The Relationship Between the Fractal Dimension of Plans And

Home , Lloyd Wright, Textile block house

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

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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)).

Such a comparison is made by plotting a log-log diagram (log[N(s)] versus log[1/s]) for each grid size. When the process is repeated a sufficient number of times, it leads to the production of an estimate of the

fractal dimension of the façade; it is in fact an estimate of the box-counting dimension (Db), which is sufficiently similar that most researchers don’t differentiate between the two.

The slope of the line (Db) is given by the following formula:

[log(N(s2))  log(N(s1))] Db  [log 1  log 1 ] ( ) ( ) s2 s1

where (1/s) = the number of boxes across the bottom of the grid. (Bovill, 1996)

It is important to note that D is an approximate fractal dimension; sometimes quoted as D(b) meaning the box-counting dimension. Because buildings, and most natural and man made forms, do not have the scalable properties of shapes generated by mathematical equations (Mandelbrot, 1982; Barnsley, 1988) they are not actually fractal and therefore any calculation is an approximation of the degree of visual complexity present in the work over multiple scales of analysis; this is commonly known as characteristic complexity. While this distinction is important, in practice the results are usually described as fractal dimensions. Furthermore, while the box-counting process was traditionally a manual exercise, TruSoft’s Benoit (vers. 1.3.1) program and Archimage (vers. 2.1), a program co-developed by the authors, automate this operation. In this paper the results of the two programs are combined, merging the robustness of the first program with the architectural calibration of the second. In practice the two software programs produce slightly different results, but the degree of difference is relatively consistent.

24 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

Figure 1 The Box-Counting method: West Façade of Wright’s Robie House (Photograph by the authors)

One final comment about the method is that there are several variations that respond to known deficiencies in the approach. The four common variations are associated with balancing “white space”, “starting image proportion”, “line width” and “scaling coefficient”. The solutions to these issues that have previously been proposed by Foroutan-Pour, Dutilleul and Smith (1999) in regards to starting image proportion and white space, by Lorenz (2003) for line width and Ostwald, Vaughan and Tucker (2008) for scaling coefficient, are adopted in the present analysis. The scaling coefficient (the ratio by which the boxes reduce in size) has been set at 2:1.41 (2:√2). This ratio is an ideal compromise between Bovill’s original 2:1 ratio (which produced too few results for a useful comparison) and most other values which produce too much white space around the image (which has a negative impact on the results). 2.2 The computational method While the box-counting method can be applied to any two dimensional architectural image, the computational technique developed by the present authors has been prepared for the testing of sets of designs by an architect. The reason for focussing on sets is that comparable mathematical methods are optimised for considering variations across bodies of data. While the box-counting approach could be used to construct a simple comparison between two architectural elevations, it would at best confirm a person’s intuitive reading of the difference. Thus, for the method to provide more meaningful and useful results a larger body of data is required. For this reason the authors have settled on comparisons of sets of five houses by the same architects. Houses have been chosen as the ideal focus for this analysis because they typically possess a similar scale, program and materiality. While it is possible to analyse smaller sets

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of works, or other building types, the focus on a minimum of five houses ensures the production of consistent, verifiable and statistically valid results. In order to maximize the potential validity of the results, past research using this method has also tended to select completed houses, in preference to unbuilt works, and designs from a similar time frame and geographic distribution. This is, once again, to reduce the possible number of external variables which can have an impact on the data. The houses of Wright present a unique opportunity for this method to be applied to a single, world- renowned architect who produced many houses during his lifetime and in several distinct periods. Never before has this method been used to analyse the plans and elevations of 15 houses by the same architect drawn from three stylistic periods in an oeuvre. In total, the computational method will analyse 1960 mathematical comparisons between detail found at different levels in the elevations as part of around 112 sets of calculations for 56 elevations and 60 sets of calculations for 30 plans. The analysis process commences with redrawing the plans and elevations of 15 of Wright’s houses using consistent graphic conventions. The plans of Wright’s work were all digitally traced from his original working drawings reproduced by Storrer (1993) and Futagawa and Pfeiffer (1984; 1985; 1985b; 1985c; 1987a; 1987b). Where the particular house was altered by Wright during construction, or only an incomplete set of working drawings is available, the measured drawings of the Historic American Buildings Survey were used to supplement the originals. Following Bovill’s convention, the lines considered in the analysis correspond to changes in form, not changes in surface or texture. Thus, in the elevations, major window reveals, thickened concrete edge beams and steel railings are all considered while brick coursing and control joints are not. In the plans, windowpanes, door openings, stairs and fixed furniture are all considered while floor surface types such as tiles or timber are not. Past research (Vaughan and Ostwald, 2009) has shown that correctly selecting the lines being analysed is critical to producing consistent, repeatable results. Ostwald and Tucker (2007) also consider the question of “significant lines” for analysis concluding that the key remains consistency which is why they recommend, despite some reservations, retaining Bovill’s original definition. Once all of the 86 images are prepared for analysis the following method is followed: 1. The elevations of each individual house are grouped together and considered as a sub-set of the data. 2. Where the house comprises several levels, the plans of each individual house are also grouped together and considered as a sub-set of the data. 3. Each elevation of the house is analysed using Archimage and Benoit programs, producing two results

which are averaged together to create a D(Elev) outcome. 4. Each plan of the house is analysed using Archimage and Benoit programs, producing two results

which are averaged together to create a D(Plan) outcome.

