The loop-the-loop is by now a standard stunt—a vehicle completes a vertical circle without losing contact with the track in a seemingly gravity-defying feat. One of the earliest mentions of the stunt is in the early 1900s when the American cyclist Conn Baker, stage name Diavolo (pictured above in action), travelled the country performing the feat on a track about 20 feet in diameter. His crowds, naturally, feared for his life.

The anxiety is understandable: most people grasp intuitively that if the speed is too low, the vehicle will come off the track at some point beyond where the track becomes vertical. But have you ever wondered about the maths behind it all? What, exactly, would happen to a dawdling Diavolo?

In order to understand how this trick works, we will need to understand something about the mechanics of the motion. In what follows we will make some modelling assumptions:

1. That the track is circular.
2. That there are no forces directed along the track, such as driving forces (ie Diavolo isn’t pedalling during the feat), or forces of resistance, such as friction.
3. That the vehicle is point-like.

Under these simplifying assumptions we can make progress using Newton’s laws. These laws are all about the concept of force. His first law says that, in the absence of a net force, an object will carry on moving in the same direction and with the same speed (which might be zero) for ever. His second law $F=ma$, tells us that an object’s acceleration, $a$ (the rate of change of its speed and direction) is determined by the forces acting upon it. His third law introduces the concept of a reaction force. This force is present between any two objects in contact. For example, when you stand barefoot on the ground your weight (the force of gravity acting on your body) is exactly balanced by the reaction forces between your feet and the ground. Newton’s first law then says that you will stay in that state indefinitely—or until a new force is applied to your body.
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Looking for structure comes naturally for human beings. Gazing over a landscape and identifying patterns; reading poems and feeling where the beat lies; staring at paintings and figuring out the stories being told. Structure both guides us and reassures us.

Structure lies at the core of mathematics. We look for abstract patterns and scaffolding in all sorts of mathematical phenomena. Counting is no exception in this hunt. When we are in elementary school, we learn rules for computing, for example, $3 + 10$, or $10 – 4$, or $1284 \times 12$. We learn that if we want to compute $(3+10)\times 4$, we can just compute $3 \times 4 + 10 \times 4$. Later, in maths at university, these kinds of properties are baked into the abstract definition of a ring. A ring is a set $R$ together with two operations: $+$ and $\times$, with special properties (which we’ll get to later) and two distinguished elements: $0$ and $1$, which are the identities of these operations. A simple example of a ring is the integers $\mathbb{Z}$, where $+$ and $\times$ are precisely the elementary school operations we’ve known for years.

To the eyes of a certain type of mathematician, however, the integers are not the wholesome and friendly object that they might seem at first.

Are we secretly linguists?

Model theorists, who live and work in the realms of mathematical logic, see mathematics as a language to be understood through the lenses of mathematics itself. While this might seem strange at first, it is not any stranger than linguists analysing English through the lenses of English itself. The bread and butter of mathematical research is proofs, which can be modelled mathematically in the same way that a fluid flowing can be modelled, the flight of a bird can be modelled, the magnetic field emanating from my Christmas lights can be modelled.

The bread and butter of mathematical research is proofs, so what is the bread and butter pudding? Image: Wikimedia Commons user codepo8, CC BY 2.0

When we say that a theorem is true, the model theorists argue, what we are really doing is playing a game. We have some rules—which mathematicians call ‘axioms’—and we reason to deduce which moves are allowed. If you have ever played Dungeons & Dragons, you may have realised that the rulebooks don’t cover all possibilities. Some reasoning is left to the players’ deduction (often wreaking havoc on friendships and destroying years-long relationships, not unlike mathematical research). Starting from the rules that are written down (the axioms), those who approach the game can try and argue that certain moves (the theorems) are possible and allowed.

What does linguistics, and logic by extension, have to do with this? Any language needs an alphabet: a set of symbols with which we can write down our phrases and statements. If we want to talk about apples, we better have the letters necessary to build the word apple in our alphabet. So, what do we want to talk about in mathematics?

Let’s consider rings again. We have already seen the ring of integers $\mathbb{Z}$, but one can also be a bit more creative, and come up with all sorts of rings. For example, $2 \times 2$ matrices with real entries form a ring $M_2(\mathbb{R})$, which is radically different from $\mathbb{Z}$: the multiplication of matrices is not commutative! For example,
\begin{align*}
\begin{pmatrix}
2 & 1 \\
1 & 1
\end{pmatrix}
\times
\begin{pmatrix}
1 & 2 \\
1 & 1
\end{pmatrix}
=
\begin{pmatrix}
3 & 5 \\
2 & 3
\end{pmatrix},
\\
\begin{pmatrix}
1 & 2 \\
1 & 1 \\
\end{pmatrix}
\times
\begin{pmatrix}
2 & 1 \\
1 & 1 \\
\end{pmatrix}
=
\begin{pmatrix}
4 & 3 \\
3 & 2 \\
\end{pmatrix}.
\end{align*}
If we want to understand rings like model theorists do, then, we need to have symbols to talk about its operations and identities: a symbol $+$ for addition, a symbol $\times$ for multiplication, a symbol $0$ and a symbol $1$. To express mathematical statements, we will need to expand our alphabet a little bit. Model theorists speak a language called first-order logic, where alphabets are assumed to also contain variables ($x,y,z,\dots$), connectives ($\land$, which represents and; $\lor$, which represents or; $\Rightarrow$, which represents then; $\neg$, which represents not), and quantifiers ($\forall$, which represents for all; $\exists$, which represents there exists) and a symbol for equality, $=$.

