Explainer: What Scaling and Fractals Are, and How Designers Can Use Them

Advanced theories of science today are all about complexity, differentiation, fractals—quite a contrast with the industrial geometries of early modernism.

With apologies to real estate agents, we’d like to say that the three most important factors in design are scale, scale, and scale. One reason is that many of the worst environmental design blunders of the 20th century have been mistakes of scale—especially our failures to come to terms with the linked nature of scales, ranging from small to large. The cumulative consequence of these failures is that the scales of the built environment have become highly fragmented, and (for reasons we detail here) this is not a good thing. Can we correct this shortcoming? Most designers know something about “fractals,” those beautiful patterns that mathematicians like Benoît Mandelbrot have described in precise structural detail. In essence, fractals are patterns of elements that are “self-similar” at different scales. They repeat a similar geometric pattern in many different sizes. We see fractal patterns almost everywhere in nature: in the graceful repetition at different scales of the fronds of ferns, or the branching patterns of veins, or the more random-appearing (but repetitive at different scales) patterns of clouds or coastlines.

We can also reproduce fractal patterns in a computer, often with strangely beautiful results. Some graphic designers use fractal methods to reproduce very realistic-looking landscapes and other natural phenomena. These, too, seem to trigger something in our perception. We somehow recognize them as being “natural” and connect with them emotionally. We seem to be wired to “read” fractals in our environment, probably for two key reasons. One is that biological structures are largely fractal in their patterning, and we are innately interested in other biological structures because they might be food, or predators, or other people, or just a key component of the biologically supportive environment. The other reason goes deeper into geometry. When we look at a long vista, structures that repeat (trees for example), repeat at smaller apparent scales when they are farther away. This fractal information helps us read distances and depth in the environment. Doing so gives us an effortless understanding of the geometrical order of our environment. We’re aware of this only as a pleasurable sense and not, coincidentally, as an important survival need, from an evolutionary point of view. Fractal structures also give us other kinds of useful information, like complex relationships among environmental elements. The order of an essential but non-graspable structure, like an ecosystem, is more intelligible to us because we can detect the symmetrical fractal patterns of its plants and animals—another important evolutionary need. In modern times we have a greater need for urban environments to be legible to us, and there is evidence that we do this by reading fractal relationships in buildings and details (after all, we have evolved with this sense). From an evolutionary point of view, it’s evident that we perceive these relationships because they are supremely useful to us. They help us understand the structure of choices that our environments present, and how the different alternatives might offer us different benefits. It is an innate skill. Importantly, fractal urban structures typically provide multiple combinations of benefits that work in synergy. And our pleasurable perception of fractals is probably related to this too. For example, the branching, layered, fractal-like paths we can take within a city help us carry out many different tasks simultaneously. People moving along such paths for the purpose of higher-level information exchange (going to a business meeting) can thus carry out lower-level information exchange (having informal “spillover” exchanges with other people, or perceiving pleasurable scenes). The time required for higher-level exchange is therefore used more effectively, and the net effect is a synergy of activities that often translate into economic, social, and other benefits.

Figure 2. The fractal pattern of self-organizing urbanism. On the left is a simple fractal pattern called a “Cantor Gasket”. On the right is a much more complex and irregular pattern with recognizably similar fractal properties, a traditional urban neighborhood in Baghdad, Iraq. Notice the similar patterning at different scales of bordering spaces and alternating patterns of indoor-outdoor space.

Drawing by Nikos A. Salingaros (left), Image: G. Eric and Edith Matson Photograph Collection, Library of Congress (right)

This “fractal loading” means that each high-level exchange carries with it simultaneous exchanges on many smaller levels. An ensemble of exchanges on different scales is supported by a physical infrastructure that permits mixed information exchanges, but does not let other competing exchanges squeeze out the weaker or lower-level exchanges. Fractal loading is important at all scales. But it becomes especially important at the scale of a human being. For instance at the scale of a region there are not that many structural choices that are relevant to an individual going about his daily activities. But as we approach the scale of a human being (in fact, a group of scales ranging from 1mm to 10m), more and more structural choices begin to crowd into the picture, so that by the time we are at that scale, the environment often presents a rich set of structural choices that a person might make on a daily, hourly, and even instantaneous basis. At this scale, the fractal loading of our environment vastly expands the structural options, and builds synergies between them. If I am in a well-connected, fractal-loaded spot at this human scale, I can read the newspaper, I can talk to a friend, I can say hello to a passerby, or I can run one errand or more. And I can easily connect these activities into a web of choices. This is very likely a key reason that, within urban systems, well-structured pedestrian networks are so important. As our work has shown, there is reason to believe that there are important synergies of economics, resource conservation, psychological health, and other benefits, which are only provided by pedestrian networks that have this key property of fractal loading.

