Category: mathematics

From the Fibonacci sequence to the Bell numbers, there is more overlap between mathematics and poetry than you might think, says Peter Rowlett, who has found his inner poet.

People like to position maths as cold, hard logic, quite distinct from creative pursuits. Actually, maths often involves a great deal of creativity. As mathematician Sofya Kovalevskaya wrote, “It is impossible to be a mathematician without being a poet in soul.” Poetry is often constrained by rules, and these add to, rather than detract from, its creativity.

Rhyming poems generally follow a scheme formed by giving each line a letter, so that lines with matching letters rhyme. This verse from a poem by A. A. Milne uses an ABAB scheme:

What shall I call
My dear little dormouse?
His eyes are small,
But his tail is e-nor-mouse.

In poetry, as in maths, it is important to understand the rules well enough to know when it is okay to break them. “Enormous” doesn’t rhyme with “dormouse”, but using a nonsense word preserves the rhyme while enhancing the playfulness.

There are lots of rhyme schemes. We can count up all the possibilities for any number of lines using what are known as the Bell numbers. These count the ways of dividing up a set of objects into smaller groupings. Two lines can either rhyme or not, so AA and AB are the only two possibilities. With three lines, we have five: AAA, ABB, ABA, AAB, ABC. With four, there are 15 schemes. And for five lines there are 52 possible rhyme schemes!

Maths is also at play in Sanskrit poetry, in which syllables have different weights. “Laghu” (light) syllables take one unit of metre to pronounce, and “guru” (heavy) syllables take two units. There are two ways to arrange a line of two units: laghu-laghu, or guru. There are three ways for a line of three units: laghu-laghu-laghu; laghu-guru; and guru-laghu. For a line of four units, we can add guru to all the ways to arrange two units or add laghu to all the ways to arrange three units, yielding five possibilities in total. As the number of arrangements for each length is counted by adding those of the previous two, these schemes correspond with Fibonacci numbers.

Not all poetry rhymes, and there are many ways to constrain writing. The haiku is a poem of three lines with five, seven and five syllables, respectively – as seen in an innovative street safety campaign in New York City, above.

Some creative mathematicians have come up with the idea of a π-ku (pi-ku) based on π, which can be approximated as 3.14. This is a three-line poem with three syllables on the first line, one on the second and four on the third. Perhaps you can come up with your own π-ku – here is my attempt, dreamt up in the garden:

White seeds float,
dance,
spinning around.

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*Credit for article given to Peter Rowlett

A mathematical model of a particle that remembers its past so that it never travels the same path twice produces stunningly complex patterns.

A beautiful and surprisingly complex pattern produced by ‘mathematical billiards’

Albers et al. PRL 2024

In a mathematical version of billiards, particles that avoid retracing their paths get trapped in intricate and hard-to-predict patterns – which might eventually help us understand the complex movement patterns of living organisms.

When searching for food, animals including ants and slime moulds leave chemical trails in their environment, which helps them avoid accidentally retracing their steps. This behaviour is not uncommon in biology, but when Maziyar Jalaal at the University of Amsterdam in the Netherlands and his colleagues modelled it as a simple mathematical problem, they uncovered an unexpected amount of complexity and chaos.

They used the framework of mathematical billiards, where an infinitely small particle bounces between the edges of a polygonal “table” without friction. Additionally, they gave the particle “spatial memory” – if it reached a point where it had already been before, it would reflect off it as if there was a wall there.

The researchers derived equations describing the motion of the particle and then used them to simulate this motion on a computer. They ran over 200 million simulations to see the path the particle would take inside different polygons – like a triangle and a hexagon – over time. Jalaal says that though the model was simple, idealised and deterministic, what they found was extremely intricate.

Within each polygon, the team identified regions where the particle was likely to become trapped after bouncing around for a long time due to its “remembering” its past trajectories, but zooming in on those regions revealed yet more patterns of motion.

“So, the patterns that you see if you keep zooming in, there is no end to them. And they don’t repeat, they’re not like fractals,” says Jalaal.

Katherine Newhall at the University of North Carolina at Chapel Hill says the study is an “interesting mental exercise” but would have to include more detail to accurately represent organisms and objects that have spatial memory in the real world. For instance, she says that a realistic particle would eventually travel in an imperfectly straight line or experience friction, which could radically change or even eradicate the patterns that the researchers found.

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*Credit for article given to Karmela Padavic-Callaghan*

Until recently, working out how three objects can stably orbit each other was nearly impossible, but now mathematicians have found a record number of solutions.

