There is something elemental in the notion of a pattern. Whether it is a decorative design from Islamic art or an intricate behavior programmed into a computer, a pattern is both concept and thing. Patterns are eternal constructions that defy categorization as mathematics or science or art.
However, patterns are more than just objects of contemplation. They are mappings of space and time, and they therefore have predictive power. So, patterns can be manipulated and put to use. They are practical products of the human mind that encapsulate our understanding of the world. Patterns are abstractions, and abstractions are knowledge.
But, for all their beauty, patterns possess a serenely authoritarian air. They are all-encompassing, almost by definition. When one is immersed in a pattern, it takes over everything in view, and there can be nothing that is not in the pattern. Therefore, patterns can seem to embody our most reductionist and dehumanizing impulses. However, I would suggest that patterns are prescriptive only if we allow them to be, if we treat them as dogma instead of description.
It is often said that mathematics differs from science in that scientific theories are validated only through measurement. However, geometric patterns illustrate how mathematics can cross the boundary into physical space, as shown by their widespread use in art, design, and architecture. Patterns also play a role in science, though more subtly and more pervasively. Scientists scour the natural world for patterns in the broadest sense of the term. Periodic recurrences, like the swinging of a pendulum or the rising and setting of the sun, are perceived as temporal patterns. Predictable phenomena, like a ball rolling down a hill, are behavioral patterns that allow quantities like speed and trajectory to be defined and calculated.
The history of physics is filled with unexpected convergences, where disparate phenomena are suddenly revealed as two sides of the same coin—Newton’s law of universal gravitation, Maxwell’s equations for electromagnetism, and electroweak unification, just to name a few. The common thread in science is the recasting of real-world complexity into a compact set of rules. Thus, in essence, scientific explanations operate like geometric patterns.
For me, as a physicist and mathematically inclined artist, the visual allure of patterns stems from their economy of expression. Geometric patterns encode complicated spatial relationships with just a small amount of information. As a concrete example, I dissect below one of the simplest patterns there is: the square grid on a sheet of graph paper. A straightforward array of boxes can be specified in various ways, but one trivial method is to robotically record the coordinates of every single square, paying no heed to the fact that all the squares are identical.
Unfortunately, not only is such a scheme terribly inefficient—the more squares there are, the more numbers are needed—but it is also not predictive because the description of one sheet does not necessarily apply to others. In contrast, an algorithm can be much more efficient. It is also universal. For instance, the following blueprint encompasses all square grids, no matter how big, in fewer than sixty words:
- Draw a square with sides of a given length, orientation, and location.
- Add squares of the same size and orientation, one by one, such that they share at least one side or corner with those from the previous step, until no new squares can be added.
- Repeat step 2 recursively, stopping at the paper’s edges.
For yet another recipe, one can dispense with squares entirely and draw evenly spaced horizontal and vertical lines across the page. Thus, a high degree of homogeneity and repetition within a system obviates the need to notate each individual element; a pattern can serve to condense the information into a much smaller package.
Geometric patterns appeared in my work when I started using photographs as material for sculptures. While experimenting with folding my prints, I was seized by a vision of a photograph shaped into a raised pattern. With this germ of an idea, I gave myself some design parameters to work with. The final folded structure would contain not only three-dimensional elements but also flat regions that could be anchored to a substrate. Also, even though cutting and pasting would be permitted (unlike with origami), I wanted to use the entire print, with no wasting of paper. The cleanest method to satisfy these constraints was to start with a grid of squares, and then fold some of them into triangular pyramids (making one cut per square). The first appropriate pattern I found was the snub square tiling, whose layout echoes the textured surface on industrial diamond plate metal sheets.
In practice, transforming a flat square grid into a three-dimensional snub square can be accomplished through brute force: cut the grid apart into separate squares, fold two-thirds of them into pyramids, and assemble the snub square painstakingly shape by shape. On a whim, I wondered if I could invent a more efficient procedure to continuously deform one pattern into the other, with whole sections of squares being manipulated en masse rather than one at a time, somewhat like a controlled and systematic wrinkling of the paper.
It was not a foregone conclusion that such a scheme could work. In fact, for the closely related elongated triangular tiling, a continuous deformation is stymied by the particular arrangement of the squares and triangles. But, for the snub square pattern, the key was to visualize each triangle as a square with two adjacent corners pinched together. Using this trick, I could see that there was indeed a set of cuts that nearly (but not completely) preserved the connections among the original squares.
The proof lay in demonstrating the concept with an actual square grid, scissors, and glue. And so, after some tentative cuts and clumsy folds, the pieces suddenly fell into place, and a snub square with triangular pyramids made of paper materialized before my eyes. It was the lifting of a veil, revealing that the two patterns had been distant cousins all along. Now, every time I make these folds, I relive the satisfaction of this modest discovery. The design challenge I had set for myself turned out to be a mathematical question in the end.
