How Mathematics Will Save the Built World!

Really, one of the very largest problems that is facing the earth just now is rarely mentioned: and that is the spread of ugliness. By the standards of the 20th century, it sounds like a sort of rather trivial and unimportant issue. It’s not.” — Christopher Alexander, in a 2016 film by Michael Mehaffy.


1. Isn’t it beauty that was supposed to save the world?


In The Idiot, Fyodor Dostoyevsky wrote, “Beauty will save the world.” Well, it hasn’t. He also wrote, “The awful thing is that beauty is mysterious as well as terrible. God and the devil are fighting there and the battlefield is the heart of man” (The Brothers Karamazov). Alexander Solzhenitsyn built an argument around this theme in his 1972 Nobel Prize Lecture. The problem is, what happens when beauty has been obliterated and forgotten? This is where mathematics comes in. It can resurrect the beauty expunged from contemporary human society. Beauty, through mathematics, can save the world.


Mathematics is largely without controversy, whereas debates on beauty are contentious and wholly subjective. Mathematics is about relationships, repeatability, and nested structures. It creates algorithms for computing results. Regular structures are discovered through thought, and then classified. Far from being an intellectual abstraction, mathematical ordering affects us viscerally because human perception relies upon information reduction through symmetries. Random (disorganized) information becomes too much for us to process, which generates anxiety.


2. Ordering via groupings and symmetries


I have written about the various mathematical constructs that underlie composition and design. Architectural elements are visible shapes, not numbers or arrow-like vectors. They need to be combined, compared, counted, grouped, and juxtaposed. Not because someone says so, but because this is what our brains do automatically: analyze and process the information presented in any composition using mathematical relations. We perceive our world by grouping adjoining geometrical elements, via symmetries, into larger wholes. This is the simplest mathematical/visual mechanism for making sense of our environment. We make our way in the world thanks to this process.


Basic symmetries have a profound effect on composition and design. Some elements need to have the same size and shape (oriented in the same way, or reflected, or rotated) and be aligned horizontally or vertically. Their repetitions should be regularly spaced, otherwise there is no symmetry. These minimal requirements already influence architectural composition to have a certain “ordered” appearance, one that echoes traditional and vernacular styles. Yet this comes from mathematics—supplemented by neuroscience, working with physics, which privilege the horizontal and vertical symmetry axes because of gravity—and has nothing to do with “style.”


Scaling symmetry is something entirely distinct from the previous type of symmetry, and links components visually when we see magnified or reduced versions of the same thing. This is the basic feature of a “fractal” (think cauliflowers), which contains a large number of substructures, all of which are self-similar at different magnifications. Again, scaling symmetry is a dominant feature in traditional and vernacular architectures, and one reason those quite different form languages appeal to our innate mathematical sense.


3. Counting and grouping reduces informational overload


Even the simplest mathematical notions turn out to be very important. Design elements can be counted when they have more or less the same size, shape, and orientation. Redundancy and similarity of shape reduce information overload. If, on the other hand, dissimilar elements appear in a composition, they need to be accounted for individually, which takes up information processing in our brains that is needed for other life tasks.


This is not the end of the story, however. The mere presence of several copies of the same element can still lead to information overload, if their positions are unrelated. Symmetries reduce this extra information needed to fix the location of elements distributed in space into a more manageable amount.


More important, information coming from numerous copies of the same repeating element is difficult to grasp, since our sensory system doesn’t count, but perceives numbers visually as patterns. The effect of monotonous repetition is perceived uncomfortably, for mathematical reasons. Psychologically, this is known as the “cognitive limit of 7,” which is the maximum number of easily remembered digits, such as a phone number. A large number of elements can be better handled cognitively by grouping them, so that we count the groups instead of the individual elements.


4. Universal (fractal) distribution of sizes


Mathematics also relates components of a whole via their relative number and size. The universal distribution law says: “In a complex system, there are few large objects, more intermediate-size objects, and many smaller objects, roughly in an inverse-power relationship.” This means that the number of elements of different sizes we perceive at the same time should be inversely proportional to their size. (More-refined versions of this law follow a scaling index that corresponds to the fractal dimension, and is not simply equal to –1). In a fractal the components are all similar through scaling (an additional geometrical relationship), which gives a fractal its coherence.


The universal (fractal) distribution is independent of simple geometric shapes and leads to the coherent structures found in the plant world, where nothing is truly straight. Artificial complex systems also evolve toward such a distribution as they acquire “emergent properties.” Examples include electrical power grids, internet links, and the structure of languages (Zipf’s Law).


5. Centers and their boundaries


Traditional architects knew some of what is discussed here in the form of intuitive rules. Nathaniel Curtis, Ernst Gombrich, George Hersey, Rob Krier, and most recently Donald Ruggles, among others, outline geometric rules of composition. For a while it was very easy to brush this material aside, especially the older books, as being relevant only to architectural history and not to contemporary practice. But recent results establish a powerful link with science, showing why these questions are essential for our well-being.


Something that has not been adequately explained is the dual relationship of a region with its border. With his Theory of Centers, Christopher Alexander established that a wide boundary is essential for organizing complexity. I distinguish between two types of centers: an “explicit” one that contains organized complexity in its interior, and another, fairly empty, “latent” type that is supported by its complex boundary. This model is validated by fractal scaling, where the boundary becomes one level in the hierarchy. It can also be understood by reference to the Stokes Theorem, which relates integrals over a region to integrals over their boundary.