5. The sub-set of D(Elev) results are averaged together to produce a D(Comp, Elev) for the house. The composite elevation result is a single D value that best approximates the two-dimensional characteristic visual complexity of the facades of the house.

26 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

6. The sub-set of D(Plan) results are averaged together to produce a D(Comp, Plan) for the house. For single

story houses, the single D(Plan) result is used for the D(Comp, Plan) output. The composite plan result is a single D value that best approximates the characteristic visual complexity of the program’s two- dimensional spatial disposition in the house. 7. Processes 1, 3, and 5 are then repeated for each set of five houses. Processes 2, 4 and 6 are also repeated for each set of five houses.

8. The five D(Comp, Elev) results for each set are averaged together to create an aggregate or D(Agg, Elev) result.

The five D(Comp, Plan) results for each set are averaged together to create a D(Agg, Plan) result. In stages 3 and 4, the settings for Archimage and Benoit, including the scaling coefficient (determining the ratio by which grids reduce in scale) and scaling limit (the smallest grid where data is collected and statistical divergence tends to occur), are preset to be consistent between the programs. The starting image

size (IS(Pixels)), largest grid size (LB(Pixels)), and number of reductions of the analytical grid (G(#)), are recorded so that the results can be tested or verified. As part of the analysis process, the Range or Gap between several sets of results are calculated. The

DRange value is the difference between the highest and lowest fractal dimensions of a set; it can also be expressed as the percentage gap (%Gap) between the results. The four different Range and Gap comparison types are described in Table 1 along with the abbreviations and definitions used in this paper. Finally, as part of the analysis a chart is produced for each set of five houses recording the standard

deviation (highest and lowest D(Elev) results) and the mean (the D(Comp) result) for both plans and elevations.

Table 1 Abbreviations and definitions Abbreviation Meaning D Approximate Fractal Dimension.

D(Elev 1-X) D result for an elevation of a house; determined by averaging the Archimage or Benoit outcomes. The subscript number refers to the elevation.

D(Plan 1-X) D result for a plan of a house; determined by averaging the Archimage or Benoit outcomes. The subscript number refers to the particular plan if there are multiple.

D(Comp) D(Comp Elev) is the average of all D(Elev) results for a house. D(Comp Plan) is the average of all D(Plan) results for a house. D(Agg) Aggregated result of five composite sets of houses reported for either plans or elevations; respectively D(Agg Plan) and D(Agg Elev). IS(Pix) The size of the starting image measured in pixels. LB(Pix) The size of the largest box or grid that the analysis commences with, measured in pixels. G(#) The number of scaled grids that the software overlays on the image to produce its comparative analysis. DRange The difference between two sets of results.

%Gap The D(Range) result expressed as a percentage of the possible maximum range being between D = 1.0 and D = 2.0. [%Gap = 100 x D(range)]

DRange (Plan) The difference between the highest and lowest results for plans in a sub-set for an individual house; may also %Gap(Plan) be expressed as a % gap. DRange (Elev) The difference between the highest and lowest results for elevations in a sub-set for an individual house; may %Gap(Elev) also be expressed as a % gap. DRange (CompPlan/Elev) The difference between the composite Plan result for a house and the composite elevation for the same house; %Gap(CompPlan/Elev) may also be expressed as a % gap. DRange (CompPlan) The difference between the highest and lowest composite results for plans in a set of houses; may also be %Gap(CompPlan) expressed as a % gap. DRange (CompElev) The difference between the highest and lowest composite results for elevations in a set of houses; may also be %Gap(CompElev) expressed as a % gap. DRange(Agg Plan/Elev) The difference/gap between the aggregate results for the plans of a set of buildings and the aggregate results for %Gap(Agg Plan/Elev) the set of elevations for the same buildings.