We now have a rather large alphabet. Let’s try to express some mathematical fact. Suppose we want to state multiplication $\times$ is a commutative operation. What this means is that whenever I pick two elements $a$ and $b$, $a\times b$ is the same as $b\times a$. In first-order logic, this looks like this:
\[
\forall a \,\, \forall b \, (a\times b = b\times a).
\]
As another example, suppose we want to state that all polynomials of degree $2$ have a root. This means that no matter how we choose coefficients $a, b, c$, there is a root of the polynomial $ax^2+bx+c$. In first-order logic, this looks like this:
\[
\forall a \,\, \forall b \,\, \forall c \,\, \exists x \, (ax^2+bx+c=0).
\]
First-order logic, together with the language for rings that we have chosen above, allows us to express many interesting properties of rings. Certain properties will be true in certain rings, and not in others: for example, the operation $\times$ is going to be commutative in the integers $\mathbb Z$, but not in $M_2(\mathbb R)$. We can collect all the statements that are true in a certain ring $R$ in the theory of that ring, denoted $\operatorname{Th}(R)$. The sentence $\forall x \,\, \forall y \, (x\times y = y\times x)$ is an element of the set $\operatorname{Th}(\mathbb Z)$, but not of the set $\operatorname{Th}(M_2(\mathbb R))$.

While first-order statements can capture many of the important facts about a mathematical object, in this case a ring, they do not typically capture everything. We think of two rings $(R_1,+_1,\times_1)$ and $(R_2,+_2,\times_2)$ as being the same if they are isomorphic, in other words if there exists a bijection $f\colon R_1 \to R_2$ which behaves well with operations, ie

1. $\forall x, y \in R_1$, $f(x+_1y) = f(x)+_2f(y)$,
2. $\forall x,y \in R_1$, $f(x\times_1 y) = f(x)\times_2 f(y)$.

Sometimes rings with the same theory are isomorphic: this is the case if we look at the complex numbers $\mathbb{C}$ and their theory $\operatorname{Th}(\mathbb{C})$. If another ring has the same theory (and the same cardinality, since we need a bijection), then it is isomorphic to $\mathbb{C}$. In a way, the theory knows everything there is to know. This does not hold true for all theories across different kinds of mathematical object. For example, there are fields that share the same theory as $\mathbb R$, but are not isomorphic to it, since they admit infinite elements (namely, the so-called Robinson hyperreals).

There are many reasons why $\operatorname{Th}(\mathbb{R})$ and $\operatorname{Th}(\mathbb{C})$ are very different in their behaviour, but there is a deep underlying one: $\mathbb R$ can define an ordering (ie, an element is non-negative if and only if it is a square), while $\mathbb C$ cannot. In this sense, $\mathbb C$ is less ‘complicated’, as it showcases less structures and patterns than $\mathbb R$.

The upshot is: the more complicated a ring $R$ is, the less its theory $\operatorname{Th}(R)$ will know about it. This leads us to a situation where rings with the same theory might not be isomorphic. Especially if the rings are complicated, containing many patterns and structures.

The hunt for forbidden patterns

Enter Shelah. Saharon Shelah was born in 1945, and graduated Tel Aviv University with a maths degree in 1964. He was awarded his PhD at the Hebrew University of Jerusalem in 1969, for research focusing on stable theories.

Saharon Shelah in front of what might be his favourite office paper. Image: Andrzej Roslanowski, CC BY-SA 2.5

The story of modern model theory begins in the 1960s, when Shelah proposed a method to map out theories based on how complicated they are, building a tentative map of the universe of mathematical theories.

In the case of rings, we think of $\mathbb{C}$ as being less complicated than $\mathbb{R}$, since the theory of $\mathbb{C}$ describes it very well (even up to isomorphism), whereas the theory of $\mathbb{R}$ does not.

Among all rings, some more complicated, some less, there is one that sits at the borders of this mapping, model theorists’ own hic sunt leones: the ring of integers $\mathbb Z$.

Long-time fans of true crime know that, whenever some deeply unsettling truth about somebody is unearthed, there will always be some neighbour commenting “Oh, but they were so kind… so polite.” No matter if the person in question has committed several gruesome murders, there is a high chance that those who knew them on a daily basis will say that they would have never expected it. While the community might disagree on which objects are easy and which ones aren’t, there is a structure that nobody would expect to create trouble: $\mathbb{Z}$, the ring of integers. After all, Kronecker wrote “God created the integers.” Model theorists, however, are able to see beyond the facade of this seemingly harmless mathematical structure. To understand why, we have to go back some 40 years before Shelah’s work.

The breaking point of the idyllic picture of the integers is a procedure known as Gödelisation. The representation was introduced, as the name suggests, by Kurt Gödel in an effort to prove his famous incompleteness theorems. If you think back to the games metaphor, Gödel wanted to argue that if a game was complicated enough, then there would always be a move that is neither allowed nor prohibited by the rules of the game.

A set of axioms is considered ‘effective’ if it can be listed by an algorithm. In the 1930s, Gödel proved that if an effective set of axioms was able to perform the arithmetic of the integers $\mathbb{Z}$, then it would not be able to prove or disprove all possible statements about $\mathbb{Z}$. By ‘able to perform the arithmetic’, we mean that it must be able to reconstruct within itself the operations $+$ and $\times$ of $\mathbb{Z}$. While his result is impressive in itself, our focus today is rather on his technique. To prove his theorems, he created a translation procedure that transformed first-order statements in the language of rings (like $2=2$, or $n+1 = m+1 \Rightarrow n = m$, or ‘every polynomial of degree $2$ has a root’) into (very big) natural numbers. For a given first-order statement, this big number is known as its Gödel number.For instance, the first-order statement $x = y \Rightarrow y = x$ is encoded with the natural number $120061121032061062032121061120$.