The motion of three objects is more complex than you might think

The question of how three objects can form a stable orbit around each other has troubled mathematicians for more than 300 years, but now researchers have found a record 12,000 orbital arrangements permitted by Isaac Newton’s laws of motion.

While mathematically describing the movement of two orbiting bodies and how each one’s gravity affects the other is relatively simple, the problem becomes vastly more complex once a third object is added. In 2017, researchers found 1223 new solutions to the three-body problem, doubling the number of possibilities then known. Now, Ivan Hristov at Sofia University in Bulgaria and his colleagues have unearthed more than 12,000 further orbits that work.

The team used a supercomputer to run an optimised version of the algorithm used in the 2017 work, discovering 12,392 new solutions. Hristov says that if he repeated the search with even more powerful hardware he could find “five times more”.

All the solutions found by the researchers start with all three bodies being stationary, before entering freefall as they are pulled towards each other by gravity. Their momentum then carries them past each other before they slow down, stop and are attracted together once more. The team found that, assuming there is no friction, this pattern would repeat infinitely.

Solutions to the three-body problem are of interest to astronomers, as they can describe how any three celestial objects – be they stars, planets or moons – can maintain a stable orbit. But it remains to be seen how stable the new solutions are when the tiny influences of additional, distant bodies and other real-world noise are taken into account.

“Their physical and astronomical relevance will be better known after the study of stability – it’s very important,” says Hristov. “But, nevertheless – stable or unstable – they are of great theoretical interest. They have a very beautiful spatial and temporal structure.”

Juhan Frank at Louisiana State University says that finding so many solutions in a precise set of conditions will be of interest to mathematicians, but of limited application in the real world.

“Most, if not all, require such precise initial conditions that they are probably never realised in nature,” says Frank. “After a complex and yet predictable orbital interaction, such three-body systems tend to break into a binary and an escaping third body, usually the least massive of the three.”

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*Credit for article given to Matthew Sparkes *

A centuries-old maths problem asks what shape a circle traces out as it rolls along a line. The answer, dubbed a “cycloid”, turns out to have applications in a variety of scientific fields.

Light reflecting off the round rim creates a mathematically significant shape in this coffee cup

Sarah Hart

The artist Paul Klee famously described drawing as “taking a line for a walk” – but why stop there? Mathematicians have been wondering for five centuries what happens when you take circles and other curves for a walk. Let me tell you about this fascinating story…

A wheel rolling along a road will trace out a series of arches

Imagine a wheel rolling along a road – or, more mathematically, a circle rolling along a line. If you follow the path of a point on that circle, it traces out a series of arches. What exactly is their shape? The first person to give the question serious thought seems to have been Galileo Galilei, who gave the arch-like curve a name – the cycloid. He was fascinated by cycloids, and part of their intriguing mystery was that it seemed impossible to answer the most basic questions we ask about a curve – how long is it and what area does it contain? In this case, what’s the area between the straight line and the arch? Galileo even constructed a cycloid on a sheet of metal, so he could weigh it to get an estimate of the area, but he never managed to solve the problem mathematically.

Within a few years, it seemed like every mathematician in Europe was obsessed with the cycloid. Pierre de Fermat, René Descartes, Marin Mersenne, Isaac Newton and Gottfried Wilhelm Leibniz all studied it. It even brought Blaise Pascal back to mathematics, after he had sworn off it in favour of theology. One night, he had a terrible toothache and, to distract himself from the pain, decided to think about cycloids. It worked – the toothache miraculously disappeared, and naturally Pascal concluded that God must approve of him doing mathematics. He never gave it up again. The statue of Pascal in the Louvre Museum in Paris even shows him with a diagram of a cycloid. The curve became so well known, in fact, that it made its way into several classic works of literature – it gets name-checked in Gulliver’s Travels, Tristram Shandy and Moby-Dick.

The question of the cycloid’s area was first solved in the mid-17th century by Gilles de Roberval, and the answer turned out to be delightfully simple – exactly three times the area of the rolling circle. The first person to determine the length of the cycloid was Christopher Wren, who was an extremely good mathematician, though I hear he also dabbled in architecture. It’s another beautifully simple formula: the length is exactly four times the diameter of the generating circle. The beguiling cycloid was so appealing to mathematicians that it was nicknamed “the Helen of Geometry”.

But its beauty wasn’t the only reason for the name. It was responsible for many bitter arguments. When mathematician Evangelista Torricelli independently found the area under the cycloid, Roberval accused him of stealing his work. “Team Roberval” even claimed that Torricelli had died of shame after being unmasked as a plagiarist (though the typhoid he had at the time may have been a contributing factor). Descartes dismissed Fermat’s work on the cycloid as “ridiculous gibberish”. And in response to a challenge from Johann Bernoulli, Isaac Newton grumpily complained about being “teased by foreigners about mathematics”.