6. Validation from our neural system


Mathematical notions of “beauty” are favored by our sensory system. I have outlined here merely the simplest beginnings of a mathematical-neurological basis for beauty. We are constantly processing the information in our immediate environment, comparing and looking for groupings, a task that consumes a lot of metabolic energy. The crucial point is that we are overloaded with environmental information and rely on built-in algorithms to organize it. If we cannot instantly classify and categorize forms and shapes surrounding us, we continue to process the information indefinitely, which tires us.


Neurophysiology supports this line of reasoning because specific brain cells are designed to recognize shapes and symmetries. Individual neurons respond to specific colors, simple geometric shapes, distinct orientations (angles), and some rather complex shapes essential to our evolutionary survival. Among the latter are “face-recognition” cells, which respond to bilateral symmetry about a vertical axis, and to a generic facial structure of “mouth” with two “eyes” above. Our brain is wired to recognize symmetric combinations of simpler elements into more complex wholes.


Violating the vertical axis and neglecting reflectional symmetry about a vertical axis create anxiety in the viewer. Building façades that lack such symmetries either repel us or simply do not register, even as we look in their direction. If a building’s entrance is not marked using our innate preference for a symmetrical, face-like design, it’s easy to miss. This defect compromises so many buildings built since the end of World War II.


7. Two distinct origins of the need for beauty


Intuitive beauty summarizes our evolved computational algorithms for survival in an informational environment; it’s the special complexity that soothes us. Beauty attracts us because we subconsciously interpret it as nourishing. “Alien” shapes in our immediate surroundings are the opposite of beauty. Because they do not remind us of natural shapes that our evolution has programmed us to interpret, they disturb us. Alarm and the “fight or flight” response take over our body until we either have enough information to judge that some object is harmless or decide to flee.


It appears that we are well equipped to handle possible conflicts in nature. When confronted with a beautifully colored and patterned snake or spider, we have inherited an automatic first response to danger. Herpetologists and entomologists rely upon a second, learned response to approach such animals based on expert knowledge.


But there is more to beauty than utilitarianism: recurring patterns are found in inanimate physical structures in the universe, hence some of our key notions of beauty originate with the structure of matter itself. This is physics, not biology. It cannot possibly have anything to do with evolutionary adaptation, because it goes far deeper and was defined before life evolved. A geometrical necessity for structural coherence is built into our body and coexists with separate aesthetic instincts arising from what is biologically “useful.” Christopher Alexander spent decades discovering those rules, condensing them into Fifteen Fundamental Properties. Applying those helps to create a visceral kind of beauty that is independent of anybody’s opinion.


8. Whatever happened to beauty in architecture?


Which brings us to architecture. By some accident of history that is too involved to go into here, the teaching of design has become focused on the opposite of what our body identifies as elements of “beauty.” Indeed, it is far easier to create the opposite of beauty than it is to create beauty itself. A simple formula—“use your body to sense anxiety and threat, but then embed those emotions into a design instead of fleeing from them”—determines forms and shapes. Fashionable design has for several decades eschewed symmetries of all types, violated gravity, and eliminated the smaller elements that could define a fractal distribution on a structure’s façade or interior.


This reversal began with a conscious attempt at innovation through breaking from traditional practices that included mechanisms for coherence. But now this negative approach to design has been internalized and is no longer questioned. Attempts by architects to include design elements I mention as necessary for our sensory well-being are interpreted as violating some absolute ethical code. Theoretical explanations within architecture avoid discussing human physiology and rely instead upon design rules that are no more than dogmatic cult beliefs.


(Leaving traditional and vernacular building aside, has beauty been suppressed in the “approved” architecture of our times? Practitioners who feel accused will deny this. The same denial comes from an intellectual community that praises new and older buildings that deliberately reject the necessary mathematical constructs, and from a smug educational system that has been teaching our young architects to create abstractions.)


9. Designing spaces adapted to human life


Some readers will counter this essay’s message by claiming that architecture is all about shaping space for habitation and movement and not primarily concerned with aesthetics. Mathematical rules from three-dimensional geometry and topology indeed determine what are the most comfortable volumes for each function and situation. A solid research basis for adaptive design exists, providing guidelines congruent with the mathematical rules for beauty discussed here. Many self-builders rely on these documented principles for their projects. This body of work is of no interest to dominant architectural culture, however, which pointedly ignores it.


Instead, the profession’s “cutting edge” adopts strictly aesthetic choices and uses them to determine a building’s design. That abstract, non-utilitarian approach to generating forms often leads to uncomfortable or dysfunctional paths and spaces. Perversely, the orientation, relative size, and spacing of architectural elements studiously avoid defining complex centers and boundaries.


10. Solzhenitsyn’s warning to the world


Finally, I come full circle to what Solzhenitsyn said in his talk. It was a very angry talk, which accused a number of people of ruining the world. Those individuals, Solzhenitsyn warned, act out of their self-interest on the basis of lies and violence. And he appealed to beauty as the weapon to fight falsehood combined with violence. I recognize that in dominant architectural culture, forms and spaces are broken without worrying whether this causes anxiety in ordinary human beings. Violence is directed toward mathematical coherence. The accompanying deception praises all those disturbing buildings as being “wonderful.”


Today’s architects don’t apply mathematical rules for building beauty because there is no market for it. On the contrary, exploiting the shock value of unnatural forms has created a worldwide market for “newness.” This is what sells today.


Nevertheless, positive trends toward more adaptive, healthier architecture are coming from outside the mainstream. Breaking out of the approved design straitjacket, the work of some smaller firms is being influenced by emotions, empathy, and feelings, concerns that simply didn’t register during the 20 century outside of the traditional and owner-built domains.


Featured image: Tomb of the Sufi Saint Sheikh Rukn-e-Alam, via Wikipedia Commons. 



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