The encoding was done in a way such that whether a statement was true or not (whether there was a proof of it or not) was a matter of the elementary arithmetic properties of its Gödel number. This means that the truth of a certain statement can be checked by verifying arithmetical identities—just like we used to do in elementary school. Think of questions like ‘is $5 \times 3 = 15$?’, or ‘is $1289$ divisible by $29$?’. This seems easy, perhaps; but try it with dozens and dozens of digits, and soon enough you’ll either grow tired or ask a computer to do it for you (or both).

Shelah’s thesis is that it is worthwhile to classify theories. In doing so, mathematicians suggest dividing lines between different theories based on good test problems, and hence are able to place ‘similar’ theories in shared regions, creating a map. Each dot is a theory. (Interactive map on forkinganddividing.com)

Gödelisation allows us to encode entire mathematical objects and theories inside the integers. Complicated theorems, emerging from all over mathematics, reduce to simply checking whether two integers divide each other—a procedure that a computer can do, given perhaps infinite time. The mathematical fact that the polynomial $X^7+X^2+3$ has a solution in $\mathbb R$ can be translated into an equality between (very big) integers, which we can compute in $\mathbb Z$.

This is a remarkable fact. Many mathematical statements could be checked by a computer performing basic arithmetic, only with numbers which have billions and billions of digits. It is also a massive problem for the theory of $\mathbb{Z}$. Gödel’s encoding means that the seemingly elementary operations of $\mathbb Z$ can understand patterns which might very well be infinitely complex, even beyond our imagination and comprehension. No matter how hard we try, $\mathbb{Z}$ will always be too complicated for us to understand with first-order logic.

There is a fine line between complex enough to be interesting and too complex to be dealt with. The integers sport an impossible, almost cosmic level of expressivity. To many mathematicians, $\mathbb Z$ is the simplest object we can think of; they are, after all, not too far from just putting one pebble after the other. They provide the base to many of our grandiose castles of ideas and theories, the foundations of many explorations into unknown mathematical lands. To model theorists, however, they reveal a different face. A twisted expression, a creepy smile. The source of all darkness, the original Pandora’s box. We don’t talk about them, hoping that in our cautious adventures we will never find ourselves alone with the ring of integers in a dimly lit alley.

In this issue’s edition of our day in the life series, we hear from three maths teachers who work in different countries:

• Elizabeth Brocklebank is a trainee secondary school teacher in England
• Karin Togaç Çağlıuzun teaches in a private middle school in Turkey
• Angelo Rosello teaches in a classe préparatoire in France

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On a bright August day in 1834, John Scott Russell, a Scottish engineer, became acquainted with a wave which would forever be associated with him within the field of fluid dynamics and beyond. He was following a boat which only just fit into the canal it was travelling down.

Suddenly the boat stopped. From the bow of the ship, a lump of water swelled and burst forward. On horseback, Scott Russell continued on, and became increasingly intrigued as the lump seemed to travel without changing shape, sustaining itself down the channel.

In this way they travelled, man and wave, at a pace of about nine miles per hour. They travelled for almost two miles before the wave disappeared among the twists and turns of the canal. This wave, standing about 40cm in height and extending about nine metres along the canal, never left Scott Russell’s thoughts. In September 1844 he gave a report on his scientific investigations into what we now call the solitary wave.

Just smile and wave boys, smile and wave…

The solitary wave is a surface wave: one that occurs at the fluid–air interface. Rudimentary surface wave theory, and usual intuition, predicts that surface waves disperse, rather than keeping their form as the solitary wave did.

Think of the last time you splashed around in a swimming pool. If you put your hands at the water level and push, a lump of water would roll off your hands. But unlike the solitary wave, it would disperse into a bunch of ripples rather than retaining its shape, stopping you from launching long-ranged splash attacks.

Mathematically, dispersion describes how the velocity of a sinusoidal wave depends on its wavelength. For simplicity, consider a fluid interface where the surface height only varies in one direction, say the $x$-direction. Typically water waves aren’t perfectly sinusoidal, but we can build up any surface profile from sine waves. The system then evolves by the sine waves of varying wavelengths travelling independently at the velocity prescribed by the dispersion relation.

Scott Russell’s discovery commemorated on the canal. Image: Jim Barton CC BY-SA 2.0

Scott Russell smiling but not waving

For a wave to keep its shape, all the constituent sine waves must travel at the same velocity. In other words, there cannot be dispersion. This holds for many waves in nature: red light and blue light differ in wavelength, but travel at the same speed. The same is true of sound at different pitches. But it’s not so for typical surface waves, where waves with longer wavelength travel faster than those with shorter wavelength. Any lump of water built from many sine waves quickly falls apart.

What saves the solitary wave is that dispersion is delicately balanced by a nonlinear effect. This is neglected in rudimentary surface wave theory, which is a linearised theory. Linearity is what allowed us to think of the wave profile as being the sum of many independent sine waves, and the nonlinearity causes the sine waves to interact in potentially complicated ways.

