Towards a Fractal Architecture by Su Wang

Towards a Fractal Architecture A study of Chaos, Complexity and Beauty.

Su Wang 05612489

Primarily, Nature furnished the materials for architectural motifs out of which the architectural forms as we know them today have been developed, and, although our practice for centuries has been for the most part to turn from her, seeking inspiration in books and adhering slavishly to dead formulae, her wealth of suggestion is inexhaustible; her riches greater than any manâ&#x20AC;&#x2122;s desire. Frank Lloyd Wright - In the Cause of Architecture, Architecture Record, March 1908

Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor does lightning travel in a straight line. Almost everything around us is non-euclidean. Benoit B. Mandelbrot - The Fractal Geometry of Nature, 1982

Abstract Formal theories in modern architecture were influenced by the writings of Le Corbusier and Adolf Loos, in which euclidean geometry and simple clear forms were suggested as an aesthetic ideal, often in stark contrast with the heavily ornamented and detailed aesthetics of the styles of previous periods. This was borne out of the cultural climate of the times and thus no empirical evidence or a quantitive theory was ever put forward to support their rhetoric. Architecture in the 20th century has been generally dominated by modernism, and later, post - modernism, which can trace their origins to these theories. Salingaros characterizes these opposing dogmas as the difference between a prison cell and a Las Vegas strip - one has a punishing lack of visual information, the other an uncoordinated barrage of random information. It has been argued that this has led to a decrease in the visual quality of the environments in which we inhabit. However, developments since the mid 20th Century in applied mathematics, evolutionary biology, environmental psychology, computer science and A.I. research, as well as architecture, landscape and urban planning, suggests the possibility of an alternative in the approach towards the aesthetic design of the built environment. This new approach recognizes fractal geometry, as opposed to euclidean geometry, as a more accurate mathematical description of our inhabited environment. Research suggests that we have an innate habitat preference that is genetically determined, including a preference for views with a fractal dimensions of between 1.3 - 1.5, This is a range of D value common to the savanna-type landscape for which there is a cross cultural preference. Information theory provide a theoretical basis for these observations, noting that requirements for information compression means that highly compressible but novel patterns are perceived to be the most beautiful and interesting. Work by Alexander and Salingaros show similarity to these theories, in particular the recognition that complexity, detail and patterns are a basic visual requirement. This new approach is not intended as a prescriptive universal design method, and does not suggest that we replace traditional architectural theory. It would, on the other hand, be irresponsible for architects, as designers of the environments which we inhabit, to ignore such formal requirements in favour of conceptual, philosophical or idealogical concerns. 3

I am very grateful to my supervisor, Daniel P. Sudhershan for his continuing advice, support and encouragement. Thanks also to Irene Yeriskin for providing additional information on the more esoteric topics covered in this dissertation.

Table of Contents 1.0

Introduction

2.0

Fractals

3.0

4.0

2.1

Alexander and Structural Wholeness

2.2

Fractal Dimension

Environmental Psychology and Biophilia 3.1

Questioning Bovill

3.2

Biophilia and Habitat Preference

Chaos and Information Theory 4.1

Iteration and The Mandelbrot Set

4.2

Algorithmic and Hierarchical Theory of Beauty

4.3

Complexity, Hierarchy and Wholeness in Historical Architecture

5.0

Conclusion

Bibliography

Less is a bore. Robert Venturi - Complexity and Contradiction in Architecture, 1966

1.0

Introduction

Modernist conception of forms were derived from an understanding of euclidean geometry. In particular there was a focus on primary forms such as “cubes, cones, spheres, cylinders, and pyramids” (Le Corbusier, 1927). Their beauty and appeal was thought to be inherent regardless of taste, for “everyone is in agreement about this: children, savages, and metaphysicians” (Le Corbusier, 1927). Le Corbusier was led by a desire “not [to pursue] an architectural idea, but [to be] simply guided by the results of calculations (derived from the principles that govern our universe)” (Le Corbusier, 1927). He was interested in a formal language for architecture that transcended local tastes and styles, a language that was universal. Adolf Loos similarly wrote against the use of “ornament” - he believed that “the evolution of culture is synonymous with the removal of ornamentation from utilitarian objects” (Loos, 1908). His writings railed against the use of superfluous detail in the creation of utilitarian objects, promoting the practicality of a craftsman that would be echoed in the later modernist mantra of forms ever following function. Whatever its effects on the design of smaller scale objects, within architecture these two modes of thought were highly influential in the development of modernist “styles” such as the International style, with its focus on simple clear primary forms devoid of extraneous detail or “ornamentation” (Frampton, 1980, pp 90 -95). To this day Loos and Le Corbusier’s theories still have an Fig. 1. The influence of ornament-free euclidean geometry on contemporary architecture in Ireland, 2010 and 2009. (Source: RIAI Irish Architecture Awards http://www.irisharchitectureawards.ie/)

effect on contemporary architecture, where ornament-free euclidean geometry is used as the de facto formal system (Fig. 1). While Le Corbusier and Loos’ intent cannot be faulted, their arguments were founded on the rhetorical and the cultural issues of their times. For example, Loos’ essay could be seen as part of a wider critique of the social conventions of late 19th century Vienna, and include, among “Loos's relatively blank exteriors in architecture”, also “the ‘silences’ of Wittgenstein's language philosophy, and Kraus's denunciations of print media conventionalism”. These “sought collectively to purge superfluous elements from a culture seen as carnivalesque and debased” (Maciuika, 2000, pp 77 - 78). Their propositions, while having interesting philosophical and cultural implications, were attempting to shape the aesthetics of a physical environment that is to be inhabited by one or more people. However, 7

