Author: IMC

Mirrors of a magical scientist: Andromeda photographed through a Newtonian telescope.

The notebooks of Sir Isaac Newton, who was famously reported to have suffered a (scientifically) earth-shaking blow to the head from an apple, are being scanned and published online by the University of Cambridge.

Newton, a Biblical numerologist when he wasn’t developing calculus or building the first reflecting telescope, founded classical mechanics with Philosophiæ Naturalis Principia Mathematica (Mathematical Principles of Natural Philosophy), which was first published in 1687. In the book that made his name, Newton set out his three laws of motion, and his theory of universal gravitation (prompted by pondering what force plummeted the fruit straight down onto his head, or so goes the apocryphal tale).

Newton studied and later held the Lucasian Chair of Mathematics at Cambridge, which was given numerous manuscripts of his in 1872 and has since bought more. The online publication has started with Newton’s mathematical works of the 1660s and more papers will become available over coming months.

Striking a light for science.

A philosopher of science at Flinders University, George Couvalis, said that Newton’s gravitational experiments – which largely corrected ancient observations of gravity – were sparked by his interest in magic and magnetism. “The idea that things might naturally attract one another is an idea that he got from magical ideas. He adapted it across to mathematical theory because it was a mystical theory,” Dr Couvalis said.

It was important to remember that scientists of Newton’s era did not have what we would consider a modern sceptical outlook and – with the exception of the “exceptional” Galileo Galilei – instead held a fusion of views that we would consider deeply irrational, Dr Couvalis said.

“It was certainly far more common in the 17th and 18th centuries for scientists to be interested in magical beliefs and alchemical beliefs and religious beliefs. Johannes Kepler, for example, had all kinds of strange views about the music of the spheres, Copernicus had strange views about the sacredness of the sun, and Newton famously had views about the mysterious numerical meanings of Biblical passages and about alchemical material, ” Dr Couvalis said.

Scientists of the period saw their work touching on many illogical and occult fields of interest, including Robert Boyle, a founder of modern chemistry, who had “an interest in doing experimental research on magical mirrors, which to us would sound bizarre but at the time it was thought to be a possibility,” said Dr Couvalis, who added that Boyle pulled back from some experiments for religious reasons. “He thought it might get him in touch with demons.”

Demonology may have fallen out of favour amongst scientists, but “the view that we’re getting everything right would be a serious mistake,” Dr Couvalis said. “To some degree science is always in the sway of the time it’s in; this is now the standard view of philosophers and historians.”

“Newton’s mechanics is in certain respects pretty much right, but in other respects it was shown by Einstein and others to be wildly wrong. By about 1900 we had people saying to their graduate students ‘You should give up physics because it’s all been done,’ but Einstein managed to show that it was wildly wrong in certain respects,” Dr Couvalis said.

The ideal of the scientific method is never met, and our beliefs and discoveries will likely on day be seen as flawed but perhaps useful stepping stones in the continuum of science, Dr Couvalis said. “People make mistakes, people have a lot of trouble leaving assumptions behind, and our tests are never rigorous enough to be absolutely certain that we’re getting things right. Future experimental studies and the sheer empirical facts will show us to be wrong in many ways that we can’t anticipate.”

“We work with what we have because we just don’t know anything better at the moment. It might turn out that Einstein’s special and general theories of relativity are wrong in some deep-seated way. It might turn out that some of our theories of the universe are wrong. It’s starting to look in biology as if neo-Darwinism isn’t completely right, so where will that go – I don’t know. Research will determine the direction. That doesn’t mean that we’re going to go back to being creationists – that view has been thoroughly debunked. Imre Lakatos wrote in the 1970s there are no good scientific theories, there’s only the best rotten theory we have.”

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*Credit For article given to Matthew Thompson*

Deciding whether a statement is true is a computational head-scratcher.

In the 1930s, Alan Turing showed there are basic tasks that are impossible to achieve by algorithmic means. In modern lingo, what he showed was that there can be no general computer program that answers yes or no to the question of whether another computer program will eventually stop when it is run.