Linearising is valid when the variations in height are small compared to the depth of the water. Our hands are small, and the lumps of water we can make aren’t big enough to bump us into the regime where the nonlinear effect becomes important. But boats are quite big, and can make swells of water large enough (relative to the depth of the canal) for nonlinearity to come into play.

The equation describing water waves which includes the nonlinear effect is known as the KdV equation, named after the Dutch duo Korteweg and De Vries. They weren’t the first to derive the equation but they found a solution that travelled with constant speed: the one-soliton solution to KdV.

The KdV one-soliton

Scott Russell’s solitary wave had been proven. He was sure the self-sustaining wave was hugely important, but it hadn’t caught the attention or imagination of many of his contemporaries. It wasn’t until the next century when the study of solitons really took off.

A two soliton solution to the KdV equation evolving in time. It starts off with two separated lumps, which combine, then pass through each other and continue on unchanged.

Solitons

The next chapter of our soliton story picks up at the Los Alamos national laboratory in 1953. Four physicists (Fermi, Pasta, Ulam and Tsingou) were puzzling over what they were seeing: an animation simulating a vibrating string with nonlinear terms in its dynamics. The simulation was programmed on the Maniac computer (great name) by the most computer-savvy among them, Mary Tsingou.

They were interested in a hypothesis—called the ergodic hypothesis—that, roughly speaking, for systems which were mostly linear but perturbed by a nonlinear term, the initial energy would eventually (potentially after a long, long time) be evenly distributed over different degrees-of-freedom of the system. For their string, this would mean the amplitudes of all different frequencies would eventually be comparable.

The Maniac had also only recently been finished. With the Maniac at their door, the physicists tried what few could do before; study a problem computationally. They were looking in particular for a problem that would be easy to formulate, but which was intractable by hand or even mechanical computer.

In order to simulate the string on a computer, it had to be modelled by a discrete, finite number of points: in their case, 64 points. This is linear in the strain, and assuming that the strains are small, the next force to consider would be a quadratic one, and this was precisely the nonlinear dynamics considered, with neighbouring forces $F = k(\delta + \alpha\delta^2)$. They started the simulation with only an excitation in the fundamental frequency of the string. Early on, they saw the behaviour they expected, with successively higher frequencies of the string starting to receive small excitations. To their surprise, only one of the frequencies would have large excitations at any one time, starting from the fundamental frequency, then passing to the second mode, and then the third. This only continued among the first few modes, then the large excitation returned to the fundamental frequency, showing quasi-periodic behaviour.

This was unresolved until around twelve years later when two Americans, Kruskal and Zabusky, began to investigate the KdV equation computationally. They noticed that the continuum limit of the system studied by the Los Alamos four was described by the KdV equation.

By simulating the KdV equation using a more direct discretisation scheme, they found something amazing. No matter what initial configuration they prescribed for the water displacement height, the surface would break into a train of solitary waves with a profile like a one-soliton, each with their own amplitude and speed. Moreover, when these solitary waves collided, they would interact, maybe raising or dipping their peaks slightly as they passed one another, but then recover the shape and speed they had before the collision.

They knew they were onto something important, and they christened these resilient lumps solitons: ‘solit-’ for solitary, ‘-on’ to denote a particle, the suffix which appears at the end of a zoo of particle physics terms: electron, proton, hadron, baryon, and so on…

The particle-like behaviour of these self-sustaining lumps of energy is their ability to enter into, then emerge from, collisions with a well-defined shape and speed. The presence of both wave- and particle-like behaviour suggests quantum shenanigans, but there is nothing quantum here, just water waves.

In 1967 a team of four scientists, including Kruskal, found a construction of exact solutions to the KdV equation which consist of precisely $n$ solitons, which start at large separation, intersect and mingle, then regain their original size and shape, and drift apart.

The details of the construction of the explicit solutions are near magical. It turned out to borrow ideas from the scattering theory of quantum particles, then requires the application of a twisted functional transform to the scattering data. This transform is something like a nonlinear version of the Fourier transform, used extensively in applied maths due to its effectiveness in analysing signals.

The forward scattering problem in quantum mechanics is to determine the reflection and transmission of different frequency particles off a given potential. It’s like a mathematical description of how we see things: the object we look at is the potential, and the amount of light of different wavelengths that gets reflected into our eyes allows us to construct an image of that object.

In quantum mechanics, the potential is a function of space, while the reflection and transmission data is packaged into a function of frequency called the spectral data. The KdV equation has an associated scattering problem, and the complicated dynamics of KdV turns into a very simple time dependence for the spectral data.

On the KdV side, the wave profile $u$ becomes the potential in the scattering problem. In the scattering picture, we have a grasp on how the spectral data evolves, so to recover a solution $u(x,t)$ for KdV, we need to reconstruct the potential from its spectral data, which is an inverse scattering problem, and where the twisted transform comes in.

This brilliant but byzantine technique, known as the inverse scattering method, allowed the team to explicitly write down $n$-soliton solutions to KdV. Kruskal and Zabusky believed it was the presence of these solitons that meant the physicists’ string didn’t obey the ergodic hypothesis. The discovery of this inverse scattering method began a flurry of research into solitons.

Beyond water waves

Shortly after the inverse scattering method was found for KdV, it was adapted to two other PDEs. Alongside the KdV equation, these PDEs have since taken on a celebrity status within the study of solitons.