architecture exists in the world not just as individual “texts” in a larger cultural landscape, but also as small parts in a much larger existing physical landscape of structure and order encompassing the natural world. For such a system, a purely cultural theory of aesthetics can not be enough, particularly when said cultural theories conflict with empirical investigations. For example, Salingaros (2003) disagrees with the assertion that ornamentation is “superfluous”, giving a physiological explanation which highlights the need of our visual sensors for contrast and edge details which is satisfied by ornamentation. The basic premise of this dissertation is to question the assumptions concerning the inherent appeal and beauty of euclidean formal systems within the inhabited environment. A possible alternative is suggested, based on an updated mathematical view of the nature of order, supported by empirical studies. Within the field of mathematics there were objects being discovered at the turn of the 19th century that fell outside the domain of euclidean geometry. These were to change how mathematics and geometry related to forms found in nature. In 1883 Georg Cantor published the Cantor set (Fig. 6), and in the subsequent years numerous other oddities such as the Peano curve (Fig. 5), the Koch curve (Fig. 3), the Sierpinski gasket (Fig. 4) and the Minkowski curve were also demonstrated (Peitgen et al. 1992). These objects displayed certain properties such as self similarity and, in the case of the Peano, Koch and Minkowski curves, being curves that are not quite one dimensional. Their discoverers “regarded them as important in showing that the world of pure mathematics contains a richness of possibilities going far beyond the simple structures they saw in nature, [but nature] has played a joke on the mathematicians. The 19th century mathematician may have been lacking in imagination, but Nature was not.” (Dyson, 1978) Similarly, architects may have been lacking in imagination in suggesting primary forms as “the most beautiful forms”, but mathematics was not. Subsequent developments have shown that these “mathematical monstrosities” are abundant in nature, (Mandelbrot, 1982) they are truly forms that are “derived from the principles that govern our universe” (Fig 2). As architecture is concerned with the design and construction of the 8

inhabited environment, the question of what kind of environments we find the most visually interesting and induce positive affective responses in us are of utmost importance. There is an evolutionary basis to this question that is not satisfied by euclidean geometry. There are of course many advantages to using simple euclidean shapes, such as simplification of design and a certain monumental impressiveness in their alien qualities compared with the natural environment (Salingaros, 2000). However the question of what really makes a good inhabited environment for people overrules arbitrary concerns such as economy and monumentality. “The human perception system has evolved over millions of years in a natural fractal environment. Only recently, by evolutionary time scales have we found ourselves in a primarily euclidean environment of straight lines and few spatial scales (Voss, 1990)”. Fig 2. - The fractal geometry of nature (Source: Creative Commons, authorʼs own photographs, Alexander, 2002)

2.0

Fractals

2.1

Alexander and Structural Wholeness

The observation that a different approach was needed to the process of design is not new - mathematics and systems theory were vigorously studied in the mid 20th Century in the hopes of discovering a formal design theory. In particular Christopher Alexander’s A Pattern Language (1979) and The Timeless Way of Building (1979) attempted to codify a generative system of design based upon patterns, which are a vocabulary of established solutions to design problems. Patterns link to each other through syntax, or a description of where the pattern Fig 3. - Iterations of the Koch curve (Source: WIkipedia Commons)

falls within a larger generic scheme. This interlinking of patterns allows the solution of large complex problems by allowing designers to focus on smaller manageable issues between several local patterns while overall coherence is maintained by the web of relationships of patterns. However, the pattern language was an attempt to create a systematic theory of design in which all possible design solutions can be solved by its application. It did not see as much adoption within architecture, instead finding more success within computer science and software engineering (Coplien & Schmidt, 1995; Gamma, et al., 1995; Gabriel, 1996).

Fig 4. - The Sierpinski Gasket (Source: WIkipedia Commons)

Alexander would refine his theories into a more generalised system that is concerned with the creation of “life” (Alexander, 2002). His theories take on a more metaphysical approach, with Alexander redefining the concept of how much “life” something has as a function of its underlying ordered structural complexity, or “wholeness”. Alexander defines wholeness thusly: “I propose a view of physical reality which is dominated by the existence of this one particular structure, W, the wholeness. In any given region of space, some subregions have higher intensity as centers, others have less. Many subregions have week intensity or none at all. The overall

Fig 5. - Iterations of the Peano curve (Source: WIkipedia Commons)

configuration of the nested centers, together with their relative intensities, comprise a single structure. I define this structure as ‘the’ wholeness of that region.” (Alexander, 2002, Book 1, p96) A recursive definition is applied to centers - “a center is a kind of entity which can

Fig 6. - The Cantor Set (Source: WIkipedia Commons)

only be defined in terms of other centers” (Alexander, 2002, Book 1 p116), but they are essentially generic structural features of focus which are themselves made up of other centers at smaller scales. 10

Centers can then be said to be scale invariant. Greater “wholeness” is achieved by the complex interrelation between all the centers in a visual system across all scales, with more wholeness resulting from each center preserving the overall structure through interrelationships. The object of good architecture, then, according to Alexander, is to create “life” in buildings. This is done by making changes to the existing structure of the physical world which preserves or enhances its overall wholeness (Fig. 7, Fig. 8). Euclidean forms, within Alexander’s system, would have relatively little amounts of structurally interrelated centers at varying scales, and thus would have low amounts of wholeness and life. Alexander’s theories, while working at the generic level, already demonstrate some of the properties which form a common thread in the critique of euclidean forms. These include the properties of scale invariance, or a cascade of centers at various scales, the importance of an ordered complexity in the interconnected structure of centers and the suggestion of the property of self similarity in the requirement for centers which preserve the overall wholeness of the system. These are all properties Fig 7. - Interrelated centers in a drawing of a doorway opening resulting in greater overall wholeness. (Source: Alexander, 2002)

of fractals.

2.2

Fractal Dimension

The term fractal was created in 1975 by French mathematician Benoit Mandelbrot, derived from the Latin fractus, and related to the English words fracture and fraction, to describe the group of odd shapes and geometries that he was studying (Gleick, 1998). They fall under a field of studies which emerged since the 1960s known as chaos theory, which deals, in a general sense with deterministic systems that are not predictable in the long term (Gleick, 1998). A more detailed explanation of chaos will be outlined below, but the key properties of fractals are that they demonstrate a cascade of scale invariant details, they show mathematical self similarity and they have dimensions that are not integers, or fractional dimensions. A graphical demonstration of these properties can be seen in the generation of the Koch curve (Fig. 3), first demonstrated in 1904 by Swedish mathematician Helge von Koch (Bovill, 1996, pp 11 - 13). A line segment of arbitrary length is divided into 3 equal segments, the middle segment is then replaced with 2 segments of equal length to the removed segment, forming an equilateral triangle. This process is then repeated for each Fig 8. - Centers that do not relate to each other resulting in less structural wholeness. (Source: Alexander, 2002)

of the new segments. After an infinite number of iterations, a curve is formed where any smaller part of the curve is similar to the whole, 11

and it is not quite a 1 dimensional curve or a 2 dimensional plane. Note 1: Derivation of the formula and example. For any shape divided into N identical pieces, if each piece can be related to the whole by a scaling factor of rN, then for euclidean shapes a power law can be established between N and rN in the form:

An important point to note is that with each iteration of the generative process, the measured length of the curve increases. For example if the original line segment of 1 unit length was divided into 3 segments, each with 1/3 unit length, after the first iteration the new curve would have 4/3 unit length. The next iteration will give a length of 16/9

D, the dimension value, evaluates to an integer in the case of euclidean shapes. D can be isolated in the equation by simple manipulation by taking the log of both sides:

units. Thus, as the number of iterations approach infinity, the measured length of the curve approaches infinity. Length therefore cannot be used to â&#x20AC;&#x153;measureâ&#x20AC;? a fractal curve, and another method of determining its distinguishing characteristic is needed. In his paper How Long is the Coast of Britain (1967), Mandelbrot, studying the possibility of a fractal structure in coastlines, highlights a formula first proposed by L.F. Richardson in 1961. This formula links the smallest geographically meaningful feature G, with the total length of a coastline, or curve, estimated based on G, denoted L(G) (Mandelbrot

Each iteration of the Koch curve is to divide the curve into 4 pieces, with each piece relating to the whole by a scaling factor of 3. Thus, N = 4, rN = 3. Substituting into the formula gives D = 1.26

1967). The relationship is given by:

Note 2: Hausdorff dimension, fractal dimension (D), self similarity dimension (Ds) and box counting dimension (Db).

this instance of a value of at least 1. D is the self similarity dimension,

Strictly speaking Mandelbrot evaluates the Hausdorff dimension of Koch curves and the coast of Britain and never explicitly states that coastlines are fractal. However, within the context of this dissertation the Hausdorff dimension, fractal dimension, self similarity dimension and box counting dimensions are used interchangeably. Another dimension defined within the wider field of chaos theory is the correlation dimension. Generally they all evaluate to the same value although there are some exceptions, but a detailed discussion falls outside the scope of this essay.

(1) Where F is a positive constant and D is another positive constant, in and Mandelbrot shows that fractional values of D exist and is related to the roughness of a curve - its fractal dimension. D is the fractal curveâ&#x20AC;&#x2122;s characteristic property and can be evaluated for any curve which can be separated into N pieces such that each piece can be related to the whole by a similarity ratio of rN (Mandelbrot 1967). This is given by the formula (Note 1):

(2) For a Koch curve, D can be evaluated to 1.26 while a Peano curve will evaluate to 2.0 (Note 2). This implies that an infinite iteration of the Peano curve will fill a 2 dimensional plane. Objects in nature do not display the same type of self-similarity as mathematical constructs like the Koch curve. However, some do show a cascade of detail like that of fractal curves. They can be considered statistically self-similar. Mandelbrot shows that the measured length of a coastline increases with an increase in the precision of the

Fig 9. - Graphs of log of measured length vs log of precision for a fractal and euclidian curve (Source: Bovill, 1996)

measuring device, due to more precise instruments taking in more details of the coastline i.e. more information, in this case, length. In 12

contrast, a euclidean shape measured with increasing precision would see its measured length converge at a fixed value. Graphing the natural log of the measured length versus the natural log of the precision highlights this increase in length property (Fig. 9), producing a linear graph which can be generalised by the equation (Bovill, 1996; Peitgen et al., 2004):

(3) Where u is the measured length and 1/s is the corresponding precision value. In this case d represents the slope of the line, as the equation is of the form: (4) d is also known as the measured dimension. It can be related to the self similarity dimension (Ds) by the equation Ds = 1 + d (Bovill, 1996; Peitgen et al., 2004). Fig 10. - The box counting method (Source: WIkipedia Commons)

Another method exists for evaluating the fractal dimension D of naturally occurring curves. This is given by plotting a graph of the log of the number of straight line measurements (a) with the log of the precision of the measuring device (1/s) at numerous scales. (Bovill, 1996, pp 35 - 40) The slope of this graph is the fractal dimension D. Mandelbrot (1967) estimates the fractional dimension of the coast of Britain to be approximately 1.25 using this method. While the above method is useful for a single curve, practical analysis of two dimensional images, such as a plan or a photograph of a view, requires another method. In these cases the fractal dimension can be estimated by measuring the box counting dimension of any significant detail in the image (Bovill, 1996, pp 41 - 43). This is achieved by superimposing square grids of increasing precision (i.e. decreasing size), denoted 1/s where s is the size of the grid, and counting the number of boxes, Ns which contain the significant detail of the image (Fig. 10). The slope of the natural log of grid precision (1/s) versus the natural log of the number of boxes is the box counting dimension. The use of the box counting dimension for an estimation of the fractal

Fig 11. - Using the box counting method to estimate D for Wright始s Robie House (Source: Bovill, 1996)

dimension of a two dimensional image gives a simple method for quantifying its cascade of detail at different scales. Bovill (1996) uses it to analyze the elevation and window detail of Frank Lloyd Wrights 13

Robie house, and compares it to a similar analysis of Le Corbusier’s Villa Savoye (Fig. 11 - 13). The Robie house maintains a relatively consistent fractal dimension at all scales, varying from 1.721 to 1.441 while Villa Savoye’s fractal dimension tend towards 1 at finer details (Bovill, 1996 pp 119 - 127, pp 136 - 144). As Mandelbrot and Richardson notes, curves with integer values of D are smooth euclidean shapes that do not display a cascade of detail. The convergence of Villa Savoye’s fractal dimension towards unity gives a quantified measurement of its lack of interesting details at varying scales. The accuracy and usefulness of the box counting dimension, however, depends on the number of box counts made as well as the nature of Fig 12. - Box counting dimension for window detail in Wrightʼs Robie house (Source: Bovill, 1996)

the drawing or image being analysed. The applicability of using the fractal dimension as a direct measurement of fit or as a generator of rhythmic variation has been questioned in literature. These will now be examined and discussed.