The amazing unsolvability of this Halting Problem contains a further perplexing subtlety. While we have no way of finding in advance if a program will halt, there is an obvious way, in principle, to demonstrate that it halts if it is a halting program: run it, wait, and witness it halting!

In other words, Turing showed that, at the broadest level, deciding whether a statement is true is computationally harder than demonstrating that it’s true when it is.

A question of efficiency

Turing’s work was a pivotal moment in the history of computing. Some 80 years later, computing devices have pervaded almost every facet of society. Turing’s original “what is computable?” question has been mostly replaced by the more pertinent, “what is efficiently computable?”

But while Turing’s Halting Problem can be proved impossible in a few magical lines, the boundary between “efficient” and “inefficient” seems far more elusive. P versus NP is the most famous of a huge swathe of unresolved questions to have emerged from this modern take on Turing’s question.

So what is this NP thing?

Roughly speaking, P (standing for “polynomial time”), corresponds to the collection of computational problems that have an efficient solution. It’s only an abstract formulation of “efficient”, but it works fairly well in practice.

The class NP corresponds to the problems for which, when the answer is “yes”, there is an efficient demonstration that the answer is yes (the “N” stands for “nondeterministic”, but the description taken here is more intuitive). P versus NP simply asks if these two classes of computational problems are the same.

It’s just the “deciding versus demonstrating” issue in Turing’s original Halting Problem, but with the added condition of efficiency.

A puzzler

P certainly doesn’t look to be the same as NP. Puzzles are good examples of the general intuition here. Crossword puzzles are popular because it’s a challenge to find the solution, and humans like challenge. But no-one spends their lunchtime checking already completed crosswords: checking someone else’s solution offers nowhere near the same challenge.

Even clearer is Sudoku: again it is a genuine challenge to solve, but checking an existing solution for correctness is so routine it is devoid of entertainment value.

The P=NP possibility is like discovering that the “finding” part of these puzzles is only of the same difficulty to the “checking” part. That seems hard to believe, but the truth is we do not know for sure.

This same intuition pervades an enormous array of important computational tasks for which we don’t currently have efficient algorithms. One particularly tantalising feature is that, more often than not, these problems can be shown to be maximally hard among NP problems.

These so-called “NP-complete” problems are test cases for P versus NP: if any one of them has an efficient algorithmic solution then they all do (and efficient checking is no harder than efficient finding).

But if even just one single one can be shown to have no efficient solution, then P does not equal NP (and efficient finding really is, in general, harder than efficient checking).

Here are some classic examples of NP-complete problems.

• Partition (the dilemma of the alien pick-pockets). On an alien planet, two pick-pockets steal a wallet. To share the proceeds, they must evenly divide the money: can they do it? Standard Earth currencies evolved to have coin values designed to make this task easy, but in general this task is NP-complete. It’s in NP because, if there is an equal division of the coins, this can be easily demonstrated by simply showing the division. (Finding it is the hard part!)
• Timetabling. Finding if a clash-free timetable exists is NP-complete. The problem is in NP because we can efficiently check a correct, clash-free timetable to be clash-free.
• Travelling Salesman. A travelling salesman must visit each of some number of cities. To save costs, the salesman wants to find the shortest route that passes through all of the cities. For some given target distance “n”, is there a route of length at most “n”?
• Short proofs. Is there a short proof for your favourite mathematical statement (a Millennium Prize problem perhaps)? With a suitable formulation of “short”, this is NP-complete. It is in NP because checking formal proofs can be done efficiently: the hard part is finding them (at least, we think that’s the hard part!).

In every case, we know of no efficient exact algorithm, and the nonexistence of such an algorithm is equivalent to proving P not equal to NP.

So are we close to a solution? It seems the best we know is that we don’t know much! Arguably, the most substantial advances in the P versus NP saga are curiously negative: they mostly show we cannot possibly hope to resolve P as different to NP by familiar techniques.

We know Turing’s approach cannot work. In 2007, Alexander Razborov and Steven Rudich were awarded the Gödel Prize (often touted as the Nobel Prize of Computer Science) for their work showing that no “natural proof” can prove P unequal to NP.