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3 Descriptions of Houses

The present paper undertakes an analysis of the fractal dimensions of sets of five of Frank Lloyd Wright’s free-standing house designs, from three significant periods of his life. During his lengthy career Wright pioneered many architectural design strategies for housing. The three distinct periods selected for analysis span 40 years, marking significant early, mid and late eras in Wright’s domestic architecture. The three periods are the Prairie style, the Textile Block and the Usonian periods. The first five of Wright’s early house designs analysed were completed between 1901 and 1910 and are from his Prairie Style period; regarded by some critics as “Wright’s greatest invention in this first phase of a long career”(Kostof, 1985). The next set of houses analysed are from Wright’s mid-career Textile Block period (1922 -1932) and are “[A]mong the most livable and attractive of Wright’s works” (Wright, 1960). The last set from Wright’s “Triangle-Plan” Usonian period (1950 - 1955), represent a “continuation of [Wright’s] lifelong quest to destroy boxlike rooms”(Lind, 1994), and display “far more flair and snap than the familiar ranch house” (Hoffmann, 1995). In some of his earliest writings, Frank Lloyd Wright set down some “propositions” regarding a method for creating an American architecture. On the subject of creating a unique character for a building, Wright (1908) stated that “I have endeavoured to establish a harmonious relationship between ground plan and elevation of these buildings considering the one as a solution and the other as an expression of the conditions of a problem of which the whole is a project”. Richard MacCormac (2005) emphasises that Wright “saw the architectural relationship between plan and section” and proposes that this understanding was gained from Wright’s use of the Froebel’s Gifts, a three-dimensional educational sequence which he undertook as a child. John Sergeant agrees with MacCormac, suggesting that Wright’s three dimensional, non-symmetrical grid planning allows for a connection between the plan and the elevation of his buildings on an experiential level. Sergeant (2005) proposes that this was achieved by Wright during his Prairie Style period where “a vocabulary of forms was used to translate or express the grid at all points - the solid rather than pierced balconies, planters, bases of flower urns, clustered piers, even built in seats were evocations of the underlying structure of a house”, and that in the Usonian houses “Wright’s skill lay in the perfect coordination of horizontal and vertical systems to manipulate the[ir] character” (Sergeant, 2005). 3.1 The five Prairie style houses The five of Wright’s Prairie style houses selected for analysis were constructed between 1901 and 1910. Four of the five are in the state of Illinois and the fifth is in Kentucky. Wright’s Prairie Style is an approach characterized by strong horizontal lines, overhanging eaves, low-pitched roofs, an open floor plan, and a central hearth. Importantly, these five houses span the period between the first publication of Wright’s Prairie Style, in the Ladies Home Journal in 1901, and what is widely regarded as the ultimate example of this approach, the Robie House (figures 2 and 3).

28 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

Figure 2 Frank Lloyd Wright’s Robie House, South Elevation

Figure 3 Frank Lloyd Wright’s Robie House, Plan 1.

The first design is the F. B. Henderson House (1901) in Elmhurst, Illinois. The house is a wooden, two-storey structure with plaster rendered elevations. A range of additions were made to the house in the years following its completion until, in 1975, the house was restored to its original form. The Tomek House (1904-1907) in Riverside, Illinois, is also a two-storey house although it possesses a basement and is sited on a large city lot. This house is finished with pale, rendered brickwork, dark timber trim and a red tile roof. Storrer notes that, in response to the Tomek family’s needs, Wright later allowed posts to be placed beneath the cantilevered roof to heighten the sense of support and enclosure (Storrer, 1993). As the posts were not required for structural reasons, and Wright found them personally unnecessary, they have been omitted from the analysis. The Robert W. Evans House (1908) in Chicago, Illinois, features a formal diagram wherein the “basic square” found in earlier Prairie Style houses is “extended into a cruciform plan”(Thomson, 1999). The house is set on a sloping site and possesses a plan similar to one Wright proposed in 1907 for a “fireproof house for $5000”. The Evans House was later altered to enclose the porch area and the stucco finish on the façade was also cement rendered. The Zeigler House (1910) in Frankfort, Kentucky has a similar plan to the Evans House. Designed as a home for a Presbyterian minister, this two storey house is sited on a small city lot and it was constructed while Wright was in Europe. After a decade of development and refinement the quintessential example of the Prairie Style, the