Fig 13. - Using the box counting method to estimate D for Le Corbusierʼs Villa Savoye (Source: Bovill, 1996)

3.0

Environmental Psychology and Biophilia

3.1

Questioning Bovill

Bovill (1996) proposed that matching the fractal dimension (D) of a building with the D of elements in its surroundings, using the box counting method, would produce a better and more visually appealing “fit”. This is known as the contextual fractal fit hypothesis and is suggested by Bovill as a possible analysis method for using fractal geometry in architecture. He proposes that the buildings in Amasya, Turkey (Fig. 14) (Bovill, 1996, pp 144 - 149) and the Sea Ranch Fig 14. - Dominant hill, housing elevation and town plan used in analysis of Amasya, Turkey (Source: Bovill, 1996)

Condominium, California, US (Fig. 15) (Bovill, 1996, pp 180 - 184) are two examples of buildings that have similar D values to their surrounding landscape and thus seem to sit better in their context,. He calculated relevant D values for both buildings and context at Amasya but only for the context at Sea Ranch. An independent study by Vaughan and Ostwald (2010) using Bovill’s original drawings for the analyses and supplemented by their own drawings produced from the architect’s originals in the case of the Sea Ranch Condominium found a relatively high variation in estimated D values between the buildings and the context. In the case of Amasya, they found an 8% variation in D between the dominant hill (the feature Bovill chose for his analysis) and the local elevation. The Sea Ranch Condominium showed a difference of 21% to 13.2% variation, depending on the drawings used

Fig 15. - Sea Ranch coastline and condominium elevation. (Source: Bovill, 1996)

between the elevations of the building and the coastline of the Sea Ranch. According to Vaughan and Ostwald (2010), “houses that appear to be genuinely visually similar will often have a gap of less than 2%. Indeed a gap of more than 8% suggests a significant difference in visual character.” Another study by Stamps (2001) found no correlation in preference among a group of 64 individuals of even division in gender for contextual fractal fit in rendered images of buildings in a landscape. The study also found that in generated skylines with “identical variances in building features (heights, widths, depths, and setbacks), but [with] one set [having] a fractal structure and the other [not, the] scenes without fractal structure were slightly preferred over the scenes with fractal structure” (Fig. 16 - 18) It is important to note that these studies show that using the fractal dimension as an absolute measurement of contextual fit does not seem to be supported by evidence. The fractal dimension is a single variable 15

in a power law describing the relationship between scaling and length, it is meaningless by itself and is merely a measure of the “fuzziness” of a geometric shape, or the degree to which a shape extends to a higher dimension. Using analysis methods such as the box counting method is a useful tool in quantifying the amount of detail in an image, but that does not mean the number can simply be “plugged back” in to a new generative system and be expected to give a meaningful result. As Stamps (2001) shows, depending on the method of generation, other variables play a greater role in modifying the final shape generated that have a greater visual impact. Vaughan and Ostwald (2010) notes that the efficacy of computational software, as well as the type of drawings/images used in the analyses have a large impact on estimated D values. Greater accuracy in analysis software and drawings actually yielded smaller differences in D values in the case of Amasya. Joye (2007) also points out that Stamps “used fractal contours, and the self-similarity was not readily perceivable in the representations”. In essence Stamps’ study used D as a modulating Fig 16. - Generated image of skyline with matching D to mountain context (top) and non matching variation (mid & bottom) (Source: Stamps, 2002)

tool for determining dimensional variations in buildings in comparison with random variation, so while it does question Bovill’s contextual fit hypothesis, it does not in general contradict Biophilian and information theory hypotheses, to be outlined below, concerning the importance of complexity and detail in built environments. A quick visual inspection of the images generated for the second Stamps study also notes that there are large amounts of background information (Fig. 16 - 18). These themselves contain a high amount of fractal detail (clouds) or elements that impact highly on one’s aesthetic judgement of the image, such as the presence of water (Ulrich, 1983). While these elements remain identical between the 2 images to be compared, the possibility exists that these features could overwhelm or impact in unaccounted for ways on the judgement between fractal or random variation. The comparison in this case is

Fig 17. - Generated skyline with fractal variation (top) and random variation (bottom) (Source: Stamps, 2002)

also between fractal variation and random variation, not between fractal variation and no variation, as in euclidean architecture, so the overall argument is not weakened. Ostwald and Tucker (2007) also question some of the basic assumptions of Bovill with regards to the usefulness of D as a tool for analysing historical buildings. In particular issues are taken with the the assumption that a cascade of detail at different perspectives is needed and with the “Kantian belief that nature is innately beautiful 16

and that people are drawn the the appreciation of natural forms because of this.” For the former part Ostwald & Tucker cites the emphasis since Roman architecture, and in particular, Greek and Renaissance architecture on a singular viewpoint. This is irrelevant to the point however, as architecture is always experienced by the user or inhabitant as a cascade of views from different angles and at different scales. The architectural theory or intent is not the issue. Salingaros (2000, 2003) also outlines a clear scientific basis for our visual need of not just cascade of details at different scales, but ornamentation. In the latter criticism, no citation is given for the claim that “the Kantian belief in the essential rightness, goodness or beauty of nature is not Fig 18. - Generated skyline with fractal variation (top) and random variation (bottom) (Source: Stamps, 2002)

supported by strong evidence and it does not stand up to close scrutiny”. On the contrary there exists evidence, as will be shown, which highlights the importance of natural environments on aesthetic preference and physiological well being, and that these effects are demonstrably universal across varying cultures. Ostwald & Tucker also take issues with Mandelbrot being “highly critical of Modern architecture while praising Beaux-Arts or Baroque buildings”, claiming that “this is problematic for a range of reasons, most notably because it places an undue positive emphasis on higher fractal dimensions while dismissing those that have relatively abstract or plain forms as being alienating”. No explanation is given for why this is problematic, other than an implicit assumption that dismissing abstract or plain forms is somehow wrong. This is a tautology.

