostwald (2010) when they constructed a fractal comparison between the elevations and plans of a small number of houses by Frank Lloyd Wright. The results of this first test, while limited in scope, suggest the possibility that a pattern might be uncovered if a larger set of data was used. The findings of that paper were the catalyst for the present research which uses both the consistent method and a statistically valid sample size. 22 The Relationship Between the Fractal Dimension of Plans and Elevations in the Architecture of Frank Lloyd Wright: Comparing the Prairie Style, Textile Block and Usonian Periods In order to test the idea that the plans and elevations of a building may be related in terms of visual complexity, this paper undertakes a computational fractal analysis of the plans and elevations of 15 houses by Frank Lloyd Wright. Wright, one of the world’s most famous architects, designed many houses during his almost 70 year professional career and he was responsible for developing three distinct stylistic approaches to housing; the Prairie Style, the Textile Block and the Usonian. Moreover, scholars such as MacCormac (1981) have argued that “the exterior forms of Wright’s buildings usually project internal spaces”; suggesting that the plan and the elevations of Wright’s buildings are visually connected. In combination, the proposed connection between plan and elevation, along with the distinct stylistic periods, makes Wright an ideal candidate for this analysis. Three sets of five houses have been selected for analysis in this paper. These include five Prairie Houses and five Usonian houses – whose elevations have been previously analysed (Vaughan and Ostwald, 2010) along with new calculations for the elevations and plans of five houses from Wright’s Textile block period. In total, this means the analysis of 56 elevations and 30 plans to determine whether any pattern exists between the two views in each of the stylistic periods. This paper commences with an overview of the box-counting method of fractal analysis and its computational variation which is used to calculate the data for the present research. A description of each of the houses analysed is then provided, followed by the presentation of the fractal dimension results. Finally, the paper reviews these results in detail and provides a conclusion regarding the comparison of the visual complexity of plans and elevations. It is important to note that this paper does not provide a detailed argument for the validity of fractal analysis. Several books and many papers have debated this question extensively with opinions ranging from the positive or sanguine to the cautious and critical. On the positive side of the debate are scholars like Bovill (1996), Bechhoefer and Appleby (1997), Makhzoumi and Pungetti (1999), Burkle-Elizondo, Sala and Valdez-Cepeda (2004) and Sala (2006). These argue that the fractal analysis method has the capacity to produce startling insights into architecture, or be used to analyse the contextual fit of a building, and even explain the psychological reactions people have to buildings (Taylor, 2005). At the other end of the spectrum, Ostwald (2001; 2003) has shown how these claims may be philosophically challenged, Lorenz (2003), Ostwald and Tucker (2007) have debated the merits of Bovill’s method and Ostwald and Wassell (2002) have corrected the flawed mathematics at the heart of several attempts to use complexity science and fractal geometry to analyze architecture. Despite this debate, it wasn’t until Lorenz (2003) demonstrated a wider application of the method, and Ostwald, Vaughan and Tucker (2008) developed a computational version, that comparable and consistent results were produced for the first time. The authors of the present paper are clearly responsible for much of the more cautious end of the spectrum of research, arguing that reliable and consistent findings must be published in order to refine the limits, and determine the long-term viability or usefulness, of the fractal method. 23 ARCHITECTURE SCIENCE, No. 4, December 2011 1.1 The box-counting method There are several methods for calculating the approximate fractal dimension of an image or object. The most common of these is the “box-counting” approach (Mandelbrot, 1982). The box-counting approach – for calculating the approximate fractal dimension of a two-dimensional image – takes as its starting point a line drawing, say the façade of a building (see figure 1). A grid is then placed over the drawing and each square in the grid is analysed to determine whether any lines from the façade are present in it. Those grid boxes that have some detail in them are then marked. This data is then processed using the following numerical values; (s) = the size of the grid N(s) = the number of boxes containing some detail 1/s = is the number of boxes at the base of the grid Next, a grid of smaller scale is placed over the same façade and the same determination is made of whether detail is present in the boxes of the grid (Figure 3). A comparison is then made of the number of boxes with detail in the first grid (N(s1)) and the number of boxes with detail in the second grid (N(s2)).
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