Mathematics, Music, Art, Poetry, Dance — It’s All Patterns
And I’ll go even further and say that mathematics, this art of abstract pattern-making — even more than storytelling, painting, or music – is our most quintessentially human art form. This is what our brains do, whether we like it or not. We are biochemical pattern-recognition machines and mathematics is nothing less than the distilled essence of who we are.
Paul Lockhart – A Mathematician’s Lament
The Infinite Guitar — A U2 classic
Sitting on my bed at my student halls of residence I remember feeling very competent playing the intro of U2’s With or Without You. I had only just started playing my first ever musical instrument. Adam Clayton repeats essentially the same four-note pattern again and again throughout the song. Yet the song does not become boring. Quite the opposite: it gradually becomes overwhelming. The bassline remains almost unchanged, but the musical landscape around it changes dramatically. The infinite guitar begins to shimmer and Bono’s vocal grows more urgent as he goes from whispered to almost screamy singing. The emotional pressure is building as the same notes that felt spacious and almost tentative at the beginning eventually feel charged with longing. The bass groove pattern has barely changed, but our experience of the pattern has.
Music is repetition, but not merely repetition
A rhythm is a sonic pattern in time. A melody is a pattern of rising and falling notes. A bassline is a recurring musical shape that glues the whole piece of music together while giving it groove. A chord progression is a pattern of tension and release. A verse establishes one musical landscape, while a chorus returns us to familiar ground. A bridge interrupts the expected structure and allows us to hear the final chorus differently when it returns. Music works because our minds gradually begin to recognise these patterns, anticipate them and inhabit them.
Much of rock and pop music is built from surprisingly simple foundations. You can even use a mathematics word to describe it, formulaic. Michael Jackson’s Billie Jean is anchored by an instantly recognisable bassline and an exceptionally restrained drum groove. AC/DC understood this almost brutally well. Many of their songs are driven by drumbeats that sound simple enough for a beginner to imitate: bass drum, snare, hi-hat and a solid backbeat. But imitation is not mastery. The challenge lies in making a simple groove feel inevitable. The best drummers know when not to add another fill. The best bass players know when not to add another note. Simplicity can require extraordinary discipline.
A weaker musician often tries to create interest by adding more. And that is true when you are first learning how to play, you want to play everything all the time. Later you learn that less is more. But more importantly, which bits to take out and what to keep in. A stronger musician can create interest by understanding the pattern so deeply that the smallest variation matters: a slightly delayed note, a pause, a drum fill at the end of a phrase, a change in intensity or a cymbal opening for a few beats. The underlying pattern remains stable, but it breathes. This is the crucial point. A pattern is not the same thing as mechanical repetition. A photocopier repeats. A musician repeats with awareness.
Repetition creates expectation
Repetition matters because it teaches us what to expect. Once we have heard a rhythm several times, we begin to anticipate the next beat. Once we have heard the chorus, some part of us waits for its return. Once a bassline has established itself (think of Staying Alive by the Bee Gees), we feel the shape of the song even before we consciously understand it. Music creates a relationship between memory and expectation. We hear what is happening now while carrying a faint trace of what happened a few seconds earlier. At the same time, we begin to predict what might happen next.
Music is the pleasure the human mind experiences from counting without being aware that it is counting
Gottfried Wilhelm Leibniz
This creates the possibility of pleasure. For a lot of pop music, the chorus itself is called a ‘hook’, and writing strong hooks or earworms is thankfully still a black art. Sometimes the expected note arrives exactly where it should. Sometimes it arrives late. Sometimes it does not arrive at all. A drummer stops for half a bar. A singer holds a note longer than expected. A chord shifts unexpectedly. The pattern is broken, and suddenly we pay attention. Without repetition, there can be no expectation. Without expectation, there can be no surprise. Without surprise, a piece of music may be technically correct but emotionally lifeless.
The tension between stability and disruption is not limited to music. It appears throughout the arts. Order and disturbance. Recognition and discovery. Continuity and change. The artist creates a world with enough structure for us to find our bearings and then decides when to disturb it, but only slightly.
For example, Leonardo Da Vinci’s The Last Supper is built on an extraordinary mathematical order. The perspective lines of the room converge towards Christ, giving the painting a powerful geometric centre. The architecture is calm, balanced and controlled. But within that ordered space, the disciples erupt into human disturbance after Christ announces that one of them will betray him. The painting works because the emotional disruption is held inside a strict structure. Geometry gives the scene its stability; human reaction gives it life.