I have presented my thought process above as a logical sequence of steps. But I distinctly recall that, in the moment, the realizations came together in a single sweeping gesture. I was not even consciously thinking—there was the impression of a vista opening itself up. In other words, whether or not one accepts the Platonic view of reality as grounded in unchanging ideal forms, it is undeniable that discoveries take place within our finite selves, and when they occur, it is one’s mind that has changed and not something external to it.
Thus, despite science’s practical reliance on experimental verification, we often judge results in math and science by an intuitive, aesthetic criterion of “elegance”. An elegant explanation of some phenomenon is both concise and far-reaching, and it employs as few ad hoc assumptions and free parameters as possible. By contrast, an inelegant, contrived explanation never stands the test of time, no matter how well it fits the data. For example, Ptolemaic astronomers went through contortions to model planetary motions as circles upon circles (deferents and epicycles) centered somewhere near the earth. It is far simpler (and more accurate) to invoke Kepler’s plain elliptical orbits around the sun.
Geometric patterns are paragons of elegance. Their constituent shapes fit together perfectly, and once a pattern is solved, the entire plane is mapped out to infinity by construction; a pattern cannot be altered locally in one spot without spoiling it somewhere else. Thus, patterns of all types elicit a sense of inevitability—a gut feeling that there could be no other solution—that lends credence to both scientific explanations and mathematical findings.
If we accept that our experience of knowledge is subjective, as I have posited above, it is logical to ask if patterns are truly inherent in the universe. Is there such a thing as objective truth? Without wading into a philosophical debate that has raged for millennia, I will stake out a middle ground: Perhaps it is merely a matter of perspective.
On the one hand, scientists have built highly successful descriptions of nature through careful observation of events, deducing cause and effect. Nature’s adherence to formulae that can be verbalized suggests strongly that it was meant to be understood, that the universe runs like clockwork according to immutable rules. On the other hand, one could dispense with the search for patterns and choose instead to live moment to moment. When every second is dissociated from the next, when the future is of no concern, there is no need for memory or knowledge. Life is just one unexplained occurrence followed by another, and the universe is a disorganized cauldron of randomness.
Both outlooks offer comfort, and I believe no one subscribes entirely to one or the other. We all oscillate between asking why and basking in wonder, and either state can be entered into at will. It all depends on how much data one processes at once, how wide is one’s field of view.
As a scientist, I always had a particular affinity for synthesizing information and working with statistical methods. To make sense our measurements, we often use computer algorithms to fit a theoretical curve through some data on a graph. The first time I tried doing this, as an undergraduate, my fit gave slightly different answers depending on how I binned my histogram. Surely, I thought, my fit was wrong because I should get identical results every time. Puzzled, I showed the problem to my mentor, who assured me, “These variations don’t matter because they’re well below your statistical uncertainty. But whenever you see real discrepancies, it can also be the curve that’s wrong, not just your fit.” I learned two things from this exchange. First, that mathematics in the real world is messy. In this case, my fit was affected by finiteness in the algorithm and the computer it ran on.
The second lesson was that science is an art filled with judgment calls. We must decide how to analyze our data, what models to use, which features to pursue, and which to ignore. This subjectivity sounds antithetical to science, but without a place to start, one cannot infer anything at all. Indeed, science is unfounded at its core because, in presuming to describe nature with patterns, we make the assumption that it is fundamentally predictable.
More broadly speaking, we all live according to unexamined postulates that we declare by fiat to hold true. Each of us is by turns rational and irrational. Reason coexists with faith. As scientists, we are obliged to conduct our business as if the universe could, in principle, be completely comprehensible (even if it is not yet so), but this does not stop us from stepping back at times to admire it in awe.
So, perhaps our capacity for knowledge is not intrinsic to our intelligence, but is forged dynamically from our limitations. We are constantly flooded with more sensory data than we can absorb. In our ancient quest for survival, perhaps patterns gave us a way to compress and activate data for future use. Knowing exactly when seasons change is more advantageous than blindly hoping for spring. So, it would not be surprising if we were somehow predisposed to detecting those facets of reality that admit description. Perhaps our perception of patterns emerged from our being in this world.
I would note that this narrative says nothing about an underlying order in nature. Indeed, science works beautifully—unreasonably well, even. But science also tells us that the patterns we have found, as convincing as they may seem, are always provisional. After all, who among us still believes that the sun orbits the earth? Our knowledge is constantly evolving, and the specific set of patterns that governs our thinking at any given time is a lens through which we view the world, an interpretation of reality that is both necessary and arbitrary.
In my early years, I showed enough technical aptitude in math and science that I majored in physics in college and went on to graduate school and a research position in particle physics. But my cultural home has always been in galleries, museums, and concert spaces. In pondering the connections between art and science, I always come back to the difference between conscious and unconscious knowing and how one’s body negotiates with both. When I played piano as a kid, learning a new piece was a tedious chore as I picked out the notes on the keyboard. But after many repetitions, muscle memory would take hold, and moving my fingers would become second nature. Progressing beyond conscious control made room for the music making to begin. The same process takes place when picking up a foreign language or a new skill—we begin with an explicit set of instructions, which is internalized through practice and first-hand experience, and eventually molded into a customized understanding that we can access immediately and automatically.