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Robie House (1908-1910), was constructed in Chicago, Illinois. Designed as a family home, the three storey structure fills most of its corner site. Unlike many of Wright’s other houses of the era, the Robie House features a façade of exposed Roman bricks with horizontal raked joints. 3.2 The Five Textile Block Houses Wright expanded his practice in California in the early 1920’s and during the following decade he designed many buildings, although only five houses were built. These five houses, which share some of the character of Wright’s famous Hollyhock House, have since become known as the Textile Block homes. Appearing as imposing, ageless structures, these houses were typically constructed from a double skin of pre-cast patterned and plain exposed concrete blocks held together by Wright’s patented system of steel rods and concrete grout. Ornamented blocks generally punctuate the plain square blocks of the houses and for each house a different pattern is employed. The first of the houses, the Millard Residence or “La Miniatura” was completed in Pasadena in 1923. This house is the only one of the Textile Block works not to feature a secondary structure of steel rods. Reflecting on La Miniatura, Wright wrote that in this project he “would take that despised outcast of the building industry – the concrete block – out from underfoot or from the gutter – find a hitherto unsuspected soul in it – make it live as a thing of beauty textured like the trees” (Wright, 1960). The second of the Textile Block Houses, the Storer Residence (figures 4 and 5), was completed in Hollywood in 1923. It is a three-storey residence with views across Los Angeles. The Samuel Freeman Residence in Los Angeles was also completed in 1923. It is regarded as the third Textile Block house and the first to use mitered glass in the corner windows of the house; all of the previous works in this style have solid corners. It is a two-storey, compact, flat roofed house made from both patterned and plain textile block and with eucalyptus timber detailing. The fourth Textile Block house, the Ennis Residence, is probably the most famous of Wright’s works of this era. Also completed in 1923 and overlooking Los Angeles, it is regarded as the “most monumental” (Storrer, 1974) of the houses of this era. It has been described as “looking more like a Mayan temple than any other Wright building except [the] Hollyhock House” (Storrer, 1974). The Ennis House is conspicuously sited and is made of neutral coloured blocks with teak detailing. Some of the windows feature art glass designed by Wright in an abstraction of wisteria plants. The final design in this sequence, the Lloyd-Jones House, is in Tulsa. Completed in 1929, it is the only non-Californian Textile Block house. Designed for Wright’s cousin, it is a large house with extensive entertaining areas and a four-car garage. The Lloyd-Jones House is notably less ornamental than the others in the sequence with Wright rejecting richly decorated blocks “in favor of an alternating pattern of piers and slots” (Frampton, 2005).

30 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

Figure 4 Frank Lloyd Wright’s Storer Figure 5 Frank Lloyd Wright’s Storer Residence, East Elevation Residence, Ground Plan.

3.3 The Five Triangle-Plan Usonian Houses More than twenty years was to pass before Wright developed his third major sequence of domestic works; the Usonian houses. Wright explains that the Usonian house is intended to be “integral to the life of the inhabitants”, be truthful in its material expression (“glass is used as glass, stone as stone, wood as wood”) and embrace the elements of nature (Wright, 1954). Hoffman describes the Usonian house as “a simplified and somewhat diluted prairie house characterized by the absence of leaded glass and the presence of […] very thin wall screens with a striated effect from wide boards spaced by recessed battens” (Hoffmann, 1995). While there were multiple variations on the Usonian house, the five works featured in the present chapter are all based on an underlying equilateral triangular grid and were constructed between 1950 and 1956. The first of the triangle-plan Usonian Houses, the William Palmer House is located in Ann Arbour, Michigan and was completed in 1950. The house is a two-storey brick structure, set into a sloping site, with wide, timber-lined eaves, giving the viewer an impression of a low, single level house. The brick walls include bands of patterned, perforated blocks, in the same colour as the brickwork. The second house, the Riesley Residence, was the last of Wright’s Usonian houses built in Pleasantville, New York; it was completed in 1951. This single level home with a small basement is constructed from local stone with timber panelling and is set on a hillside site. To accommodate the Riesley’s expanding family, five years later Wright designed an addition for the children’s bedrooms. In contrast, in the same year the Chahroudi Cottage was built on an island in Lake Mahopac, New York, and constructed using Wright’s desert masonry rubblestone technique with some timber cladding and detailing. Wright originally designed the cottage as the guest quarters for the Chahroudi family home, however only the cottage was built and subsequently used as the primary residence. The Dobkins House was built in 1953 for Dr John and Syd Dobkins in Canton, Ohio (figures 6 and 7). This small house is constructed from brick with deeply raked mortar joints. However, unlike the Robie House, the mortar colour contrasts with the bricks

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in the vertical as well as the horizontal joints. Finally, the Fawcett Residence, completed in 1955, had an unusual brief for Wright to design a home for a farming family. The house is set on the large flat expanse of the Fawcett’s walnut farm in Los Banos, California. The single storey house is constructed primarily of grey concrete block with a red gravel roof.

Figure 6 Frank Lloyd Wright’s Dobkins House, Figure 7 Frank Lloyd Wright’s Dobkins North Elevation. House, Floor Plan.

4 Results and Discussion

The results for the fractal dimensions of the plans and elevations of all 15 houses analysed are reported and discussed here. For the five Prairie Houses a total of 20 elevations and 12 plans produced 696 data points, the five Textile Block houses with 18 elevations and 12 plans produced 640 data points and 18 elevations and 6 plans for the Usonian Triangle-Plan houses produced 624 data points. While the data for the Textile block houses is entirely new, the data for the Prairie Houses and the Usonian houses have been in part previously published although not to this level of accuracy (Ostwald, et al., 2008; Vaughan and Ostwald, 2010). Additionally, for all three sets, this paper includes calculations of the

fractal dimension range (DRange) and percentage gap (%Gap). Other studies drawing comparisons from

fractal dimension data use a DRange result as a different way of viewing the information that has gone into

a D(Agg) result. Generally, the D(Range) value is the difference between the highest and lowest fractal

dimensions of a set and when expressed as a percentage the D(Range) value is known as the percentage gap (%Gap). There are six types of Range and Gap calculations used in the present paper. The first three are concerned with the individual house; the consistency of its (1) plans, (2) elevations and (3) the difference between the plans and elevations. The next three are concerned with comparisons across the full set of five houses to measure the relative consistency of the composite (4) plans, (5) elevations, and (6) the gap between these two sets. The six calculations are as follows.