3.2

Biophilia and Habitat Preference

There have been several studies carried out on the aesthetic qualities of fractal patterns and their effects on human physiology. Firstly, a study by Hagerhall et al. (2008) using quantitative Fig 19.1 - Fractal silhouette of D value 1.14 (Source: Hagerhall et al., 2008)

electroencephalography (qEEG) measured the alpha, beta and delta wave responses of 32 adults in the frontal, parietal and temporal lobes while viewing a black and white silhouette diagram generated with varying fractal dimensions (Fig. 19.1 - 19.4). The results showed a statistically significant response in alpha and beta waves to fractals of mid D dimension. Hagerhall et al. note that alpha wave components of the EEG show “a wakefully relaxed state” (Ward, 2003), while beta components relate to external focus and alertness (Kolb & Whishaw, 2003). Thus the results are “consistent with the hypothesis that [mid

Fig 19.2 - Fractal silhouette of D value 1.32 (Source: Hagerhall et al., 2008)

range D values] are [the] most restorative and relaxing” and “[generate the] most activation in the processing of the pattern’s 17

spatial properties”. The advantage of this study is that it directly measures physiological responses rather than relying on individual preference judgements as in the case of studies by Stamps (2001) qEEG is generally a “good indicator of cortical arousal” (Hagerhall et al. 2008). Hagerhall’s work is consistent with previous landscape preference studies they have carried out (Hagerhall et al. 2004). The results are also consistent with an independent study which showed that people have a preference for fractals of mid range D value (1.3 to Fig 19.3 - Fractal silhouette of D value 1.51 (Source: Hagerhall et al., 2008)

1.5) and that images of natural scenes of D ranges between 1.6 and 1.4 played a role in reducing stress (Taylor, 2006). The images calculated to have mid range D values in this study are of savanna scenes, while forest scenes have comparatively higher D and less stress reduction. Further support is given by studies from Aks & Sprott (1996) using computer generated fractals that also showed a preference for fractals with a D value of 1.3. Taylor (2001) proposes that the preference for these values of D could be due to their abundance in natural fractals, noting that clouds have a D of 1.3. The suggestion from these studies

Fig 19.4 - Fractal silhouette of D value 1.70 (Source: Hagerhall et al., 2008)

is that have an innate preference for fractals within a certain range of D as these patterns are what is most common within the environment in which we evolved. At the general level, within the field of environmental psychology there have been several studies that address human’s innate preference for certain natural structures. A recent cross cultural study by Falk & Balling (2010) looked at landscape preference between different biomes (savanna, rain forest, deciduous forest, coniferous forest and desert) among people of Nigerian descent, as a follow up to a similar study (Falk & Balling, 1982) among people of American descent. The studies found that preference for a savanna like setting was highest in both samples with differences in preference dependent on the local biomes in which the sample was exposed to as well as other social and cultural factors. The studies noted that younger children seemed to have a higher level of preference for savanna landscapes and that preference for other biome types never surpassed preference for savanna biomes (see also Orians & Heerwagen, 1992). The implication seems to be that there is an innate preference among people of different cultural, educational and social systems for a certain visual structure, which, however can be shaped over time by personal experiences. The savanna preference is telling in relation to work by Taylor which links mid range fractal values with savannas. 18

Whether or not fractal properties play a role in determining this preference is a matter of speculation but the fact that there is a certain universality to the visual preference of environments is encouraging. Indeed there is a large body of evidence that habitat preference is genetically determined. (Holt, 1987; Jaenike & Holt, 1991; MacCallum et al. 1998; Martin, 1998). Falk & Balling argue for a possible explanation of such common properties among people as a result of evidence that suggests that “it is because it was present in this [a] founder population of Homo sapiens” consisting of “perhaps no more than a few thousand closely related individuals, living in Africa some 70,000 years ago”. Mithen (1996) also suggests that since the Holocene, approximately 10,000 years ago, the brain has remained structurally identical to that of modern humans. If all of modern humans are descended from a small founder population, then it is natural that their common genetic habitat preference still exists within us. Dutton (2003) gives evidence for this by noting that savanna type landscapes are prevalent in certain types of art worldwide, independent of culture, such as in calendar art and in the design of public parks. Finally, a large amount of research has been carried out which have shown that people have a preference for natural scenes over urban scenes (Ulrich, 1981) and that natural environments play a key role in stress reduction (Ulrich, 1983; Ulrich et al., 1991). This is consistent with Taylor’s (1996) findings regarding stress reduction and fractal dimensions. Joye (2007) gives a comprehensive review of the evidence for a focus on biophilia in architecture, noting in particular the possibility of fractal geometry as a method for structurally replicating natural patterns in order to create environments that induce positive affective responses in people.

Note 3: A simple mathematical demonstration of chaos Chaos, it needs to be noted, is an inherent property that arises from certain relationships and is not due simply to noise, randomness or lack of retrievable information. It is completely possible to have a simple recursive algorithm that is fully understood mathematically, but for certain parameters, long term prediction of general behaviour is impossible. (Gleick 1998) Consider the logistic difference equation (May 1974, 1976):

4.0

Chaos and Information Theory

4.1

Iteration and The Mandelbrot Set

Chaos as understood in mathematics is different from the general meaning of the term which is often used synonymously with randomness or disorder. Chaos theory deals with systems that are completely deterministic in so far as the next immediate state of the system depends on the previous state, but any long term prediction of the system is impossible. We may also completely understand all the rules which govern the system at a local level, however we are unable

This equation was used as a simple model for population growth over discreet periods of time. Here 0 < x < 1 and it represented the population of a system, 0 being extinction, 1 being the maximum population the given environment can support. The population for each period of time (xn+1) depends on the previous period of time (xn). The rate of growth r can be varied and the evolution of the system can be computed for changing values of r by running the equation through an arbitrarily large number of time periods. May showed that as the value of r increased the system progressed from extinction to stable equilibrium, dynamic equilibrium and finally chaotic behaviour. A graph of population versus r graphically highlights this transition to chaos, with successive bifurcations of final equilibrium level past r = 3.

to make any meaningful predictions about the system at the global level based on these rules, except for the immediate future (Cartwright 1990, Gleick 1998). Conversely, any deductions about the initial state of the system is also impossible based on its current state (Note 3). The implication for design as Cartwright (1991) notes is that for systems that are chaotic (Cartwright is referring specifically to planning, however it applies equally to design), as such systems are sensitive to the “cumulative effects of various kinds of feedback” and that “on an incremental or local basis, the effects of feedback from one time period into the next are often perfectly clear”, the best strategy may be one which is “incremental rather then [sic] comprehensive in scope and that rely on a capacity for adaptation rather than on blueprints of results”. Essentially due to the very nature of chaotic systems, if architecture is chaotic then the only logical process of design is an iterative process from the ground up, rather than a top down systematic design. While speculative, anecdotal experience suggests that the very nature of design is chaotic, with its