An amazing property of the cycloid was discovered by Christiaan Huygens, who designed the first pendulum clock. Pendulums are good for timekeeping because the period of their motion – the time taken for one full swing of the pendulum – is constant, no matter what the angle of release. But in fact, that’s only approximately true – the period does vary slightly. Huygens wondered if he could do better. The end of a pendulum string moves along the arc of a circle, but is there a curved path it could follow so that the bob would reach the bottom of the curve in the same time no matter where it was released? This became known as the “tautochrone problem”. And guess which curve is the solution? An added bonus is its link to the “brachistochrone problem” of finding the curve between any two points along which a particle moving under gravity will descend in the shortest time. There’s no reason at all to think that the same curve could answer both problems, but it does. The solution is the cycloid. It’s a delightful surprise to find it cropping up in situations seemingly so unrelated to where we first encountered it.

When you roll a circle along a line, you get a cycloid. But what happens when you roll a line along a circle? This is an instance of a curve called an involute. To make one, you take a point at the end of a line segment and roll that line along the curve so it’s always just touching it (in other words, it’s a tangent). The involute is the curve traced out by that point. For the involute of a circle, imagine unspooling a thread from a cotton reel and following the end of the thread as it moves. The result is a spiralling curve emerging from the circle’s circumference.

When a line rolls along a circle, it produces a curve called an involute

Huygens was the first person to ask about involutes, as part of his attempts to make more accurate clocks. It’s all very well knowing the cycloid is the perfect tautochrone, but how do you get your string to follow a cycloidal path? You need to find a curve whose involute is a cycloid. The miraculous cycloid, it turns out, has the beautiful property that it is its own involute! But those lovely spiralling circle involutes turn out to be extremely useful too.

A circle with many involutes

My favourite application is one Huygens definitely couldn’t have predicted: in the design of a nuclear reactor that produces high-mass elements for scientific research. This is done by smashing neutrons at high speed into lighter elements, to create heavier ones. Within the cylindrical reactor cores, the uranium oxide fuel is sandwiched in thin layers between strips of aluminium, which are then curved to fit into the cylindrical shape. The heat produced by a quadrillion neutrons hurtling around every square centimetre is considerable, so coolant runs between these strips. It’s vital that they must be a constant distance apart all the way along their curved surfaces, to prevent hotspots. That’s where a useful property of circle involutes comes in. If you draw a set of circle involutes starting at equally spaced points on the circumference of a circle, then the distances between them remain constant along the whole of each curve. So, they are the perfect choice for the fuel strips in the reactor core. What’s more, the circle involute is the only curve for which this is true! I just love that a curve first studied in the context of pendulum clocks turns out to solve a key design question for nuclear reactors.

We’ve rolled circles along lines and lines along circles. Clearly the next step is to roll circles along circles. What happens? Here, we have some choices. What size is the rolling circle? And are we rolling along the inside or the outside of the stationary one? The curve made by a circle rolling along inside of the circle is called a hypocycloid; rolling it along the outside gives you an epicycloid. If you’ve ever played with a Spirograph toy, you’ll almost have drawn hypocycloids. Because your pen is not quite at the rim of the rolling circle, technically you are creating what are called hypotrochoids.

A cardioid (left) and nephroid (right)

Of the epicycloids, the most interesting is the cardioid: the heart-shaped curve resulting when the rolling circle has the same radius as the fixed one. Meanwhile, the kidney-shaped nephroid is produced by a rolling circle half the radius of the fixed one. Cardioids crop up in the most fascinating places. The central region of the Mandelbrot set, a famous fractal, is a cardioid. Sound engineers will be familiar with cardioid microphones, which pick up sound in a cardioid-shaped region. You might also find cardioid-like curves in the light patterns created in coffee cups in some kinds of lighting. If light rays from a fixed source are reflected off a curved mirror, the curve to which each of those reflected rays are tangent will be visible as a concentrated region of light, called a caustic. It turns out that a light source on the circumference of a perfectly circular mirror will result precisely in a cardioid!

Of course, in our coffee cup example, usually the light source isn’t exactly on the rim of the cup, but some way away. If it were very far away, we could assume that the light rays hitting the rim of the cup are parallel. In that situation, it can be shown that the caustic is actually not a cardioid but another epicycloid: the nephroid. Since a strong overhead light is somewhere between these two extremes, the curve we get is usually going to be somewhere between a cardioid and a nephroid. The mathematician Alfréd Rényi once defined a mathematician as “a device for turning coffee into theorems”. That process is nowhere more clearly seen than with our wonderful epicycloids. Check them out if you’re reading this with your morning cuppa!