1. The difference between the highest and lowest results for plans for an individual house; DRange (Plan) and

%Gap(Plan). 2. The difference between the highest and lowest results for elevation results for an individual house;

DRange (Elev) and %Gap(Elev).

32 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

3. The difference between the composite plan result for a house and the composite elevation for the same

house; DRange (Comp Plan/Elev) and %Gap(Comp Plan/Elev).

4. The difference between the highest and lowest composite plan results for a set of five houses; DRange

(Comp Plan) and %Gap(Comp Plan). 5. The difference between the highest and lowest composite elevation results for a set of five houses;

DRange (Comp Elev) and %Gap(Comp Elev). 6. Finally the difference between the aggregate results for the plans of a set of buildings and the

aggregate results for the set of elevations for the same buildings; D Range (Agg Plan/Elev) and %Gap(Agg

Plan/Elev).

The DRange and %Gap results are especially important in the present study because they describe the limits within which Wright worked at each of the three stylistic periods in his life. A tight and consistent set of limits might imply that Wright had a highly developed sense of the relationship between visual complexity in plan and in elevation. If such a pattern could be uncovered then a case could be made that, for each period, Wright’s architecture could be considered to possess a unique design signature. A previous paper (Ostwald and Vaughan, 2010) suggested that a particular characteristic of Peter Eisenman’s House VI was that the plans and elevations had very similar levels of visual complexity

(%Gap(Comp Plan/Elev) = 1.8%) while an analysis of John Hedjuk’s House 7 had a different result; %Gap(Comp

Plan/Elev) = 6%. 4.1 The Prairie Houses

The most visually complex of the Prairie Houses is the Evans House (D(comp,Elev) = 1.5771,

D(comp,Plan)= 1.483) and the least visually complex, by a clear margin, is the Zeigler House (D(comp,Elev)=

1.503, D(comp,Plan)= 1.435). All five houses have an average visual complexity for their elevations of D(Agg

Elev) = 1.5388. The average visual complexity of the typical set of elevations was higher than the result for

the average plan; D(Agg Plan) = 1.4595. The difference between these results is a DRange(Agg Plan/Elev) of 0.07926

or a % Gap(Agg Plan/Elev) of 7.9%. This means that the typical Prairie house plan was almost 8% less visually

complex than the typical elevation. The range of % Gap(Comp Plan/Elev) results for each house in the set was between a low of 6.4% (Tomek house) and a high of 9.6% (Henderson house). Overall, this implies that while the plans and elevations have quite different visual qualities (an 8% gap is a low level of correlation), there is a higher degree of consistency in the size of the gap itself, (a 3.2% range), in the

range of elevations (DRange (Comp Elev) = 7.4%) and in the range of plans (DRange(Comp Plan) = 4.8%). This is a reasonably consistent pattern for a set of results of this scale. However, it is apparent from the results that the Zeigler House does not fit very well with the rest of the results, and if it was removed from the set, the

revised results would be (DRange (Comp Elev) = 4.7%) and in the range of plans (DRange (Comp Plan) = 3.3%); a highly consistent set of results.

33 ARCHITECTURE SCIENCE, No. 4, December 2011

Table 2 D Values of the five Prairie Style Houses House Data Henderson Tomek Evans Zeigler Robie All Five Elevations D(Elev 1) 1.565 1.509 1.5995 1.4855 1.505 D(Elev 2) 1.550 1.518 1.5975 1.5015 1.573 D(Elev 3) 1.5495 1.540 1.556 1.517 1.5625 D(Elev 4) 1.5215 1.5535 1.5555 1.5085 1.5095 D(Comp,Elev) 1.5465 1.530 1.5771 1.503 1.5375 D(Agg, Elev) 1.5388 Plans D(Plan 0) 1.4765 1.476 1.475 1.420 1.415 D(Plan 1) 1.4245 1.465 1.491 1.450 1.498 D(Plan 2) - 1.4575 - - 1.4785 D(comp,Plan) 1.450 1.466 1.483 1.435 1.4638 D(Agg, Plan) 1.45956 Range/Gap Analysis DRange (Elev) 0.0435 0.0445 0.044 0.0315 0.068 % Gap(Elev) 4.3% 4.4% 4.4% 3.1% 6.8% DRange (Plan) 0.052 0.0185 0.016 0.030 0.083 % Gap(Plan) 5.2% 1.8% 1.6% 3.0% 8.3% DRange 0.0965 0.0640 0.0941 0.068 0.0737 (Comp Plan/Elev) % Gap(Comp Plan/Elev) 9.65% 6.4% 9.41% 6.8% 7.37% DRange (Comp Elev) 0.0741 % Gap(Comp Elev) 7.4% DRange (Comp Plan) 0.048 % Gap(Comp Plan) 4.8% DRange (Agg Plan/Elev) 0.07926 % Gap(Agg Plan/Elev) 7.9%

