Fractals have significance in many fields and are an active field of research. From building nanoscale fractal architectures in gold (HianáTeo, 2010) to efficient principles in programing biological form in genetic design

(Weibel, 1991), to universal theories in physics (El-Showk, Poland et al., 2014), fractals are receiving much attention. Here I focus on the relationship between fractals, and architecture and the larger built environment.

Fractal geometry and fractal dimension have been used in generative architecture and design and have become increasingly popular in analyzing natural and artificial objects. With mathematical tools at our disposal it is now possible to analyze the fractallity of architecture. I will review below the historical background of fractals in architecture, current research, and applications currently being developed.

Fractals are self-similar geometric patterns that repeat at multiple scales. Benoit Mandelbrot famously applied the concept of fractals to the measurement of the coastline of Great Britain. If you measure Britain’s very jagged

coastline with increasingly smaller measuring devices, first at the level of kilometers, then with a meter stick, and even at the level of centimeters and below, you’ll continue to see a self-similar jagged structure. As you use smaller and smaller measuring devices, the length you measure gets increasingly longer.

Architecture has long employed fractals as an organizational principle. Architecture is perhaps unique in the arts in

that a geometric pattern could represent something at many different spatial scales. In architecture especially, a geometric pattern may be designed to occur at many discrete scales from the pattern or motif in window muntins to

the general layout of the plan and even the organization of the larger urban fabric. A good example of this idea is Frank Lloyd Wright’s plan for the Palmer House (Fig. 2.2.1). In this plan the repeating figure of an equilateral

triangle is evident at 7 different spatial levels (Eaton, 1998. Joye, 2011).

This technique follows patterns found in natural objects. The leaves of a fern are composed of smaller replicas of themselves for instance. Fig. 2.2.2 shows drawings on the 15th century Topkapi Scroll found in Turkey. These

drawings represent a quarter of a dome in plan. The dome is composed of a tiling pattern that forms threedimensional shapes or muqarnas. Muqarnas are often seen in Islamic architecture and are used to transition from a square to a circle or from an orthogonal space to a dome in 3 dimensions.

**Figure 2.2.2. 15th century Topkapi Scroll showing a quarter section of a dome in plan that is further subdivided into miniature muqarnas which reflect the geometry of the whole at a smaller scale (By Unknown architect) – http://kilyos.ee.bilkent.edu.tr/~history/geometry.html, Public Domain, https://commons.wikimedia.org/w/index.php?
curid=17697287**

Fig. 2.2.3 shows an example of how this technique was used to ornament the corbelling of an apse when

transitioning from orthogonal to circular shapes. Notice that each individual unit is a smaller version of the larger apse itself. Additionally, the muqarnas are often adorned with smaller tiling patterns, creating another level of self similar detail. Similarly, the apses, domes, colonnades, etc. could be repeated to create an entire building or a number of buildings in a complex. 2

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2 Self-similarity might suggest a certain worldview. The idea that the whole is reflected in each part is sometimes associated with a hologram because a holographic image if shattered will re-appear complete in the broken fragments. Whereas, the notion that the whole is greater than the sum of its parts (superadditivity) is more associated with the notion of emergence, where the macro behavior of a system is different and more complex than the parts themselves or the interactions between the parts.

Figure 2.2.3. Detail of Red Mosque in Safed, Israel 1276. By Bukvoed (Wikipedia Commons).

Although multiscale self-similarity has been evident in architecture for a long time the name “fractal” and a mathematical model was only recently applied rigorously to them by Benoit Mandelbrot (Mandelbrot, 1983) although they had been studied in certain mathematical circles prior to him. Most notably by Gaston Julia who

published a paper in 1918 that was influential to Mandelbrot entitled, ‘Memoire Sur L’iteration des fonctions rationnelles’ (Julia, 1918). This formulated the basic idea that has been influential in non-linear dynamical systems and deterministic chaos, namely that from a simple set of rules an infinitely complex object can be created. It was

the advent of modern computing that gave Mandelbrot the ability to iterate the Julia set and see ‘fractals’ for the first time (Mandelbrot, 1979). Mandelbrot researched and appropriated many mathematical models and collected them under a common framework. One tool, which Mandelbrot borrowed from Felix Hausdorff, is a method for determining the fractional dimension of an object called Hausdorff dimension or fractal dimension. Another method Mandelbrot used for approximating the fractal dimension of real objects such as coastlines or mountains etc. is also based on Hausdorff dimension and will be called box-counting dimension. Box-counting dimension is the primary analytical tool used in this study.