Poetry is patterned language
Poetry is not simply beautiful language arranged in shorter lines. It is language shaped by pattern. A poem may use rhyme, metre or repetition. Certain sounds return as they rhyme and sound pleasing when repeated. A phrase appears several times and gathers a different meaning on each appearance. An image introduced near the beginning comes back at the end, transformed by everything that has happened in between. Even poetry without a formal rhyme scheme has rhythm. Words have weight, they are containers of ideas. Syllables create movement. Line breaks create pauses. Silence becomes part of the pattern.
A poet may establish a flowing rhythm and then interrupt it with a short sentence. The interruption matters because the rhythm came first. A broken pattern is only meaningful when there was something to break. This is why structure is not the enemy of creativity. Structure makes creativity possible. A sonnet gives the poet a form to work within. A musician works within a key, a tempo and a time signature. The 12 bar blues progression is always the same family of chords played in the same way over and over again in 12 bar containers. A tango dancer works within the music, the embrace and the physical possibilities of two bodies moving together. Limits do not necessarily crush expression, most often they create it. Derek Sivers puts it so beautifully, restrictions will set you free.
Art is pattern in space
Music and poetry unfold through time. Visual art arranges pattern through space. The human eye notices repetition almost involuntarily. We see symmetry, proportion, contrast and balance. We notice when a shape appears again in another part of a painting. We sense when an image feels stable, crowded, harmonious or disturbed. Architecture often depends on rhythm too. Windows repeat across a building. Columns create intervals. Arches establish a visual beat. A staircase rises through space almost like a musical scale.
Nature is full of patterns: the branching of a tree, the spiral of a shell, the veins of a leaf, the symmetry of a face, the repeated forms within a flower and the waves of the sea. Yet natural beauty is rarely perfectly mechanical. The branches are not identical. The waves do not repeat with absolute precision. The left side of a human face is not an exact mirror image of the right. Often, the beauty lies in the slight irregularity. Order is present, but it is not rigid. The pattern is recognisable, but alive.
Dance is movement of physical patterns
Dance makes pattern physical by taking pattern and placing it inside the body. Long before dance becomes a named art form, technique or genre, it begins as patterned movement. The animal world is full of this: birds, insects and other creatures use repeated sequences of posture, rhythm, colour and gesture to signal courtship, vitality, coordination or desire. The pattern makes the signal intelligible. The variation within the pattern gives it life.
Human dance seems to grow from something equally primal. Before there is ballet, tango, salsa or flamenco, there is the body responding to rhythm, pulse, attraction, group energy and ritual. A crowd begins to clap, sway or stamp together. A child hears music and moves before anyone has explained technique. A dance genre refines this primal material. It gives instinct a vocabulary.
This is something I have come to understand through Argentine tango. From the outside, tango can appear complex and mysterious. Two people move together without fixed choreography. Every partner is different. Every piece of music creates a different atmosphere. Yet tango is not random. It is built from patterns: pulse, phrase, pause, walking, turning, weight transfer and embrace.
The same basic walk can feel entirely different depending on timing, intention and connection. One step can be calm, playful, suspended or intense. This resembles the bassline in With or Without You: the notes remain simple, but their meaning changes because the context changes. A beautiful walk, deeply connected to the music, may say more than a complicated sequence performed without feeling. A simple pattern, inhabited fully, can contain a world.
Mathematics is the study of patterns
This brings us back to mathematics. Many people leave school believing that mathematics is mainly about calculation. They remember times tables, fractions, algebra, equations and exam questions. They remember rules they were expected to follow and methods they were supposed to reproduce. Some succeed while others conclude that mathematics is simply not for them.
But mathematics is not fundamentally a collection of procedures. It is the study of patterns, relationships and structure. What repeats? What changes? What stays the same? How does one quantity relate to another? What happens when we vary one part of a system? Can we describe a pattern precisely? Can we predict what comes next?
A young child arranging objects into groups is beginning to encounter multiplication. A student noticing that fractions, decimals and percentages are different ways of expressing the same relationship is beginning to see structure. A mathematician investigating symmetry, probability, geometry or prime numbers is doing something far more advanced, but the underlying instinct is similar: look carefully, notice the pattern and ask what lies beneath the surface.
Mathematics is not merely about obtaining the answer. It is so much more than that, and a large part of it is spotting and connecting various patterns.