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*Credit for article given to Sarah Hart*

What does “equals” mean? For mathematicians, this simple question has more than one answer, which is causing issues when it comes to using computers to check proofs. The solution might be to tear up the foundations of maths.

When you see “2 + 2 = 4”, what does “=” mean? It turns out that’s a complicated question, because mathematicians can’t agree on the definition of what makes two things equal.

While this argument has been quietly simmering for decades, a recent push to make mathematical proofs checkable by computer programs, called formalisation, has given the argument new significance.

“Mathematicians use equality to mean two different things, and I was fine with that,” says Kevin Buzzard at Imperial College London. “Then I started doing maths on a computer.” Working with computer proof assistants made him realise that mathematicians must now confront what was, until recently, a useful ambiguity, he says – and it could force them to completely redefine the foundations of their subject.

The first definition of equality will be a familiar one. Most mathematicians take it to mean that each side of an equation represents the same mathematical object, which can be proven through a series of logical transformations from one side to the other. While “=”, the equals sign, only emerged in the 16th century, this concept of equality dates back to antiquity.

It was the late 19th century when things began to change, with the development of set theory, which provides the logical foundations for most modern mathematics. Set theory deals with collections, or sets, of mathematical objects, and introduced another definition of equality: if two sets contain the same elements, then they are equal, similar to the original mathematical definition. For example, the sets {1, 2, 3} and {3, 2, 1} are equal, because the order of the elements in a set doesn’t matter.

But as set theory developed, mathematicians started saying that two sets were equal if there was an obvious way to map between them, even if they didn’t contain exactly the same elements, says Buzzard.

To understand why, take the sets {1, 2, 3} and {a, b, c}. Clearly, the elements of each set are different, so the sets aren’t equal. But there are also ways of mapping between the two sets, by identifying each letter with a number. Mathematicians call this an isomorphism. In this case, there are multiple isomorphisms because you have a choice of which number to assign to each letter, but in many cases, there is only one clear choice, called the canonical isomorphism.

Because a canonical isomorphism of two sets is the only possible way to link them, many mathematicians now take this to mean they are equal, even though it isn’t technically the same concept of equality that most of us are used to. “These sets match up with each other in a completely natural way and mathematicians realised it would be really convenient if we just call those equal as well,” says Buzzard.

Having two definitions for equality is of no real concern to mathematicians when they write papers or give lectures, as the meaning is always clear from the context, but they present problems for computer programs that need strict, precise instructions, says Chris Birkbeck at the University of East Anglia, UK. “We’re finding that we were a little bit sloppy all along, and that maybe we should fix a few things.”

To address this, Buzzard has been investigating the way some mathematicians widely use canonical isomorphism as equality, and the problems this can cause with formal computer proof systems.

In particular, the work of Alexander Grothendieck, one of the leading mathematicians of the 20th century, is currently extremely difficult to formalise. “None of the systems that exist so far capture the way that mathematicians such as Grothendieck use the equal symbol,” says Buzzard.

The problem has its roots in the way mathematicians put together proofs. To begin proving anything, you must first make assumptions called axioms that are taken to be true without proof, providing a logical framework to build upon. Since the early 20^th century, mathematicians have settled on a collection of axioms within set theory that provide a firm foundation. This means they don’t generally have to use axioms directly in their day-to-day business, because common tools can be assumed to work correctly – in the same way you probably don’t worry about the inner workings of your kitchen before cooking a recipe.

“As a mathematician, you somehow know well enough what you’re doing that you don’t worry too much about it,” says Birkbeck. That falls down, however, when computers get involved, carrying out maths in a way that is similar to building a kitchen from scratch for every meal. “Once you have a computer checking everything you say, you can’t really be vague at all, you really have to be very precise,” says Birkbeck.

To solve the problem, some mathematicians argue we should just redefine the foundations of mathematics to make canonical isomorphisms and equality one and the same. Then, we can make computer programs work around that. “Isomorphism is equality,” says Thorsten Altenkirch at the University of Nottingham, UK. “I mean, what else? If you cannot distinguish two isomorphic objects, what else would it be? What else would you call this relationship?”

Efforts are already under way to do this in a mathematical field called homotopy type theory, in which traditional equality and canonical isomorphism are defined identically. Rather than trying to contort existing proof assistants to fit canonical isomorphism, says Altenkirch, mathematicians should adopt type theory and use alternative proof assistants that work with it directly.