1.65

1.6

) 1.55 D (

1.5

1.45 Dimension

1.4

Fractal 1.35

1.3

1.25 Elev. Pl. Elev. Pl. Elev. Pl. Elev. Pl. Elev. Pl.

Henderson Tomek Evans Zeigler Robie Set of Prairie Houses D(Agg, Plan) D(Agg, Elev) D(Comp, Elev) D(Comp, Plan)

Figure 8 Chart for Prairie house results showing range and mean for each house.

34 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

4.2 The Textile Block Houses The composite values for the elevations of the five houses of Wright’s Textile Block period range

from D(Comp,Elev) = 1.500 (Lloyd Jones House) to D(Comp,Elev) = 1.614 (La Miniatura), while the composite

values of the plans of the Textile Block houses range from D(Comp,Plan) = 1.384 (Lloyd Jones House) to

D(Comp,Plan) = 1.5215 (Ennis House). The percentage gap between the aggregate values of the plans and of

the elevations of these five Textile Block houses is DRange (Agg Plan/Elev) = 9.9%; marginally higher than the

equivalent statistic for the Prairie houses of DRange (Agg Plan/Elev) = 7.9%. The percentage gap between the average or composite results for the plans and elevations (%

Gap(Comp Plan/Elev)) of the houses of Wright’s Textile Block period vary from 3.2% (Ennis House), to 6.4% (Freeman House), 8.8% (Storer House), 11.6% (Lloyd Jones House) and up to 19.5% (La Miniatura).

The percentage gap for the overall aggregate results for the Textile Block houses, (% Gap(Agg Plan/Elev)

= 9.9%), does not suggest a striking level of similar visual complexity between the plans and elevations. Considering the data for the individual houses in this set, only one house, the Ennis House, could be

considered to have a close relationship in plan and elevation, with a percentage gap of % Gap(Comp Plan/Elev) = 3.2%. The results for this house are generally consistent, with the fractal dimensions of all four of the

elevations between D(Elev) = 1.519 and D(Elev) = 1.5755, and the only plan result D(Plan) = 1.5215. The

Freeman House has a % Gap(Comp Plan/Elev) = 6.4% and may be considered to have a limited visual complexity connection in plan and elevation. The other 3 houses in this set, however, do not provide a convincing

case for similarity between plan and elevation. La Miniatura, in particular, has a wide % Gap(Comp Plan/Elev) = 19.5%, suggesting little or no similarity between the degree of detail found in the elevation and in the plan. The data set for this house however, is not complete, with only 2 elevations available for analysis.

However these two results are close (D(Elev) = 1.6265 and D(Elev) = 1.602), and the 4 plan results are quite

close, ranging from D(Plan) = 1.3805 to D(Plan) = 1.463, suggesting by the consistency of results that the percentage gap is likely quite an accurate projection.

35 ARCHITECTURE SCIENCE, No. 4, December 2011

Table 3 D Values of the five Textile Block Houses House Data La Storer Ennis Freeman Lloyd All Five Miniatura Jones

Elevations

D(Elev 1) 1.6265 1.5175 1.564 1.444 1.459

D(Elev 2) 1.602 1.5545 1.5755 1.5705 1.498

D(Elev 3) - 1.5225 1.5575 1.5175 1.5275

D(Elev 4) - 1.439 1.519 1.494 1.517

D(Comp,Elev) 1.61425 1.508375 1.554 1.5065 1.500375

D(Agg, Elev) 1.5367 Plans

D(Plan 0) 1.463 1.489 1.5215 1.46565 1.496

D(Plan 1) 1.4325 1.3505 - 1.419 1.3545

D(Plan 2) 1.401 - - - 1.3015

D(Plan 3) 1.3805 - - - -

D(comp,Plan) 1.41925 1.41975 1.5215 1.442325 1.384

D(Agg, Plan) 1.43736 Range/Gap Analysis

DRange (Elev) 0.0245 0.1155 0.0565 0.1265 0.0685

% Gap(Elev) 2.4% 11.5% 5.6% 12.6% 6.8%

DRange (Plan) 0.0825 0.1385 0 0.0466 0.1945

% Gap(Plan) 8.2% 13.8% 0% 4.6% 19.4%

DRange (Comp Plan/Elev) 0.195 0.0886 0.0325 0.064 0.1163

% Gap(Comp Plan/Elev) 19.5% 8.8% 3.2% 6.4% 11.6%

DRange (Comp Elev) 0.1139

% Gap(Comp Elev) 11.39%

DRange (Comp Plan) 0.1375

% Gap(Comp Plan) 13.75%

DRange (Agg Plan/Elev) 0.9933

% Gap(Agg Plan/Elev) 9.9%

36 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

1.65

1.6

1.55 ) D ( 1.5

1.45 Dimension

1.4 Fractal

1.35

1.3

1.25 Elev. Pl. Elev. Pl. Elev. Pl. Elev. Pl. Elev. Pl.

La Miniatura Storer Ennis Freeman Lloyd Jones Set of Textile Block Houses D(Agg, Plan) D(Agg, Elev) D(Comp, Elev) D(Comp, Plan) Figure 9 Chart for Textile Block house results showing range and mean for each house.