Of course, we’ll keep looking!

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

*Credit for article given to Marcel Jackson*

How fluids move has fascinated researchers since the birth of science.

Among the seven problems in mathematics put forward by the Clay Mathematics Institute in 2000 is one that relates in a fundamental way to our understanding of the physical world we live in.

It’s the Navier-Stokes existence and uniqueness problem, based on equations written down in the 19th century.

The solution of this prize problem would have a profound impact on our understanding of the behaviour of fluids which, of course, are ubiquitous in nature. Air and water are the most recognisable fluids; how they move and behave has fascinated scientists and mathematicians since the birth of science.

But what are the so-called Navier-Stokes equations? What do they describe?

The equations

In order to understand the Navier-Stokes equations and their derivation we need considerable mathematical training and also a sound understanding of basic physics.

Without that, we must draw upon some very simple basics and talk in terms of broad generalities – but that should be sufficient to give the reader a sense of how we arrive at these fundamental equations, and the importance of the questions.

From this point, I’ll refer to the Navier-Stokes equations as “the equations”.

The equations governing the motion of a fluid are most simply described as a statement of Newton’s Second Law of Motion as it applies to the movement of a mass of fluid (whether that be air, water or a more exotic fluid). Newton’s second law states that:

Mass x Acceleration = Force acting on a body

For a fluid the “mass” is the mass of the fluid body; the “acceleration” is the acceleration of a particular fluid particle; the “forces acting on the body” are the total forces acting on our fluid.

Without going into full details, it’s possible to state here that Newton’s Second Law produces a system of differential equations relating rates of change of fluid velocity to the forces acting on the fluid. We require one other physical constraint to be applied on our fluid, which can be most simply stated as:

Mass is conserved! – i.e. fluid neither appears nor disappears from our system.

The solution

Having a sense of what the Navier-Stokes equations are allows us to discuss why the Millennium Prize solution is so important. The prize problem can be broken into two parts. The first focuses on the existence of solutions to the equations. The second focuses on whether these solutions are bounded (remain finite).

It’s not possible to give a precise mathematical description of these two components so I’ll try to place the two parts of the problem in a physical context.

1) For a mathematical model, however complicated, to represent the physical world we are trying to understand, the model must first have solutions.

At first glance, this seems a slightly strange statement – why study equations if we are not sure they have solutions? In practice we know many solutions that provide excellent agreement with many physically relevant and important fluid flows.

But these solutions are approximations to the solutions of the full Navier-Stokes equations (the approximation comes about because there is, usually, no simple mathematical formulae available – we must resort to solving the equations on a computer using numerical approximations).

Although we are very confident that our (approximate) solutions are correct, a formal mathematical proof of the existence of solutions is lacking. That provides the first part of the Millennium Prize challenge.

2) The second part asks whether the solutions of the Navier-Stokes equations can become singular (or grow without limit).

Again, a lot of mathematics is required to explain this. But we can examine why this is an important question.

There is an old saying that “nature abhors a vacuum”. This has a modern parallel in the assertion by physicist Stephen Hawking, while referring to black holes, that “nature abhors a naked singularity”. Singularity, in this case, refers to the point at which the gravitational forces – pulling objects towards a black hole – appear (according to our current theories) to become infinite.

In the context of the Navier-Stokes equations, and our belief that they describe the movement of fluids under a wide range of conditions, a singularity would indicate we might have missed some important, as yet unknown, physics. Why? Because mathematics doesn’t deal in infinites.

The history of fluid mechanics is peppered with solutions of simplified versions of the Navier-Stokes equations that yield singular solutions. In such cases, the singular solutions have often hinted at some new physics previously not considered in the simplified models.

Identifying this new physics has allowed researchers to further refine their mathematical models and so improve the agreement between model and reality.

If, as many believe, the Navier-Stokes equations do posses singular solutions then perhaps the next Millennium Prize will go to the person that discovers just what new physics is required to remove the singularity.