Why students struggle with mathematics
Sadly, our present system of mathematics education is precisely this kind of nightmare. In
Paul Lockhart – A Mathematician’s Lament
fact, if I had to design a mechanism for the express purpose of destroying a child’s natural
curiosity and love of pattern-making, I couldn’t possibly do as good a job as is currently being
done— I simply wouldn’t have the imagination to come up with the kind of senseless, soulcrushing ideas that constitute contemporary mathematics education.
A great deal of mathematics teaching goes wrong because students are given rules before they can see the patterns that make those rules meaningful. They learn to borrow in subtraction without understanding place value. They learn to invert and multiply when dividing fractions without understanding what division means. They memorise procedures for algebra without seeing the relationships beneath the symbols. The result is fragile knowledge. A student may reproduce a method while the question looks familiar, but change the wording slightly and the whole structure collapses.
Real understanding is different. A student begins to recognise the mathematical architecture beneath the surface. They see that multiplication is connected to repeated addition, arrays, scaling and proportion. They see that division can mean sharing, grouping or comparing. They see that fractions are not strange symbols but relationships between quantities. They see that algebra is not an arbitrary collection of letters but a language for expressing patterns.
This is where mathematics can learn something from music. A musician does not become fluent by hearing an explanation once. A pianist practises scales. A guitarist practises chord changes. A drummer practises grooves. A tango dancer practises walking. But meaningful practice is not mindless repetition. The pattern is repeated until it becomes stable. Then it is varied, combined with other patterns, tested in a different context and used expressively.
Mathematics should work in the same way. Understand the pattern. Repeat it. Vary it. Connect it. Use it. Eventually, what once felt awkward becomes natural.
Patterns within patterns
One reason this subject fascinates me is that patterns are often nested inside larger patterns. In a song, an individual drumbeat forms a pattern. Several bars form a phrase. Several phrases form a verse. The verses and choruses create the architecture of the song. The song itself may sit inside the larger arc of an album.
The same is true in mathematics. Individual numbers form sequences. Small relationships combine into larger structures. A simple idea reappears in a more complex form years later. A child who understands equal groups is laying foundations for multiplication. Multiplication supports fractions and ratios. Ratios lead into proportion, gradients, trigonometry and rates of change. The later mathematics may look completely different, but the earlier patterns have not disappeared. They remain underneath.
Dance works like this too. An individual step matters, but so does the phrase. So does the entire song. So does the relationship between two people over the course of the dance. A dancer who thinks only about the next step may lose the music. A mathematician who thinks only about the next calculation may lose the structure.
Not everything can be reduced to mathematics
There is a danger here. When people speak about mathematics and the arts, they sometimes become overexcited and make claims that flatten the very thing they are trying to celebrate. Music is not merely mathematics made audible. A painting is not merely geometry with colour added. A poem cannot be reduced to syllable counts and rhyme schemes. A dance cannot be explained fully by timing, balance and angles.
Patterns matter enormously, but they are not the whole story. A technically perfect singer can leave us cold. A simple, imperfect voice can break our hearts. Two tango dancers can perform the same steps, and one dance may feel empty while the other feels unforgettable. The structure matters, but so does presence. So does intention. So does human experience. Patterns give art its architecture. They do not explain away its mystery.
The deeper connection
I have spent much of my life moving between worlds that are often treated as separate. I trained as an engineer. I have taught mathematics for years, often working with students whose foundations need to be rebuilt carefully. I have sung in rock bands for more than two decades. I have spent countless hours dancing tango and trying to understand why one movement feels natural while another feels forced.
The longer I continue, the less separate these worlds appear. The same questions keep returning. What is the underlying structure? What should remain stable? What needs to change? When does repetition become mastery? When does repetition become lifeless? When should we introduce variation? When should we resist the temptation to add more?
These are mathematical questions. They are musical questions. They are artistic questions. They are questions about teaching. They may even be questions about life.
We need patterns. They help us make sense of the world. They create continuity. They allow us to learn. They give us something to return to. But we also need variation. Without variation, a pattern becomes a prison. Without pattern, variation becomes noise. Somewhere between the two lies expression.
The bassline repeats. The drummer holds the groove. The poet returns to an image. The painter repeats a shape. The dancers keep walking. The mathematician notices a relationship. And then something changes.
A note arrives late. A chord shifts. A line breaks. A dancer pauses. A pattern that was underneath all this time suddenly reveals itself. For a moment, we see the structure beneath the surface, and the world feels richer because we noticed.
Note : This post was generated with ChatGPT and my own knowledge