Of the three, the sine-Gordon equation is the one which has been studied for the longest time. It began its life in the field of classical differential geometry. Surprisingly, solutions to the sine-Gordon equation correspond one-to-one with pseudospheres, which are surfaces of constant negative curvature immersed in three-dimensional space. Here, immersed means the surfaces might do funky things like intersect themselves or have sharp cusps. But that’s an article for another day.

The sine-Gordon equation has also drawn attention in several parts of physics, from particle physics and statistical physics to material science, where it was used to study screw dislocations in crystals. The one-soliton solution to sine-Gordon has the interesting property that it limits to $2\pi$ instead of zero as $x$ tends to infinity. That is, if we plot the sine-Gordon soliton as a function of position, it does not look like a lump, like the KdV soliton, but has an S shape.

Soliton solution to the sine-Gordon equation

The sine-Gordon soliton is an example of a topological soliton. In fact, all solutions of the sine-Gordon equation have the property that the difference of limiting values between $x$ at positive and negative infinity is an integer multiple of $2\mathrm{\pi}$. This integer is called the winding number of the solution, and offers a glimpse of the interplay between topology and soliton theory.

The last of the three gold standard soliton PDEs is the nonlinear Schrödinger equation. The solitons of this PDE were the closest to having technological applications. Light signals in optical fibres are well modelled by solitons of this equation. Their ability to keep their shape was desirable for sending signals over vast distances, and solitons made it to lab trials before other technological advances in the field made them obsolete. In an alternate universe, the online edition of this magazine may have been brought to you courtesy of solitons.

There is a dizzyingly rich array of PDEs with soliton solutions. The majority of known PDEs admitting solitons are defined with only one spatial dimension, although of course more realistic systems incorporate two or three spatial dimensions. One such PDE is the Kadomtsev–Petviashvili equation, which describes a two-dimensional model of shallow water waves and generalises the KdV equation.

From humble beginnings in the Union canal, our soliton story has taken us worldwide, where they were one of the first phenomena explored using mathematical computing, and even into the materials and optics labs. The rich theory of solitons has sustained itself now for almost 200 years and solitons are now ubiquitous in mathematical physics, fulfilling Scott Russell’s vision. And there is still so much left to explore.

Having turned 18, I have now swapped out the impromptu school parties with their questionable music, eccentric concoctions which come about after a few drinks—Shloer, vodka and salad dressing, anyone?—and soon-to-be sticky carpets, to attend city centre institutions with equally interesting music, drinks and floor-al adhesion.

I and a group of close friends (who will be referred to as A, B, C, D throughout, leaving me as E) decided to paint the town red and celebrate the arrival of my newfound rights to drink, drive a forklift truck, and go bungee jumping (though not simultaneously). We decided unanimously on a venue whose name I will refrain from mentioning (as I am certain I saw the bouncer doing the Chalkdust prize crossnumber, and hence may be liable to read this article).

We approached the bar, and the bartender—thrilled for what must have been their first clientele of the night—took our orders with great gusto. Although the bartender was blessed with many things, short-term memory was not among them: each order was perfectly procured, but much like the pianist who could play all the right notes but not necessarily in the right order, the hapless bartender had no clue which order was whose. In fact, by some combinatorial miracle, he managed to give everyone the wrong drink: whereas it ought to have gone ABCDE, instead, he went DAECB.

“Just our luck,” said C, exasperatedly.

“A derangement no less.” said D.

“Who are you calling deranged?” responded B indignantly.

“No—a derangement—a kind of permutation,” I explained to defuse the situation.

“What’s a permutation?” pondered D thoughtfully.

“It’s simply a bijection $\pi: \{1, \dots, n\} \rightarrow \{1, \dots, n\}$: a map sending each element of $\{1, \dots, n\}$ to a different element,” I explained.

“Right-o. But a derangement?” enquired B.

“A derangement is a permutation without fixed points: so $\pi(i) \neq i$ for all $1 \leq i \leq n$.” I answered.

Deranged or otherwise, we needed to work out how to return the beverages to their rightful owners. Our attempt to slide them across, wild west style, was scuppered by one inadvertent collision and the stern eye of the bouncer (who was getting started on the across clues) so we decided to work it out rigorously. While most people would get off their seats and swap drinks, we mathematicians know that no night out is complete until it becomes a maths problem. We may make things difficult for ourselves—but at least we’ll get a paper out of a night out.

“Let’s set some ground rules,” said C, enjoying his newfound role as leader. “Rule one: we must return our drinks in some series of swaps. Rule two: people may only swap with their immediate neighbour to prevent any collisions. Rule three: people may not engage in two swaps at once or have more than one drink at once. That constitutes cheating, and will result in the aforementioned cheater being required to buy everyone a round of drinks. And rule four: we will carry this out with a minimal number of swaps to prevent anyone from ‘sampling’ someone else’s drink. I know who you are,” he said, eyeing A suspiciously.

With this, we took our seats and set to work, drawing frenzied diagrams on cocktail napkins with whatever we had to hand.

Ticket to ride

I stared quizzically at my sketches, resisting the urge to sip the Guinness which had been placed before me. “We can guarantee we will only need to do at most 10 swaps if we don’t need to swap each drink more than once.”

“Actually, only six pairs of drinks are in the wrong order,” added B.

“What do you mean, ‘the wrong order’?” I asked.

“I mean that for our permutation, there are only six pairs of people X and Y such that X sits to the left of Y but X’s drink is to the right of Y’s—like A and D. So, at some point, the drinks will have to cross past each other.”