Overall, the difference between the composite plans and elevations is % Gap(Comp Plan/Elev) = 16.3%; a

very high score. The range for the composite plans is D Range (Comp Plan) = 0.137 or %Gap(Comp Plan) = 13.7%

and the range for the composite elevations is D Range (Comp Elev) = 0.113, or %Gap(Comp Elev) = 11.3%. In summary, whereas the Prairie houses had a strong pattern of results, the textile block houses do not have such a clear relationship to each other.

4.3 The Usonian Triangle-Plan houses The spread of the average visual complexity of the façades of the five Usonian houses (1.350 <

D(Comp, Elev) < 1.486) begins with the least complex façade set of the Fawcett House and is capped by the more complex set of façades of the Palmer House. The complexity of the plans of these five houses is

lowest in the Riesley House (D(Comp, Plan) = 1.36575) and highest in the Chahroudi House (D(Comp, Plan) = 1.489). The percentage gap between the aggregate values of the plans and of the elevations of these five

Usonian Triangle- Plan houses is % Gap(Agg Plan/Elev) = 1.5%; a very close result.

37 ARCHITECTURE SCIENCE, No. 4, December 2011

Table 4 D Values of the five Usonian Triangle-Plan Houses House Data Palmer Riesley Chahroudi Fawcett Dobkins All Five Elevations

D(Elev 1) 1.4805 1.4405 1.466 1.3465 1.374

D(Elev 2) 1.4695 1.439 1.491 1.428 1.521

D(Elev 3) 1.485 1.435 1.4605 1.2745 1.400

D(Elev 4) 1.5055 1.4585 - - 1.473

D(Comp,Elev) 1.486 1.434 1.403 1.350 1.442

D(Agg, Elev) 1.423 Plans

D(Plan 0) 1.4095 1.352 1.489 1.3965 1.3765

D(Plan 1) - 1.3795 - - -

D(comp,Plan) 1.4095 1.36575 1.489 1.3965 1.3765

D(Agg, Plan) 1.4074 Range/Gap Analysis

DRange (Elev) 0.036 0.0235 0.0305 0.1535 0.147

% Gap(Elev) 3.6% 2.3% 3.0% 15.3% 14.7%

DRange (Plan) 0 0.0275 0 0 0

% Gap(Plan) 0% 2.7% 0% 0% 0%

DRange (Comp Plan/Elev) 0.0765 0.06825 0.086 0.0465 0.0655

% Gap(Comp Plan/Elev) 7.6% 6.8% 8.6% 4.6% 6.5%

DRange (Comp Elev) 0.136

% Gap(Comp Elev) 13.6%

DRange (Comp Plan) 0.12325

% Gap(Comp Plan) 12.3%

DRange (Agg Plan/Elev) 0.0156

% Gap(Agg Plan/Elev) 1.5%

The percentage gap between the aggregate values of the plans and of the elevations of these five Usonian Triangle-Plan houses is significantly lower than the other two house sets, and suggests a high degree of similarity between the visual complexity of the plans and elevations of the houses of this set. However this overall percentage gap does not clearly reflect the individual house calculations for

percentage gap. Only the Fawcett House (% Gap(Comp Plan/Elev) = 4.6%) is under 6%, actually suggesting some degree of closeness between the degree of detail in plan and elevation, while the Dobkins House (%

Gap(Comp Plan/Elev) = 6.5%), Reisely House (% Gap(Comp Plan/Elev) = 6.8%), Palmer House (% Gap(Comp Plan/Elev) =

7.6%) and Chahroudi House (% Gap(Comp Plan/Elev) = 8.6%) are all higher.

38 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

1.65

1.6

1.55 ) D ( 1.5

1.45 Dimension

1.4 Fractal

1.35

1.3

1.25 Elev. Pl. Elev. Pl. Elev. Pl. Elev. Pl. Elev. Pl.

Palmer Riesley Chahroudi Fawcett Dobkins Set of Usonian Houses D(Agg, Plan) D(Agg, Elev) D(Comp, Elev) D(Comp, Plan)

Figure 10 Chart Usonian house results showing range and mean for each house.