Buzzard isn’t a fan of this suggestion, having already spent considerable effort using current tools to formalise mathematical proofs that are needed to check more advanced work, such as a proof of Fermat’s last theorem. The axioms of mathematics should be left as they are, rather than adopting type theory, and existing systems should be tweaked instead, he says. “Probably the way to fix it is just to leave mathematicians as they are,” says Buzzard. “It’s very difficult to change mathematicians. You have to make the computer systems better.”

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*Credit for article given to Alex Wilkins*

Third time’s a charm: just weeks after cracking an elusive problem involving the number 42, mathematicians have found a solution to an even harder problem for the number 3.

Andrew Booker at Bristol University, UK, and Andrew Sutherland at the Massachusetts Institute of Technology have found a big solution to a maths problem known as the sum of three cubes.

The problem asks whether any integer, or whole number, can be represented as the sum of three cubed numbers.

There were already two known solutions for the number 3, both of which involve small numbers: 1³ + 1³ + 1³ and 4³ + 4³ + (-5)³.

But mathematicians have been searching for a third for decades. The solution that Booker and Sutherland found is:

569936821221962380720³ + (-569936821113563493509) ³ + (-472715493453327032) ³ = 3

Earlier this month, the pair also found a solution to the same problem for 42, which was the last remaining unsolved number less than 100.

To find these solutions, Booker and Sutherland worked with software firm Charity Engine to run an algorithm across the idle computers of half a million volunteers.

For the number 3, the amount of processing time was equivalent to a single computer processor running continuously for 4 million hours, or more than 456 years.

When a number can be expressed as the sum of three cubes, there are infinitely many possible solutions, says Booker. “So there should be infinitely many solutions for three, and we’ve just found the third one,” he says.

There’s a reason the third solution for 3 was so hard to find. “If you look at just the solutions for any one number, they look random,” he says. “We think that if you could get your hands on loads and loads of solutions – of course, that’s not possible, just because the numbers get so huge so quickly – but if you could, there’s kind of a general trend to them: that the digit sizes are growing roughly linearly with the number of solutions you find.”

It turns out that this rate of growth is extremely small for the number 3 – only 114, now the smallest unsolved number, has a smaller rate of growth. In other words, numbers with a slow rate of growth have fewer solutions with a lower number of digits.

The duo also found a solution to the problem for 906. We know for sure that certain numbers, such as 4, 5 and 13, can’t be expressed as the sum of three cubes. There now remain nine unsolved numbers under 1000. Mathematicians think these can be written as the sum of three cubes, but we don’t yet know how.

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*Credit for article given to Donna Lu*

I work on the mathematics of sharing resources, which has led me to consider emotions such as envy, behaviour such as risk-taking and the best way to cut a cake.

Like, I suspect, many women, my wife enjoys eating dessert but not ordering it. I therefore dutifully order what I think she’ll like, cut it in half and invite her to choose a piece.

This is a sure-fire recipe for marital accord. Indeed, many mathematicians, economists, political scientists and others have studied this protocol and would agree. The protocol is known as the “cut-and-choose” procedure. I cut. You choose.

Cut-and-choose

Cut-and-choose is not limited to the dining table – it dates back to antiquity. It appears nearly 3,000 years ago in Hesiod’s poem Theogeny where Prometheus divides a cow and Zeus selects the part he prefers.

In more recent times, cut-and-choose has been enshrined in the UN’s 1982 Convention of the Law of the Sea where it was proposed as a mechanism to resolve disputes when dividing the seabed for mining.

To study the division of cake, cows and the seabed in a more formal way, various mathematical models have been developed. As with all models, these need to make a number of simplifying assumptions.

One typical assumption is that the people employing the cut-and-choose method are risk-averse. They won’t adopt a risky strategy that may give them less cake than a more conservative strategy.

With such assumptions in place, we can then prove what properties cake cutting procedures have and don’t have. For instance, cut-and-choose is envy free.

You won’t envy the cake I have, otherwise you would have taken this piece. And I won’t envy the piece you have, as the only risk-averse strategy is for me to cut the cake into two parts that I value equally.

On the other hand, the cutting of the cake is not totally equitable since the player who chooses can get cake that has more than half the total value for them.

With two players, it’s hard to do better than cut-and-choose. But I should record that my wife argues with me about this.

She believes it favours the second player since the first player inevitably can’t divide the cake perfectly and the second player can capitalise on this. This is the sort of assumption ignored in our mathematical models.

My wife might prefer the moving-knife procedure which doesn’t favour either player. A knife is moved over the cake, and either player calls “cut” when they are happy with the slice.