As I struggled to make sense of this, A made a suggestion. “Let’s use some shorthand to make it slightly easier. How about we use $\sigma_i$ to denote the swap between the $i$th and $(i + 1)$th person, counting from A to E.”

“So a total of $n-1$ kinds of swap for $n$ people,” I added.

“Right,” replied A. “And when we do more than one swap, we work from left to right—because we’re indexing positions of drinks, not the drinks themselves.” We agreed this was a splendid idea.

“We can actually draw out an exact map of how to do this, and which exchanges to do… like a tube map!” said D, drawing the following diagram:

“All you need to do is draw lines between the drink and its destination. All the points at which the drinks should ‘collide’ give you the exact order in which we should exchange our drinks. So we can actually trace which drinks go where and when—like a tube map for cocktails!” said C, brimming with excitement.

“And lo and behold… only six exchanges, like you said!” chimed in B.

“Lower bound and upper bound: since the lines will only cross precisely when their drinks start on the wrong side, it’s optimal!” A surmised.

“And we can read off the orders of drinks as follows: DAECB, ADCEB, ADCBE, ADBCE, ABDCE, ABCDE. Hey presto—six swaps, like you said!” exclaimed C.

“Or, with our newfound notation, simply $\sigma_1 \sigma_3 \sigma_4 \sigma_3 \sigma_2 \sigma_3$.” B added.

We tried it, and lo and behold, our drinks were returned to the right order.

I continued: “So we can write a formula for the number of swaps in general: for any permutation $\sigma$, where the $i$th element is notated $\sigma(i)$, the number of exchanges we need to use is
\[|\{i < j: \sigma(i) > \sigma(j)\}|\text{.}\]

“And so for five drinks, we’ll never need more than 10 swaps—because we only have $\left(\begin{smallmatrix}5\\2\end{smallmatrix}\right) = 10$ pairs $\{i, j\}$ where $1 \leq i < j \leq 5$," parried B. "Which could happen if we started out with EDCBA,” added A.

They’re the same picture

While we supped our drinks, C opined the following.

“I think the diagram is rather splendid, because we can actually prove things with the diagrams themselves.”

“How so?” I asked.

“Well, say we wanted to multiply out two permutations. We can just stack them up like so:”

“But then how would you undo it?” asked A, unsatisfied.

“By reflecting the diagram, top-to-bottom,” C countered.

Yet D looked suspicious. “But surely your diagrams aren’t unique—you can change the diagram, but the permutation remains the same. How else could we have done this?”

“If you do a swap twice, it undoes itself: that checks out from the diagram!” said B, drawing the following:

“For some pairs of swaps, like you mentioned, it doesn’t matter which order we do them—they may as well have been in either order, or at the same time,” said A.

“So we could say $\sigma_i \sigma_j = \sigma_j \sigma_i$,” I surmised. “And that would look a little like this:” I added, displaying my proof:

“But that only happens if $i$ and $j$ are at least 2 apart. So what if they are next to each other?” A asked trenchantly.

“There’s an even cleverer one. If we have a permutation like $\sigma_i \sigma_{i + 1} \sigma_i$, we can ‘pick up’ the middle strand and place it over the other two without changing the result of the permutation, and get $\sigma_{i + 1} \sigma_i \sigma_{i + 1}$ back.” intoned C. “Like this:”

“Or equivalently, $(\sigma_i \sigma_{i + 1})^3$ should return everything to where it started, since both $\sigma_i \sigma_{i + 1} \sigma_i$ and $\sigma_{i + 1} \sigma_i \sigma_{i + 1}$ swap drinks $i$ and $i + 2$,” I added.

In the heat of excitement, C urged A and B to show him in order to demonstrate this. The drinks had returned like clockwork.

“So we now have a set of rules for our permutations: we have permutations $\sigma_i$ for $1 \leq i \leq n$ such that:

• $\sigma_i^2 = 1$ (where $1$ is just the identity permutation);
• $\sigma_i \sigma_j = \sigma_j \sigma_i$ if $|i-j| > 1$; and
• $\sigma_i \sigma_{i + 1} \sigma_i = \sigma_{i + 1} \sigma_i \sigma_{i + 1}$,”

summed up C. “Or, alternatively, $(\sigma_i \sigma_{i + 1})^3 = 1$ and $(\sigma_i \sigma_j)^2 = 1$ otherwise.”

“And we can get to any diagram of any permutation we want this way?” B enquired.

“Using C’s diagram idea.” I offered.

“So we’ve given a full description of the collection of permutations!” D said, eagerly awaiting their drink.

We sat in awe of our realisation.

You’ve really made the braid

Then C once again saw fit to throw the cat among the pigeons. “Hold on. Let’s go back to the original diagrams. What if they were actual pieces of string?”

“So the strands would go over and under each other?” asked D.

“Right. You’d multiply and take inverses just like we did earlier—so it’s as well-defined as our diagrams,” replied C.

“If you did a swap twice, you wouldn’t get the same thing back—it would be all tangled!” I said, drawing a version of our original map:

“Braided!” suggested C.

“Ah—but we’d still have our previous rules, that $\sigma_i \sigma_j = \sigma_j \sigma_i$ if $|i-j| > 1$, and $\sigma_i \sigma_{i + 1} \sigma_i = \sigma_{i + 1} \sigma_i \sigma_{i + 1}$.” A elicited.

“In fact, I don’t think we can say anything more—these are the only ways we can move any of the strands without tangling our strings. So that’s it—that’s our new algebra,” I concluded.

To be or knot to be

Then D picked up his bar napkin and made it into a loop: “What if we joined the ends together?” he asked.