Figure 3. The pedestrian networks of medieval Rome have a fractal structure, extending into the buildings and even the rich ornamental details of the buildings themselves. These “place networks” offer pedestrians a dense and overlapping set of choices of movement, views, and other enriching experiences.

Drawings/Photos by Michael Mehaffy

Fractal loading is one example of a “scaling phenomenon” in complex network structures like cities, and an active area of urban research. Another related phenomenon is that as the scale of a structure like a network increases, the phenomena that happen at a smaller scale often do not increase at a linear (proportional) rate. Often they are “super-linear” (they increase more than proportionally) or “sub-linear” (they increase less than proportionally). These phenomena, such as economic growth and resource use per person, are very important to us. If we get more economic growth per person at a larger scale, or less resource use per person, then our quality of life can improve. This may be one important reason why people are attracted to large cities. Dense settlements really do offer more quality of life for proportionally less cost than sprawl does. And by understanding scaling, we can deal better with challenges like resource depletion and climate change. But notice that this phenomenon occurs as a result of the specific network structure of the city, and its “metabolic” interactions and synergies (such as fractal loading). A collection of entirely separate individuals all “doing their own thing” would likely not benefit from such scaling phenomena. It is in the multi-scale interactions that these phenomena, and the synergetic benefits they bring, come about. Interestingly, this characteristic of fractal loading tends to emerge spontaneously within urban systems that are allowed to self-organize within the natural processes of human culture—that is, within traditional urban environments. We all recognize this intuitively in the fractal-rich environments of popular tourist destinations like Bruges or Edinburgh. (And we recognize its absence in engineered environments that are decidedly not tourist destinations, like London’s Docklands, or Paris’ La Defense.)

Figure 4. On the left is the highly fractal structure of urbanism in Bruges, Belgium. On the right, a much more sparse, fractal-free environment in the modern suburbs of Bruges—which is also far less walkable, and has other negative impacts.

Photos by Michael Mehaffy

What does this tell us? Are fractal urban structures just nostalgic remnants of an obsolete pre-modern era? Or do they offer crucial lessons for designers today? While there are certainly ideologically dogmatic theories of style and history that support the nostalgic remnant proposition, they are unsupported by real scientific evidence. And, critically, there is important evidence for the crucial lessons for today’s designers proposition. To see what these lessons might be, we will discuss how fractal structures are formed in nature—and, it appears, in human nature—and why they might be such important attributes of a well-functioning environment. Fractals have two related characteristics: They show complexity at every magnification. Their edges and interfaces are not smooth, but are either crinkled or perforated.

Fractal patterns tend to form naturally for one simple reason: there is a “generative process” that creates the geometric pattern, and it does so at more than one scale. For example, in a blooming flower, the genetic code that creates the pattern does so in a time sequence, while the previously generated patterns grow larger. In a computer-generated fractal, the generative process is called an “algorithm,” a bit of code that generates the pattern from a complex interaction with what has been generated previously. In a city the generative processes are carried out by people doing what people do in making cities. They articulate spaces with boundaries that are shared to varying degrees. They create spaces that have degrees of publicness, somewhere along the spectrum ranging from public to private, from the most public streets and squares to the most private bedrooms and baths.

The boundaries of living spaces are not simple structures either, but complex membrane-like structures offering their own set of structural choices, either to maximize privacy (by closing a curtain) or publicness (by opening a door). These boundaries are wonderfully complex structures in themselves and self-organize into larger patterns (doors or windows that become shared types over time, and neighborhoods that develop characteristic interface patterns of porches or colonnades). How are the different scales linked? Just as biological structures and computer algorithms spontaneously repeat their geometric patterns at different scales, so do we, unless we’re forced to do otherwise either by legislation or by ideology. Individuals might make small repetitions of a pattern (a rectangular room shape) while groups might make larger versions of the same pattern (a courtyard) and larger groups might make a still larger one (an urban plaza).

Figure 5. Some essential properties of fractals. (a) Fractal loading uses a basic scale as a carrier for other successively smaller mechanisms and structures. Far from being monofunctional and simplistic, every structure becomes richly complex and carries information on several distinct scales. (b) Longitudinal compression forms a “folded” fractal, creating a crinkled line that then generates crinkles on its crinkles. This interface can catalyze urban interactions, mimicking the non-smooth surface of a chemical catalyst. (c) Longitudinal tension and breaking along the entire line form a “perforated” fractal, here shown at its first stage. This is a natural mechanism for defining an urban colonnade and any semi-permeable urban boundary, such as a row of bollards that protect pedestrian from vehicular traffic.