Again, this will divide the cake in such a way that neither player will envy the other (else they would have called “cut” themselves).

Three’s a crowd

Unfortunately, moving beyond two players increases the complexity of cutting cake significantly.

With two players, we needed just one cut to get to an envy free state. With three players, a complex series of five cuts of the cake might be needed. Of course, only two cuts are needed to get three slices.

The other three cuts are needed to remove any envy. And with four players, the problem explodes in our face.

An infinite number of cuts may be required to get to a situation where no one envies another’s cake. I’m sure there’s some moral here about too many cake cutters spoiling the dessert.

There are many interesting extensions of the problem. One such extension is to indivisible goods.

Suppose you have a bag of toys to divide between two children. How do you divide them fairly? As a twin myself, I know that the best solution is to ensure you buy two of everything.

It’s much more difficult when your great aunt gives you one Zhu Zhu pet, one Bratz doll and three Silly Bandz bracelets to share.

Online

More recently, I have been studying a version of the problem applicable to online settings. In such problems, not all players may be available all of the time. Consider, for instance, allocating time on a large telescope.

Astronomers will have different preferences for when to use the telescope depending on what objects are visible, the position of the sun, etcetera. How do we design a web-based reservation system so that astronomers can choose observation times that is fair to all?

We don’t want to insist all astronomers log in at the same time to decide an allocation. And we might have to start allocating time on the telescope now, before everyone has expressed their preferences. We can view this as a cake-cutting problem where the cake is made up of the time slots for observations.

The online nature of such cake-cutting problems poses some interesting new challenges.

How can we ensure that late-arriving players don’t envy cake already given to earlier players? The bad news is that we cannot now achieve even a simple property like envy freeness.

No procedure can guarantee situations where players don’t envy one another. But more relaxed properties are possible, such as not envying cake allocated whilst you are participating in the cutting of the cake.

Ham sandwich

There’s a brilliantly named piece of mathematics due to Arthur H. Stone and [John Tukey](http://www.morris.umn.edu/~sungurea/introstat/history/w98/Tukey.html, the Ham Sandwich Theorem which proves we can always cut a three-layered cake perfectly with a single cut.

Suppose we have three objects. Let’s call them “the top slice of bread”, “the ham filling” and “the bottom slice of bread”. Or if you prefer “the top layer” of the cake, “the middle layer” and “the bottom layer”.

The ham sandwich theorem proves a single slice can always perfectly bisect the three objects. Actually, the ham sandwich theorem works in any number of dimensions: any n objects in n-dimensional space can be simultaneously bisected by a single (n − 1) dimensional hyperplane.

So, in the case of the three-layered cake, n = 3, and the three-layered cake can be bisected (or cut) using a single, two-dimensional “hyperplane”. Such as, say, a knife.

Who would have thought that cutting cake would lead to higher dimensions of mathematics by way of a ham sandwich?

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*Credit for article given to Toby Walsh*

Maths tells us the best way to cover a surface with copies of a shape – even when it comes to jigsaw puzzles, says Katie Steckles.

WHAT do a bathroom wall, a honeycomb and a jigsaw puzzle have in common? Obviously, the answer is mathematics.

If you are trying to cover a surface with copies of a shape – say, for example, you are tiling a bathroom – you ideally want a shape like a square or rectangle. They will cover the whole surface with no gaps, which is why these boring shapes get used as wall tiles so often.

But if your shapes don’t fit together exactly, you can still try to get the best coverage possible by arranging them in an efficient way.

Imagine trying to cover a surface with circular coins. The roundness of the circles means there will be gaps between them. For example, we could use a square grid, placing the coins on the intersections. This will cover about 78.5 per cent of the area.

But this isn’t the most efficient way: in 1773, mathematician Joseph-Louis Lagrange showed that the optimal arrangement of circles involves a hexagonal grid, like the cells in a regular honeycomb – neat rows where each circle sits nestled between the two below it.

In this situation, the circles will cover around 90.7 per cent of the space, which is the best you can achieve with this shape. If you ever need to cover a surface with same-size circles, or pack identical round things into a tray, the hexagon arrangement is the way to go.

But this isn’t just useful knowledge if you are a bee: a recent research paper used this hexagonal arrangement to figure out the optimal size table for working a jigsaw puzzle. The researchers calculated how much space would be needed to lay out the pieces of an unsolved jigsaw puzzle, relative to the solved version. Puzzle pieces aren’t circular, but they can be in any orientation and the tabs sticking out stop them from moving closer together, so each takes up a theoretically circular space on the table.

By comparing the size of the central rectangular section of the jigsaw piece to the area it would take up in the hexagonal arrangement, the paper concluded that an unsolved puzzle takes up around 1.73 times as much space.