“We get some kind of knot—or maybe a link if there happens to be more than one connected component,” said B:

“Could we get any kind of knot like that? And which braids give us the same thing?” I asked.

“Well, I suppose if you did braids $\sigma \tau$, it would be the same as doing $\tau \sigma$—we would just loop round and get the same thing back, because it doesn’t matter where you start,” said C:

“And if we add an extra strand, $\tau \sigma_n$ on $n + 1$ strands should look the same as $\tau$ on $n$ strands,” I added:

“Wait—this all feels an awful lot like physics.” D said. “Imagine we watch how our drinks proceed at some point in time. Your original diagram, C, shows us the worldlines of the individual drinks as they pass through time from top to bottom, like frames in a movie:”

“But originally, if we swapped them twice, we would get the same thing back.” A replied. “Now, if we swap them twice, the particles would remember what they were swapped with! In fact, you could make a computer with these—they would store the way in which they were exchanged!”

“No more—you’re giving me a headache!” implored B.

Hexagonator the almighty

“But how could we do this?” I asked.

“Say we have 5 particles, say, A, B, C, D and E. We denote their states by $|A\rangle$ and so on.” said C. “We’d write $|A\rangle \otimes |B\rangle$ for the superposition of two particles.”

“And this would be different from $|B\rangle \otimes |A\rangle$?” I enquired.

“Yes,” replied C, “but we’d need some kind of operator $b_{A, B}$. One which would do \[|A\rangle \otimes |B\rangle \rightarrow |B\rangle \otimes |A\rangle.\text{“}\]

“And if you did it twice with our braided particles, you wouldn’t get the same configuration back,” added A.

“What about if we have three particles? Wouldn’t \[(|A\rangle \otimes |B\rangle) \otimes |C\rangle\] give us the same thing as \[|A\rangle \otimes (|B\rangle \otimes |C\rangle)\text{?”}\] I asked.

“Ah, but they’re not—be careful,” said D, toyingly. “We’d need some kind of map, $a_{A, B, C}$ which would carry out \[(|A\rangle \otimes |B\rangle) \otimes |C\rangle \rightarrow |A\rangle \otimes (|B\rangle \otimes |C\rangle)\] for us.”

“Call it the associator! I like the sound of that—like a superhero,” I said.

“There’s another way of seeing this,” added A. “This all sounds an awful lot like a category—a collection of objects (like our particles) and maps, or morphisms (like our $a$ and $b$ maps) between them.

“And since this superposition malarkey is associative, we have a monoid.”

“So our particles form a braided monoidal category!” I surmised.

“I remember reading about this—there’s a couple of results it should always satisfy… Ah, here’s one:” said D, scrawling across two napkins and a beer mat (overleaf).

“A hexagon diagram!” I added.

“What about the three equivalences C found earlier?” raised B, gesturing to the discarded diagram. “Couldn’t we encode this in our new categorical form?”

“Just considering the strands, swapping the first pair and fixing the third is simply $b_{A, B} \otimes 1_C$. That’s like our $\sigma_1$,” said C.

“Right… and similarly if we swap the second two, we get $1_A \otimes b_{B, C}$ —like our $\sigma_2$,” returned B.

“So,” said D, “our relation looks a little like this, up to association:
\begin{align*}
(1_A\otimes b_{B, C})(b_{A, C} \otimes 1_B)(1_C \otimes b_{A, B}) \hspace{2cm} \\ \hspace{1.6cm} = (b_{A, B} \otimes 1_C)(1_B \otimes b_{A, C})(b_{B, C} \otimes 1_A)\text{.”}
\end{align*}

“And that’s just the Yang–Baxter relation from theoretical physics!” A added.

“But what does that mean?” B asked.

“Having this relation in 2 dimensions (1 spacelike plus 1 timelike) means that a system will have certain quantities fixed—such as the momentum remaining the same in each particle—which is called being integrable,” answered A.

“Moreover, if we have a system in which 3 or more particles are scattering we can always reduce it to an equivalent system of 2 particles,” added C. “In fact, one can do a similar trick in higher dimensions—but unfortunately this beer mat is too narrow to contain such an explanation.”

“Isn’t it just remarkable how many different ways there are of seeing the same thing, just from considering our drinks?” I remarked, satisfied with the night’s work.

“Isn’t it just,” said B. “Now, whose round is it next?”

* I was planning on naming this “Merry Twist-mas” after the oft-overlooked masterpiece by The Marcels, which is objectively the best Christmas song of all time. Alas, the timing was not to be.

As a maths teacher, I often find that setting seemingly ordinary tasks for my students lead to surprisingly interesting mathematics. Recently, an investigation that on the surface seemed quite unremarkable ended with me taking a deep dive into areas of mathematics I had never seen before. It started with us thinking about the spread of a forest fire.

The trees in the forest in question are perceived to be squares on a sheet of squared paper. At the start, a random tree catches fire and grid looks this:

After a unit of time, all trees (squares) that touched a tree that is already on fire catch fire too. This leads to nine trees being on fire:

The spread of the fire repeats. After a second unit of time, there are 25 trees on fire:

If you continue, then the next numbers in the sequence are 49, 81, and 121. From immediate inspection, you can spot that these are the odd square numbers. Part of the task was to then take this and write a quadratic for the $n$th term in the sequence. Providing you know that $2n + 1$ is an algebraic way of writing odd numbers, the sequence jumps out immediately:

\begin{align*} a(n)&=(2n+1)^2\\ &=4n^2 + 4n + 1. \end{align*}

A forest of hexagons

Following the same process as for squared paper, we can look at the fire spreading but using differently tessellated paper. Squared paper produced a quadratic sequence, but it is a little harder to predict what would happen when the tiling changes.