Batty has explored the relationship between fractals, natural phenomena and the form of cities and has used fractal geometry to explain the form and characteristics of cities. Much of his research is covered in ‘Complexity and Cities’ (Batty, 2005). Others have used fractal geometry to research the growth of cities such as Sara Encarnacao et al., ‘Fractal Cartography of Urban Areas’ (Encarnacao, 2012) who discusses five types of urban environments in Lisbon Portugal based on their fractal dimension. The present author has also looked at the relation between fractal dimension and various attributes of cities, City Population Dynamics and Urban Transport Networks. Here we measured various cities worldwide and discovered FD to peak in large complex urban environments like New York and Tokyo. FD was observed to plateau between 1.7 and 1.8 (Abundo, Driscoll et al., 2013).

Fractals are found historically in art and architecture, from before the conceptualization of fractals (Joye, 2008). Fractal geometry has likewise been used to analyze individual buildings and the architectural ideas associated with

them. Bovill published a book titled, ‘Fractal Geometry in Architecture and Design’ in which he analyses two significant works by Frank Lloyd Wright (FLLW) and Le Corbusier respectively using box counting dimension. In Bovill’s analysis he concludes that Wright’s work is more complex than Corbusier’s and in some ways more related to nature than modernist architecture (Bovil, 1996). These results and conclusions are challenged by Ostwald et al. who analyze five buildings by Wright and five by Corbusier and determine their box counting dimension is not as different as Bovill suggests (Ostwald, 2008) in some fashion Eaton also applies fractal dimension to the work of Wright (Eaton, 1998).

Another use of fractal geometry as a means for assessing man-made objects is undertaken by the physicist Richard Taylor from the University of Oregon who used fractal dimension to evaluate the authenticity of drip paintings attributed to the painter Jackson Pollack (R.P. Taylor et al., 2006). This study was tested and

elaborated on by Jim Coddington et al. (Coddington, 2008). In addition to exploring the use of fractal geometry in analyzing architecture, a number of authors have also interpreted fractal geometry and made various assertions regarding its theoretical significance. Al Goldberger discusses a dichotomy between the Romanesque and Gothic styles that he characterizes with fractal dimension and links to the fractal qualities of the brain itself. He makes this bold claim in ‘Fractals and the Birth of the Gothic, reflections on the biological basis of creativity’ (Goldberger, 1996). Others have picked up this theme of linking more complex architecture characterized by fractal dimension to nature. Again the juxtaposition between Frank Lloyd Wright’s Organic architecture and the modernists is a common thread. Joye has linked complexity in architecture to nature, suggesting that a certain range of fractal dimension relates to our evolutionary development and the natural landscapes that were favorable to our survival and well being: ‘Fractal Architecture Could Be Good for You’ (Joye, 2008). This idea is related to what has been termed ‘biophilic’ or ‘biomimetic’ design as discussed above.

A more theoretical approach to applying fractal geometry in nature and urbanism is provided by Geoffrey West (West, 2005) and Louis Bettencourt (Bettencourt 2013) respectively. West and Bettencourt study scale free behavior in a variety of settings. West theorizes that fractal geometry is efficient in distribution and uptake systems and therefore selected for in biological evolution. West and Bettencourt’s work is inspired by Kleiber’s Law pertaining to

biological allometry as well as Zipf’s Law in terms of population dynamics. Zipf’s law approximates the covariance of populations of cities with their rank within individual countries (Zipf, 1942). Zipf’s law is an inverse proportion between frequency and rank, so subsequent ranks of 1, 2, 3, 4, 5, etc. have populations of 1, 1/2, 1/3, 1/4, 1/5 etc. Mandelbrot generalized Zipf’s law by adding two constants of proportionality which are parameters allowing for the fine tuning of the model to fit a particular distribution more accurately. This equation has become a standard one for

comparing variables in a city such as population and area. A power law relationship also exists for biological scaling originating with the work of Max Klieber. This log-linear relationship correlates body mass (horizontal axis) to metabolism (vertical axis) over 27 orders of magnitude with a scaling exponent of 3/4. West’s theory relates this law to fractal structures (Fig. 2.2.4) (West, 2002). Additional power laws have been discovered relating many variables in city dynamics and remain an active area of research. Bettencourt has applied West’s ideas to cities and urban infrastructure. To name a few scaling laws found in cities–when compared to population density: roads, cables, numbers of gas stations and post offices are sub-linear; patents, income, real estate and crime are super-linear (Bettencourt, 2013).

FIG 2.2.4. Biological scaling showing 3/4 power law over 27 orders of magnitude (West, 2002).