Bifurcation diagram for r=2.4 to r=4.0 (Image source: Wikipedia Commons) All the information needed to generate the graph is encapsulated in the above difference equation, however the final graph is infinitely complex. Successive zooms on the chaotic region shows a fractal like structure of regions of chaos and stability. There is a sort of order within the disorder. Feigenbaum (1980) showed that there was a “universality” to the way in which systems tended towards chaos that does not depend on its particular details. A Poincaré map of time n+1 versus time n highlights a structure towards which the system would tend to evolve in the chaotic regions. This structure, a strange attractor, is infinitely complex, but nonetheless it is a layer of order within the chaos.

various possible inputs at every moment of the process which give an unpredictable but deterministic impact on the final design. This is directly analogous to the process of hierarchical design proposed by Salingaros (2000). Similarity is also found in Alexander’s (2002) process of iterative feedback at each step and each scale of the structure, in particular, his description of a step by step recursive process for the design of a column which creates a “center” while preserving and enhancing the overall structural “wholeness” (Fig. 28 30) (Alexander, 2002, pp 128 - 131). Further, the observation that it is impossible to make deductions about the initial state of a systems from its current state is related to the concept of “emergent properties” that could be common to all successful works of architecture 20

(Salingaros, 2000), i.e. that properties can appear in complex systems that cannot be derived from “the sum of its parts”, or its initial state and governing rules. Recursive iteration is a key component of chaotic systems, fractal Poincaré map of time n+1 vs. time n (Image source: Cartwright, 1991)

shapes can also be generated through the use of iterated function systems (Bovill, 1996, pp 47 - 55). The Barnsley Fern (Barnsley, 1988) can be generated through a set of affine transformations, i.e. transformations that allow shear, rotation, and reflection of an original form (Fig. 20, Fig. 21). It is notable as it is a complex pattern that is similar to a natural structure but was produced with a set of iterated transformations defined by 24 numbers. It is a highly complex structure which emerges from a simple set of equations and numbers. Another example of a structure generated by iteration, and an image often used as a graphical representation of chaos is the Mandelbrot set, discovered by Benoit Mandelbrot in 1979 (Bovill, 1996, pp 68 70). It displays an infinite amount of complexity on zooming in to its details, and while each pattern is statistically similar to the whole, no two patterns are strictly identical as in a Koch curve or Sierpinski triangle. The detail displayed is only limited by computation power. Its generation and definition is the result of an attempt by Mandelbrot to find for what values of C in the Julia set would produce a connected set (Fig 23, Fig 24), in which C was a complex number. Julia sets, at

Fig 20. Barnsley Fern (Source: Bovill, 1996)

the simple level, involves the iteration: Z(n+1) = Z(n)2 + C Where Z and C are complex numbers. The Julia set is defined as the boundary between values of Z that tends towards infinity and towards the origin, or the boundary between the basin of attraction towards zero and towards infinity. The Julia set produces a fractal shape for values of C not equal to 0 on the Argand plane. The Mandelbrot set involves the iterative sequence: C, (C2 + C), [(C2 + C)2 + C], … If the above sequence is bounded, then it falls within the Mandelbrot

Fig 21. Generation of Barnsley Fern through affine transformations (Source: Bovill, 1996)

set. A graph of the Mandelbrot set on the Argand plane gives a highly complex fractal shape that at the same time can be fully described 21

Fig. 22.1 The Mandelbrot Set (Generated by the author with GNU XaoS, free software licensed under GPL. http://xaos.sourceforge.net)

Fig. 22.2 Zoom of details within the Mandelbrot Set (Generated by the author with GNU XaoS, free software licensed under GPL. http://xaos.sourceforge.net)

Fig. 22.3 Zoom of details within the Mandelbrot Set (Generated by the author with GNU XaoS, free software licensed under GPL. http://xaos.sourceforge.net)

Fig. 22.4 Zoom of details within the Mandelbrot Set. Statistical self-similarity and infinite complexity (Generated by the author with GNU XaoS, free software licensed under GPL. http://xaos.sourceforge.net)

with the above computations. It can be considered a highly complex but coherent shape with a simple underlying structure (Fig 22.1 22.4). This a demonstration of an ideal, of a shape with infinite depth and detail, but which is fully encapsulated by a short description. This property of compressibility ties into an information theory perspective on our subjective perception of â&#x20AC;&#x153;beautyâ&#x20AC;?.

4.2

Algorithmic and Hierarchical Theory of Beauty

While evidence has been shown that people do prefer natural or complex fractal views and an evolutionary biological basis has been Fig 23. Connected Julia Sets, including set of Z2 - 1, or where C = -1 (Bottom). (Source: Wikipedia Commons)

put forward for how this may have come about, there still exists the question of why complexity should in any sense be inherently interesting. The interestingness of nature is not due to our recognition and identification of real natural forms such as trees or mountains, but as the peak shift effect shows (Ramachandran & Hirstein, 1999), is due to some perception of its underlying structure. In the case of natural forms this is its recursiveness that could be embodied in a quantitive description such as the fractal dimension. Why then is this specific type of structure interesting at all? Jurgen Schmidhuber (1997, 1998, 2007, 2009) proposes an information theory solution to the question, based on the Kolmogorov complexity of a piece of information. This piece of information can be

Fig 24. Unconnected Julia Set. (Source: Wikipedia Commons)

visual information from our retina. The Kolmogorov complexity of a finite piece of string within computer science is defined as the shortest program that can compute it on a universal Turing machine (Note 4) and halt. (Schmidhuber, 1997) The shortest program refers to the smallest piece of information that can be used to compute or calculate all of the whole. For example at a general level the Mandelbrot set as explained above would have a very low Kolmogorov complexity compared to its overall complexity as all the information required to replicate the image is encapsulated within its defining equation. Similarly the bifurcation diagram has a low Kolmogorov complexity compared with its whole as it can be computed with the single logistic difference equation as shown in Note 3. Schmidhuber proposes that our notions of beauty is related to 2 concepts, compressibility and interestingness. For the former, as our eyes must transmit and our brains store raw visual information, a compression algorithm is utilized in order to maximize efficiency of these steps. A piece of 24