Looking first at the hexagonal paper, we see that, as it did for squared paper, each iteration looks like an enlarged version of the previous one:

Listing out the sequence we have 1, 7, 19, 37, 61, 91, 127, and so on. Perhaps surprisingly, this too turns out to be a quadratic sequence. Quadratic sequences are sequences that have a general rule of $$a(n)=c_1n^2+c_2n+c_3,$$ where $c_1$, $c_2$, and $c_3$ are real numbers. To calculate $c_1$, we look at the second difference between terms of the growing sequence:

The second difference is a constant, and can be halved to obtain $c_1$. Next, you can subtract the sequence $c_1n^2$ from the original sequence. If your original sequence is quadratic and you’ve done it right so far, this should leave you with a linear sequence that you can then work out the rule for. For our sequence this ends up being:

\begin{align*} a(n) &= 3n(n+1) + 1\\ &= 3n^2 + 3n + 1. \end{align*}

A forest of triangles

Finally, the students looked at the growth on triangular paper. The spread here ends up being a little different to the previous two. For both the square and hexagonal paper the shape grows and replicates the original shape. However, for the triangular paper, as it spreads the triangle shape gets smoothed at the edges.

Continuing the sequence we get 1, 13, 37, 73, 121, 181, and so on. Once again, this is a quadratic sequence. Using a similar method as we did for hexagons, we obtain:

\begin{align*} a(n) &= 6n(n-1)+1\\ &= 6n^2-6n + 1. \end{align*}

Polygonal numbers

These results may look like just three quadratic sequences, but there’s more that connects them: they’re all examples of a type of number sequence called the centred polygonal numbers (also called the centred $k$-gonal numbers where the choice of $k$ gives a specific sequence). These sequences are created by drawing increasingly larger polygons made of dots around a central dot, hence the name.

The centred polygonal numbers are closely related to, but not the same as, another type of number sequence: the polygonal numbers. These include the well known triangular and square numbers, and are also created by drawing increasingly large polygons of dots, but in this case the polygons all share a vertex rather than all being around a central point. You can see the difference by looking at the two sequences for a square.

The first four square numbers: 1, 4, 9, and 16.

The first three centred square numbers: 1, 5, and 13.

The centred polygonal number can also be produced using the triangle numbers: you can start with the central dot, then make the centred $k$-gonal numbers by putting $k$ copies of the triangle numbers around the point. For the centred square and hexagonal numbers, that looks like this:

The centred square numbers can be built from a central dot and four copies of a triangle number.

The centred hexagonal numbers can be built from a central dot and six copies of a triangle number

If we call the central dot the 0th centred $k$-gonal number, then we see that the $n$th $k$-gonal number is built from the $(n-1)$th triangular number. This means that the formula for the $n$th term of the centred $k$-gonal number is:

\begin{align*} a_k(n) &= k\times(\text{$n$th triangular number}) + 1\\ &= \frac{kn(n+1)}2 + 1. \end{align*} Despite our sequence of fire on squared paper being the odd square numbers, the sequence is actually the centred octagonal numbers (ie $k = 8$). This is because we can rearrange an odd square number of dots to make increasingly larger octagons from a central point. It is also interesting to note that the sum of the reciprocals of this sequence is convergent: \[ \sum_{n=0}^\infty \frac1{(2n+1)^2} = \frac{\pi^2}8. \] The sequence we observed on hexagonal paper is much more aptly named: the centred hexagonal numbers (ie $k = 6$). Hexagons are notable shapes and can be seen in the natural world in such oddities as bee honeycombs or in popular media such as the board game Settlers of Catan.

Settlers of Catan’s board is based on hexagons.
Image: Catan Wikimedia Commons user Yonghokim, CC BY-SA 4.0

The final sequence was created from the triangular paper. This produces a sequence called the centred dodecagonal numbers (ie $k = 12$). Theses dots that make up the dodecagonal numbers can also be rearranged to produce evergrowing stars and so are also called star numbers. Like the centred hexagonal numbers, star numbers have been used in board games such as in a Chinese checkers board.

A Chinese checkers board set up for a three player game.
Image: Chinese-checkers Wikimedia Commons user Splattne, CC BY-SA 3.0

Centred dodecagonal numbers can be rearranged to make stars.

The centred polygonal numbers are interesting in their own ways for many other values of $k$, and we could talk about them for hours, but I’m going to return to my students and to something really interesting that they noticed.

The three choices of paper we used are the only three examples of regular tessellation. This is because squares, triangles and hexagons are the only three regular shapes whose interior angles are factors of 360°: six equilateral triangles, four squares, or three regular hexagons can fit around a single point. We’ve seen these numbers before: they are the coefficients of $x^2$ and $x$ in the general rules for the three sequences! This shows a remarkable link between the geometric property of tessellation and the algebraic representation of the sequences we produced.

This is what I love about mathematics. A task that on the surface seemed like a nice little investigation for students ended with varied classroom discussions as well as my own deep dive down a mathematical rabbit hole. Despite spending years learning and teaching, I still discover new ways of looking at things that I thought I had previously mastered. I hope that as I get older, I still get the joy of discoveries like this and can continue to share these with the students I teach.