This is the square root of three (√3), a number with close connections to the regular hexagon – one with a side length of 1 will have a height of √3. Consequently, there is also a √3 in the formula for the hexagon’s area, which is 3/2 × √3 × s², where s is the length of a side. This is partly why it pops out, after some fortuitous cancellation, as the answer here.

So if you know the dimensions of a completed jigsaw puzzle, you can figure out what size table you need to lay out all the pieces: multiply the width and height, then multiply that by 1.73. For this ingenious insight, we can thank the bees.

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*Credit for article given to Katie Steckles*

Knot theory is linked to many other branches of science, including those that tell us about the cosmos.

The mathematical study of knots started with a mistake. In the 1800s, mathematician and physicist William Thomson, also known as Lord Kelvin, suggested that the elemental building blocks of matter were knotted vortices in the ether: invisible microscopic currents in the background material of the universe. His theory dropped by the wayside fairly quickly, but this first attempt to classify how curves could be knotted grew into the modern mathematical field of knot theory. Today, knot theory is not only connected to many branches of theoretical mathematics but also to other parts of science, like physics and molecular biology. It’s not obvious what your shoelace has to do with the shape of the universe, but the two may be more closely related than you think.

As it turns out, a tangled necklace offers a better model of a knot than a shoelace: to a mathematician, a knot is a loop in three-dimensional space rather than a string with loose ends. Just as a physical loop of string can stretch and twist and rotate, so can a mathematical knot – these loops are floppy rather than fixed. If we studied strings with free ends, they could wiggle around and untie themselves, but a loop stays knotted unless it’s cut.

Most questions in knot theory come in two varieties: sorting knots into classes and using knots to study other mathematical objects. I’ll try to give a flavour of both, starting with the simplest possible example: the unknot.

Draw a circle on a piece of paper. Congratulations, you’ve just constructed an unknot! This is the name for any loop in three-dimensional space that is the boundary of a disc. When you draw a circle on a piece of paper, you can see this disc as the space inside the circle, and your curve continues to be an unknot if you crumple the paper up, toss it through the air, flatten it out and then do some origami. As long as the disc is intact, no matter how distorted, the boundary is always an unknot.

Things get more interesting when you start with just the curve. How can you tell if it’s an unknot? There may secretly be a disc that can fill in the loop, but with no limits on how deformed the disc could be, it’s not clear how you can figure this out.

Two unknots

It turns out that this question is both hard and important: the first step in studying complicated objects is distinguishing them from simple ones. It’s also a question that gets answered inside certain bacterial cells each time they replicate. In the nuclei of these cells, the DNA forms a loop, rather than a strand with loose ends, and sometimes these loops end up knotted. However, the DNA can replicate only when the loop is an unknot, so the basic life processes of the cell require a process for turning a potentially complicated loop into an unknotted one.

A class of proteins called topoisomerases unknot tangled loops of DNA by cutting a strand, moving the free ends and then reattaching them. In a mathematical context, this operation is called a “crossing change”, and it’s known that any loop can be turned into the unknot by some number of crossing changes. However, there’s a puzzle in this process, since random crossing changes are unlikely to simplify a knot. Each topoisomerase operates locally, but collectively they’re able to reliably unknot the DNA for replication. Topoisomerases were discovered more than 50 years ago, but biologists are still studying how they unknot DNA so effectively.

When mathematicians want to identify a knot, they don’t turn to a protein to unknot it for them.  Instead, they rely on invariants, mathematical objects associated with knots. Some invariants are familiar things like numbers, while others are elaborate algebraic structures. The best invariants have two properties: they’re practical to compute, given the input of a specific knot, and they distinguish many different classes of knots from each other. It’s easy to define an invariant with only one of these properties, but a computable and effective knot invariant is a rare find.

The modern era of knot theory began with the introduction of an invariant called the Jones Polynomial in the 1980s. Vaughan Jones was studying statistical mechanics when he discovered a process that assigns a polynomial – a type of simple algebraic expression – to any knot. The method he used was technical, but the essential feature is that no amount of wiggling, stretching or twisting changes the output. The Jones Polynomial of an unknot is 1, no matter how complicated the associated disc might be.

Jones’s discovery caught the attention of other researchers, who found simpler techniques for computing the same polynomial. The result was an invariant that satisfies both the conditions listed above: the Jones Polynomial can be computed from a drawing of a knot on paper, and many thousands of knots can be distinguished by the fact that they have different Jones Polynomials.