In summary, the range between the simplest and the most complex of the composite plans was D

Range(Comp Plan) = 0.123 or %Gap(Comp Plan) = 12.3%. The range between the lowest and higher composite

elevations is D Range(Comp Elev) = 0.136, or %Gap(Elev) = 13.6% and the difference between the closest gap (between plans and elevations in the one house) and the largest gap (between plans and elevations in another house) is 4%. Overall, this suggests a partial pattern in both range and gap that is not as clear as the pattern of relations for the Prairie houses, but is much more apparent than the data for the Textile Block houses.

6 Conclusion

The first observation arising from the research is that the elevations in general produced higher fractal dimensions than the plans for all 15 houses. This may be particular to the architecture of Frank Lloyd Wright but it is more likely a reflection of the minimum size and dimensionality of rooms required to accommodate human inhabitation; that is, the primary force shaping the plan. In contrast, elevations are shaped by materials, outlook, environmental affects and privacy needs all of which may occur across a wider range of scales.

39 ARCHITECTURE SCIENCE, No. 4, December 2011

Next, given the data recorded in this paper, none of the 15 houses demonstrated a high level of correspondence between the visual complexity of the elevations and of the plans. Only 2 houses had a

level of similarity lower than a % Gap(Comp Plan/Elev) = 6%; one each from the Textile Block and Usonian periods. However the results over 6% did generally cluster, with 9 houses fitting within the percentage gap between 6% and 9%. Three of these houses were from the Prairie Style, two from the Textile Block period and four from the Usonian period. Of the four houses over the 9% percentage gap, two Prairie style houses were just over at 9.4% and 9.65%. The two remaining houses from the Textile Block period produced outlying results (Lloyd Jones 11.6% and La Minatura 19.5%). These results suggest that if a plan is a direct reflection of the elevations that define its boundaries, then that plan might be expected to have a fractal dimension result of between 6% and 9% less than its elevations; this is the first time this figure has been calculated. Finally, the close analysis of the Prairie house results uncovered a clear pattern in the relationship between visual complexity in plans and in elevations. Indeed, if one of the data sources (the Zeigler house) was removed, an exceptionally clear pattern would have been revealed for the other four. That such a pattern is visible from a data set of this size is especially informative and suggests that this pattern may be a reflection of key concerns or strategies in a particular architect’s work. A related, but less statistically clear pattern may be traced in Wright’s Usonian Houses. In both cases these patterns could be seen to represent a quantification of Wright’s visual style; his formal signature. In the final case, the textile block houses, no clear pattern was revealed. Even the process of limiting the data, to remove any outliers, fails to provide a convincing pattern. There may be several reasons for this. First, perhaps no pattern exists connecting these spatially and programmatically complex works. Alternatively, the method used for fractal analysis may be inconsistent in its handling of the visual complexity of multiple types of textile cladding systems and multiple floor levels and conditions. Whereas there were many Prairie and Usonian Houses that adhered to clear formal principals, there were only six textile works and they differed significantly. Perhaps this lack of scope to formalise the design style undermined Wright’s capacity to develop a consistent visual signature for this intermediate period of work.

Acknowledgments

An ARC Fellowship (FT0991309) and an ARC Discovery Grant (DP1094154) supported the research undertaken in this paper.

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法蘭克‧洛伊‧萊特建築之平面與立面碎形維度的關係：比較草原風格、紡織積木與美國風時期

喬瑟芬‧佛漢1* 麥可‧奧斯沃德 2

1 澳洲紐卡斯爾大學學術研究員 2 澳洲紐卡斯爾大學建築學系教授 *通訊作者 Email：[email protected] （2010 年 6 月 15 日投稿，2011 年 3 月 4 日通過)

摘要碎形幾何學最早是在 1990 年代中葉被用來分析建築之視覺複雜性的一種量化分析法。爾後許多學者即重複使用各種手算及電腦計算的碎形分析法，以數學方式來分析建築的立面。此法廣泛應用於描述知名建築作品立面的視覺複雜性，結果產生大量的相關比較資料。然而，碎形分析應用在建築平面的相關研究極少，更少的是應用在探討平面與立面視覺複雜性的相關性，或這樣的相關性究竟是否存在。本文以碎形分析法比較十五件法蘭克‧洛伊‧萊特設計的獨立住宅作品，測試建築平面與其立面的視覺複雜性之間的相互關係是否有脈絡可循。重要的是，這些獨立住宅來自萊特三個不同風格的時期，意味著倘若建築平面和立面之間真有顯著的關係，那每一個時期的作品都應體現出明顯不同的趨勢。本文除了介紹標準的碎形分析法，也說明了用來計算取得本研究結果的電腦計算法。文中分別描述每一棟住宅及其碎形維度的分析結果。最後詳細分析本研究結果，結論中嘗試提出萊特的住宅建築平面與立面視覺複雜性之間的關係。

關鍵詞：法蘭克‧洛伊‧萊特，碎形分析法，電腦計算分析，視覺複雜性