Note 4: Turing Machines The definition and discussion of Turing machines falls outside the scope of this text, however it is sufficient to say that all modern computers are approximations of a Turing machine and that any and all calculations that can be made on modern computers can, with time, be made on a Turing machine. With the application of the invariance theorem and compiler theorem (Schmidhuber, 1998), this means that so long as the brain functions as a Turing machine, then the principles of complexity and interestingness can be applied. Whether the human brain functions as a Turing machine is unclear, however there are some evidence that at least on the structural level there are some similarities, specifically “the need for robust active maintenance and rapid updating of information in the prefrontal cortex appears to be satisfied by bistable activation states and dynamic gating mechanisms.” and that “these mechanisms are fundamental to digital computers”. (OʼReilly, 2006)

information’s Kolmogorov complexity defines its maximum compressibility. Objects that tend to have a higher compressibility, i.e. complex objects that can be described in simple terms, are perceived to be more subjectively “beautiful” as these results in greater efficiencies in informations storage. For the latter, Schmidhuber notes that as our brain is constantly seeking out new compression algorithms, or new “patterns”, what we find “interesting” are views in which there are new patterns to be discovered and that once the full compressibility of something has been discovered, we no longer find it interesting (in the immediate short term perceptual sense). This is supported by the idea that our attention is diverted towards areas which shows the highest bayesian surprise, or new novel patterns, compared with existing models within our brain of patterns to be expected (Baldi & Itti, 2005). Thus beauty as defined by Schmidhuber is a careful balance between order and surprise. Fractal geometry, in comparison to euclidean geometry, satisfies these requirements better as fractal geometry can have a much lower Kolmogorov complexity compared with its overall complexity - it is more compressible. It also has more “interestingness” compared with euclidean geometry, due to the observation that more precise measurement of fractal curves yields greater length, or more information, and in the case of shapes like the Mandelbrot set, more patterns that are similar but not identical to the whole. This results in the possibility of new compression algorithms to be discovered compared to euclidean geometry, which, beyond a certain point, yields no additional information with more precise

Fig 25. “Beautiful” face generated with fractal squares (Source: Schmidhuber, 1997)

measurements. The triangle is less visually interesting than the Sierpinski gasket, which is less interesting than the Julia and Mandelbrot set. Schmidhuber’s theory has echoes in Ulrich’s (1983) list of visual cues associated with positive responses - complexity, patterns, depth, ground surface and texture, and deflected vista, as well as in Kaplans’ (R. Kaplan & Kaplan, 1989; S. Kaplan, 1987, 1988) model of involvement and understanding of an environment, in particular the reference to complexity, coherence, legibility and mystery. In both these models there are requirements for a complex but understandably compressible environment (coherence, legibility, patterns) with a high amount of “interestingness” (mystery, deflected vista). As a

Fig 26. An example of low complexity art generated with legal circles (Source: Schmidhuber, 1997)

demonstration of his theories Schmidhuber puts forward two examples of what he labels “low Kolmogorov complexity 25

art” (Schmidhuber, 1997, 1998). The first is an image of a female face that was rated as beautiful among a sample of 14 people (Schmidhuber 1997), generated according to his theories. The second are a series of illustrations, similarly generated. In both these cases a fractal system was used to ensure high complexity while maintaining low Kolmogorov complexity, the former with 3 superimposed grids of fractal squares based on powers of 2 (Fig. 25), the latter on a fractal pattern of legal circles (Fig. 26, Fig. 27). Fig 27. The generative grid for low complexity art. (Source: Schmidhuber, 1997)

Schmidhuber notes that while the images are aesthetically appealing, and can be compressed to a simple algorithm, the process of discovering the correct pattern in the first place was difficult. This is due to the non computability of Kolmogorov complexity (Schmidhuber, 2007), which means that there is no general method to find “the shortest program computing any given data” (Schmidhuber, 2009, p8). The significance of this observation is that the creative act, which may be approximate by the discovery or creation of “novel, non-random, non-arbitrary patterns with surprising, previously unknown regularities” (Schmidhuber, 2009, p9), can not be generalised into a computation method from which all programs can be generated i.e. there is no singular universal method of creativity.

Fig 28. Recursive generation of column (Source: Alexander, 2002)

Salingaros (2000, 2003) makes a similar argument to Schmidhuber, but gives further elaborations on our physiological requirement for detail. In particular he notes that the anatomical makeup of the eye is such that we require high amounts of detail in our visual information, and the importance of symmetry and patterns to allow us to organize visual information (decrease computational overhead) (Salingaros, 2003). He also notes the inherent necessity of ornamentation as part of our visual language due to our requirement for “a lower threshold of visual information” (Salingaros, 2003). He cites the fact that environments that do not provide adequate visual information similar to the environment in which we have adapted over our evolutionary timeline can induce physiological stress in us. He proposes a “hierarchical” system of formal generation, not too dissimilar to the “living system” of “centers” proposed by Alexander (2002). Salingaros notes that all complex systems eventually adopt a

Fig 29. Final generated column (Source: Alexander, 2002)

hierarchical organisation of a cascade of details and an emergent complexity built upon the interaction of simple initial rules, noting such structures in nature as well as computer programming. This 26

hierarchical system would be able to generate a cascade of details at different scales, a fractal structure, that would better satisfy our need for visual information. Architecturally speaking the key component of his theory is the importance of significant features at a cascade of scales. For example, colonnades and fenestration can define one scale, or “center”, in Alexander’s terms, while within the colonnade, column sizing, spacing, capitals, and ornamentation can each define other scales. Trims, frames, and baseboards define smaller scales internally, and texture and materials define further smaller scales (Salingaros, 2000). The importance is in a coherent relationship between the varying scales - in particular that the larger scales depend on the existence of smaller scales. The minimalist ornament free architecture of euclidean geometry generally lack this web of hierarchy - often scales jump from that of massing to that of texture and material. According to Fig 30. Column in context (Source: Alexander, 2002)

Salingaros (2000) - “The greatest buildings (Parthenon; Hagia Sophia; Dome of the Rock; Palatine Chapel; Phoenix Hall, Kyoto; Konarak Temple, Orissa; Salisbury Cathedral; Baptistry, Pisa; Alhambra; Maison Horta; Carson, Pirie, Scott store; etc.) succeed in good part because they integrate their different subdivisions into a hierarchy of interconnected scales.” Buildings generated from fractal geometry would, according to him satisfy this hierarchical requirement for scales, but this must be tempered to a degree by relating the scales to that of human perception (Salingaros, 2001) (Fig. 31).