However, there are still many things we don’t know about the Jones Polynomial, and one of the most tantalising questions is which knots it can detect. Most invariants distinguish some knots while lumping others together, and we say an invariant detects a knot if all the examples sharing a certain value are actually deformations of each other. There are certainly pairs of distinct knots with the same Jones Polynomial, but after decades of study, we still don’t know whether any knot besides the unknot has the polynomial 1. With computer assistance, experts have examined nearly 60 trillion examples of distinct knots without finding any new knots whose Jones Polynomials equal 1.

The Jones Polynomial has applications beyond knot detection. To see this, let’s return to the definition of an unknot as a loop that bounds a disc. In fact, every knot is the boundary of some surface – what distinguishes an unknot is that this surface is particularly simple. There’s a precise way to rank the complexity of surfaces, and we can use this to rank the complexity of knots. In this classification, the simplest knot is the unknot, and the second simplest is the trefoil, which is shown below.

Trefoil knot

To build a surface with a trefoil boundary, start with a strip of paper. Twist it three times and then glue the ends together. This is more complicated than a disc, but still pretty simple. It also gives us a new question to investigate: given an arbitrary knot, where does it fit in the ranking of knot complexity? What’s the simplest surface it can bound? Starting with a curve and then hunting for a surface may seem backwards, but in some settings, the Jones Polynomial answers this question: the coefficients of the knot polynomial can be used to estimate the complexity of the surfaces it bounds.

Joan Licata

Knots also help us classify other mathematical objects. We can visually distinguish the two-dimensional surface of sphere from the surface a torus (the shape of a ring donut), but an ant walking on one of these surfaces might need knot theory to tell them apart. On the surface of a torus, there are loops that can’t be pulled any tighter, while any loop lying on a sphere can contract to a point.

We live inside a universe of three physical dimensions, so like the ant on a surface, we lack a bird’s eye view that could help us identify its global shape. However, we can ask the analogous question: can each loop we encounter shrink without breaking, or is there a shortest representative? Mathematicians can classify three-dimensional spaces by the existence of the shortest knots they contain. Presently, we don’t know if some knots twisting through the universe are unfathomably long or if every knot can be made as small as one of Lord Kelvin’s knotted vortices.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Joan Licata*

Whether you are sharing a cake or a coastline, maths can help make sure everyone is happy with their cut, says Katie Steckles.

One big challenge in life is dividing things fairly. From sharing a tasty snack to allocating resources between nations, having a strategy to divvy things up equitably will make everyone a little happier.

But it gets complicated when the thing you are dividing isn’t an indistinguishable substance: maybe the cake you are sharing has a cherry on top, and the piece with the cherry (or the area of coastline with good fish stocks) is more desirable. Luckily, maths – specifically game theory, which deals with strategy and decision-making when people interact – has some ideas.

When splitting between two parties, you might know a simple rule, proven to be mathematically optimal: I cut, you choose. One person divides the cake (or whatever it is) and the other gets to pick which piece they prefer.

Since the person cutting the cake doesn’t choose which piece they get, they are incentivised to cut the cake fairly. Then when the other person chooses, everyone is satisfied – the cutter would be equally happy with either piece, and the chooser gets their favourite of the two options.

This results in what is called an envy-free allocation – neither participant can claim they would rather have the other person’s share. This also takes care of the problem of non-homogeneous objects: if some parts of the cake are more desirable, the cutter can position their cut so the two pieces are equal in value to them.

What if there are more people? It is more complicated, but still possible, to produce an envy-free allocation with several so-called fair-sharing algorithms.

Let’s say Ali, Blake and Chris are sharing a cake three ways. Ali cuts the cake into three pieces, equal in value to her. Then Blake judges if there are at least two pieces he would be happy with. If Blake says yes, Chris chooses a piece (happily, since he gets free choice); Blake chooses next, pleased to get one of the two pieces he liked, followed by Ali, who would be satisfied with any of the pieces. If Blake doesn’t think Ali’s split was equitable, Chris looks to see if there are two pieces he would take. If yes, Blake picks first, then Chris, then Ali.

If both Blake and Chris reject Ali’s initial chop, then there must be at least one piece they both thought was no good. This piece goes to Ali – who is still happy, because she thought the pieces were all fine – and the remaining two pieces get smooshed back together (that is a mathematical term) to create one piece of cake for Blake and Chris to perform “I cut, you choose” on.

While this seems long-winded, it ensures mathematically optimal sharing – and while it does get even more complicated, it can be extended to larger groups. So whether you are sharing a treat or a divorce settlement, maths can help prevent arguments.

For more such insights, log into www.international-maths-challenge.com.

*Credit for article given to Katie Steckles*