These instructions that transmit information from person to person are only patterns to be followed. They are monochrome templates for recreating someone else’s thoughts and actions in full living color inside our minds. Building knowledge is not a spectator sport; it is a physical activity. Nowhere is this more apparent than in scientific research, where the creation of knowledge is operationally tied to memory, since data must be remembered to be analyzed. In connecting the dots in our data, we might organize the passage of time into months and years, or we might realize that isolated severe weather events add up to a pattern of climate change. When we do so, we become part of the world we seek to apprehend.
Thus, knowledge resides as much within the individual as in the collective psyche, as much in real space as in the equations that model it. It is for this reason that objects like paper and scissors can illuminate the mathematics of patterns. Admittedly, actual objects, being finite and flawed, can only provide partial answers. But every time a pattern appears in art and design, it resonates with us yet more deeply.
Conveying one’s experience of knowing is a task tailor-made for art. And although science ostensibly traffics in objectivity, it too can be a vehicle for creativity and personal artistry. When we are touched by artistry of any kind, the moment of transcendence is escorted by an undercurrent of mystery. We often speak of the unknown as an undiscovered country, a Platonic ideal oblivious to our prying eyes. However, by continually depicting knowledge as the vanquisher of mystery, I suspect we are promulgating a false opposition. Mystery is a somatic sensation, just like the experience of knowing; therefore, mystery and knowledge are cut from the same cloth. Perhaps mystery does not stem from the absence of knowledge, but is, in fact, concurrently generated with it, the way a single fold divides a sheet of paper in two.
In science, mystery goes by the name of uncertainty, and it springs directly from the act of perception. Statistical fluctuations and unseen systematic effects can throw off our measurements in unpredictable ways. And at microscopic length scales, quantum noise puts a fundamental limit on instrumental resolution. So, any measured value is meaningless without an estimate of its uncertainty. And when we analyze what we see, we are layering mystery upon mystery.
In his book, Meta Math!, Gregory Chaitin relates an insight about randomness that he attributes to Emile Borel. According to Chaitin, Borel realized that it is impossible to devise a singular, definitive criterion for randomness because, in its barest form, randomness stands for a complete lack of order that necessarily evades description. At best, we can only define some measurable property that resembles randomness, by which we can rank different datasets. But there is no basis for privileging this somewhat arbitrary definition over any other.
We might think of reality as being as elusive as Borel’s randomness, which implies that there is no magic pattern that explains everything. Science, art, music, literature—these are all valid and complementary ways to encircle, without ever pinning down, the nature of our world. And so, despite their well-earned authority, patterns—or, really, any truth amenable to human expression—can only be part of the whole story. Even within a field as rigorous as mathematics, every concept can be rendered in multiple dialects. Thus, the universe does not force us to choose between knowledge and mystery, between the crystalline order of patterns and the warmth of human values. The grand total of our existence is wrapped up not in a cosmic “either/or,” but in an enfolding “and also”.
Patterns are not math, nor science, nor art; they are entities unto themselves. They are both fact and fiction. They are vessels for mystery. They are the just-so stories of our age.
In the end, the patterns we construct are only approximations of reality. Patterns conceived in the mind are like airtight explanations, but when put into practice, they show their limitations. The snub square pattern that I use in my art never achieves geometric perfection. The thickness of the paper and the vagaries of the hand invariably produce gaps in the pattern. Similarly, writing cannot reflect the nuances of spoken words, and musical notation is not the performance.
However, imperfections do not ruin a pattern; quite the opposite. When a weaver executes a pattern composed on graph paper or envisioned in the mind’s eye, the corresponding pixels made of malleable fiber are not exactly square, and the fabric itself never hangs flat. But these gentle excursions from the ideal are what imbue the woven material with a tactile specificity powerful enough to anchor the beholder in the present. It is precisely the contingency in the physical manifestations of patterns—the unique world-views, the handcrafted objects striving for mathematical perfection, and the palpable effort of making them—that marks our time on this earth.
In the end, each of us is an unfathomable black box. There is no penetrating the indescribable, unquantifiable universe of thoughts and stimuli that resides within an isolated brain. At the same time, being social, self-perpetuating creatures, we find meaning in the company of others. We exist through our imprints on the world. And patterns, both idealized and real, being the (necessarily reductive) means by which we mediate and communicate our experiences, are not sterile symbols frozen in amber, but rather, the very stuff of our common shared vitality.
 Chaitin, Gregory. Meta Math! : the quest for omega. Vintage Books, 2005, pp. 123-125.
All images copyright and courtesy of Werner Sun
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