4.3

Complexity, Hierarchy and Wholeness in Historical Architecture

For all works of architecture there can be significant cultural and historical contexts attached to it. It was stated earlier that buildings are not just works of text to be interpreted within a cultural landscape but also exist as structures within a much larger physical landscape. The converse of this is just as true. It is therefore difficult to make a value judgement on the aesthetic appeal of a building while ignoring any possible effects of culture, history, philosophy etc. on such a judgement. Thus there are some difficulties in giving specific Fig 31. Pattern at human and inhuman scale (Source: Salingaros, 2001)

architectural examples as to do so can attract focus away from the main point that regardless of context, there is an innate requirement for a certain level of complexity within our built environment. 27

However there is a consistent theme among Salingaros, Alexander, Bovill and Joye that historical works of architecture of the premodernism era display a certain level of visual complexity and integrated fractal geometry. Salingaros (2000, 2001) theorizes that people have a subconscious tendency to create fractal details and that it was only with the conscious cultural decision towards the minimalism of, for example, Bauhaus aesthetics or the monumentality of euclidean geometry that fractal properties in architecture were removed. At the extreme end of the scale Joye (2007) cites Gothic Fig 32. Generation of a Hindu spire through self-similar iteration, resulting in the creation of a fractal object (Source: Joye, 2007)

architecture as a general example of highly complex fractal structure in architecture, and that similarly complex forms can be generated through self similar iteration (Fig. 32). Alexander gives examples of a wider range of works which he deems has a large degree of “life” or structural wholeness, including such works as “the Parthenon, Notre Dame or Chartres, the mosques at Isfahan, the Alhambra… the earliest buddhist temples in Japan like Tofuku-ji” (Fig. 35, Fig 36), but also unexpected examples such as the underside of the elevated tracks in Brooklyn, New York (Fig. 34), the slums of Bangkok (Alexander, 2002, Book 1, pp 40 - 62) or even the distribution of electricity pylons near Dumbarton bridge, San Francisco (Fig. 37) (Alexander, 2002, Book 2, p164). With regards to contemporary architecture, Bovill (1996) and Alexander (2002, Book 2, pp 154 - 155) in particular cites Frank Lloyd Wright as an example of an architect whose works exhibit high fractal complexity. Indeed the general style of architecture classified as “organic” seem to demonstrate this tendency,

Fig 33. Detailed fractal structure of the Sagrada Familia, Barcelona (Source, accessed 15/04/11: http:// www.flickr.com/photos/oksidor/3351776629/)

for example the works of Antoni Gaudí (Fig. 33).

Fig 34. View of underneath the tracks at Brooklyn (Source: Alexander, 2002)

5.0

Conclusion

Architecture, and its extensions, landscape architecture and planning, has always been about shaping the environment around us, but it is only recently on the evolutionary timeline that we have had the capability to shape the environment on a mass scale. Landscape preference is common and innate among all people and habitat preference is genetically determined. Before any design can happen it is imperative that we first understand the type of landscape to which our visual, perceptive and cognitive systems have adapted, and how they have adapted, so that we can make an informed decision about how to create a positive environment. Contrary to established architectural theory, it has been shown that Fig 35. The Alhambra (Source: Alexander, 2002)

euclidean forms are not inherently beautiful, at least as part of our visual landscape. The environment in which we have evolved and to which we have adapted, instead, displays a highly complex but ordered structure that can be described by the mathematics of fractals. Studies of our preference for fractals is consistent with landscape preference studies. To design and build in this environment requires an understanding of this existing physical structure, at the intuitive level if not the mathematical so that we may make interventions which preserve or enhance its overall structural â&#x20AC;&#x153;wholenessâ&#x20AC;?. Concurrently an understanding of information theory can give an explanation from first principles of the type of complexity and pattern which our visual field requires. Studies of chaos theory, as well as the works of Alexander and Salingaros, suggests that a certain methodology of

Fig 36. Indian arch (Source: Alexander, 2002)

design can best give rise to complex structures - this methodology tends towards a more generative approach, with step by step iterative feedback of the design of forms being favoured. Ultimately this simple to describe process, analogous to the iterated functions which produce the chaos of bifurcation diagrams, and the fractal shapes of Barnsley Ferns and Mandelbrot sets, gives rise to complex emergent shapes which at the same time have a low Kolmogorov complexity. It is important to note that no prescription has ever been made with

Fig 37. Pylons at Dumbarton, San Francisco (Source: Alexander, 2002)

regards to architectural style or theory. A generative design process is not mutually exclusive to any other architectural design consideration such as culture, economy, history, structure, energy requirements, etc. It is simply a new frame of reference that puts important constraints on the process. Historical buildings have been shown to demonstrate a 29

certain fractal complexity, but the idea is not to replicate their designs, form or style at the superficial level to create kitsch. The attempt is instead to understand what, at the fundamental level, gives certain environments a positive quality for us so that we may induce such a quality on future buildings. As Salingaros (2000) states, â&#x20AC;&#x153;style is a matter of choice, but architectural order is of profound importance to the human experience.â&#x20AC;? In conclusion, the key requirement to be derived from all this is the importance, visually, of a cascade of significant details at varying scales, which interlink in a coherent, hierarchical and fractal structure. It is interesting to note that the field of architecture itself sits at a particular scale in the cascading structure of our entire designed environment. This structure extends from the small scales of industrial design to the large scale of urban planning (Evolved complexity and fractals are known in urban planning, see Marshall, 2009). In a sense, the designed environment is itself a complex fractal. Should architecture then not work to preserve and enhance this structure?

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