Drawings by Nikos A. Salingaros

But as with biological and computer structures, the story does not end at any particular scale. The boundary of a room is perforated with smaller structures like rectangular doors and windows. The boundary of larger spaces might be perforated with colonnades (we are talking about living spaces and not the dead spaces characteristic of post-war architecture and urbanism). These repetitive perforations at smaller scales—the fractal loading that results from the characteristic “generative algorithm” of fractal structure—will often continue on down to the scales of detail and ornament. Why is this? It seems likely that we, the users, making our way through these places find such complex environments (complex in a very precisely ordered sense) easier to comprehend, more intelligible, more usefully organized, and more beautiful. We are very good at reading the multiple scales of these “place networks”.

But there is a serious problem. If we are not users, but designers educated in our industrial/artistic culture, we might have another agenda: to impose another kind of order on the built environment. And that agenda might come from a very different set of criteria than the environmental experience of humans. Such is indeed the case. To put it simply, our current methods of making cities are over-reliant on economies of repetition and scale, which do offer narrow advantages but are also extremely limited, and from a human perspective, very crude and destructive. Natural systems never use those strategies in isolation, but are always combined with economies of differentiation and adaptation. Surprisingly, we haven’t really figured out how to employ these in our current strategies (though many people are working on this problem, and our own work takes up this challenge).

Choosing to work with a severe technological limitation, modernist designers argued that a more sophisticated approach was to strip down buildings into “minimalist” compositions, much easier and cheaper to produce under the crude industrial processes of the early 20th century. It was the compositions of these elementary “Platonic” solids that were most beautiful, postulated architects like Le Corbusier, because they were “pure” expressions of form. The old Gothic cathedrals, with their fractal tracery, were “not very beautiful,” he said infamously. Nor were the lively streets that he despised!

Indeed, Corb and other designers made a strong ideological case (still persuasive today) that the old ornamented designs were bourgeois, contemptible, even (in the famous words of Adolf Loos) “a crime.” In this ideologically driven design movement, we have come to accept the incorrect idea that fractals are somehow primitive, whereas smooth, undifferentiated “Platonic” forms are “modern” and sophisticated. Ironically, the opposite is the case: The most advanced theories of today’s science are all about complexity, differentiation, networks, and fractals—a dramatic contrast with the straight, smooth industrial geometries of early modernism.

Recognizing this, many architects and urban designers are speaking in terms of fractals, scaling laws, and “morphogenetic design.” But the question remains: Are these individuals really engaging such principles to create human-adaptive structure? Or are they only using them to create attention-getting aesthetic schemes, tacked onto what is essentially the same failing industrial model of design? These questions are at the heart of the debate on the future of the built environment. What, then, are the lessons to be drawn? Fractal structure is not just an aesthetic gimmick. It is an important characteristic of sustainable human environments. And this structure does not arise from the well-meaning top-down schemes of old-mode art-designers, but from those with a skilled application of processes of self-organization, as part of a new way of thinking about what it is to design. And yet, we designers have been exceedingly stubborn in taking on this lesson. Under a misguided theory of environmental structure that confuses simplicity with order, we have been stripping away the critical connected scales and fractal relationships within our environment. We have replaced a world of richly connected urbanism with a disordered geography of artfully packaged, catastrophically failing art-products.

Michael Mehaffy is an urbanist and critical thinker in complexity and the built environment. He is a practicing planner and builder, and is known for his many projects as well as his writings. He has been a close associate of the architect and software pioneer Christopher Alexander. Currently he is a Sir David Anderson Fellow at the University of Strathclyde in Glasgow, a Visiting Faculty Associate at Arizona State University; a Research Associate with the Center for Environmental Structure, Chris Alexander’s research center founded in 1967; and a strategic consultant on international projects, currently in Europe, North America and South America.

Nikos A. Salingaros is a mathematician and polymath known for his work on urban theory, architectural theory, complexity theory, and design philosophy. He has been a close collaborator of the architect and computer software pioneer Christopher Alexander. Salingaros published substantive research on Algebras, Mathematical Physics, Electromagnetic Fields, and Thermonuclear Fusion before turning his attention to Architecture and Urbanism. He still is Professor of Mathematics at the University of Texas at San Antonioand is also on the Architecture faculties of universities in Italy, Mexico, and The Netherlands.