Then nature can, as all fluid mechanists already do, come to delight in the equations handed down to us by Claude-Louis Navier and George Gabriel Stokes.

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*Credit for article given to Jim Denier*

Building polyhedral models is a nice way to explore a lot of significant mathematics. The above models were made by printing patterns onto card-stock, cutting them out, and gluing them together. For these models, only triangular faces were used. These can give you a wide variety of cumulated (or augmented) polyhedra. The triangular faces are circumscribed to provide tabs that you glue together. You can fold and glue the tabs so that they are inside the models, but it is easier to leave them out, and they look nice this way (I think).
1. Decide on a model that you would like to make, and figure out how many faces you will need.
2. Copy and paste the images below into a document or presentation slide (PowerPoint works well) for printing. Choose the right ones for your model, and fit as many as you can on a single sheet.
3. Print out onto card-stock. Most desk ink-jet printers can take card-stock instead of printer-paper.
4. Cut out the units, fold the tabs, and assemble and glue.
Throughout this process, it helps if you have pictures of the polyhedra that you want to construct. Poly is a nice software package for browsing through families of polyhedra.
I’ve found that it works well to bend the tabs using a ruler, that glue-sticks provide the best gluing, and that it helps to hold the model together with binder-clips while assembling.

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*Credit for article given to dan.mackinnon*

The humble pigeon mightn’t look smart, but it’s no bird-brain.

We humans have long been interested in defining the abilities that set us apart from other species. Along with capabilities such as language, the ability to recognise and manipulate numbers (“numerical competence”) has long been seen as a hallmark of human cognition.

In reality, a number of animal species are numerically competent and according to new research from psychologists at the University of Otago in New Zealand, the humble pigeon could be one such species.

Damian Scarf, Harlene Hayne and Michael Colombo found that pigeons possess far greater numerical abilities than was previously thought, actually putting them on par with primates.

More on pigeons in a moment, but first: why would non-human animals even need to be numerically competent? Would they encounter numerical problems in day-to-day life?

In fact, there are many reports indicating that number is an important factor in the way many species behave.

Brown cowbirds are nest parasites – they lay their eggs in the nests of “host” species; species that are then landed with the job of raising a young cowbird.

Cowbirds are sensitive to the number of eggs in the host nest, preferring to lay in nests with three host eggs rather than one. This presumably ensures the host parent is close to the end of laying a complete clutch and will begin incubating shortly after the parasite egg has been added.

Crows identify individuals by the number of caw sounds in their vocalisations, while lionesses appear to evaluate the risk of approaching intruder lions based on how many individuals they hear roaring.

But numerical competence is about more than an ability to count. In fact, it’s three distinct abilities:

• the “cardinal” aspect: the ability to evaluate quantity (eg. counting the number of eggs already in a nest)
• the “ordinal” aspect: the ability to put an arbitrary collection of items in their correct order or rank (eg. ordering a list of animals based on the number of legs they have, or ordering the letters of the alphabet)
• the “symbolic” aspect: the ability to symbolically represent a given numerical quantity (eg. the number “3” or the word “three” are symbols that represent the quantity 3).

We know that humans are capable of all three aspects of numerical competence, but what about other animals?

For a start, we already know that the cowbird, lion and crow possess the cardinal aspect of numerical competency – they are all able to count. Pigeons possess the cardinal aspect too (as was reported as early as 1941) as do several other vertebrate and invertebrate species.

And in 1998, Elizabeth Brannon and Herbert Terrace showed that rhesus monkeys have the ability to order arrays of objects according to the number of items contained within these arrays. After learning to order sets of one, two and three items, the monkeys were able to order any three sets containing from one to nine items.

This discovery represented a clear progression in complexity, since ranking according to numerical quantity is an abstract ability – the ordinal aspect.

The new research by Scarf, Hayne and Colombo – which was published in Science in late December – has pushed, even further, our understanding of numerical abilities in the animal kingdom.

So what did they do?

Well, first they trained pigeons to peck three “stimulus arrays” – collections of objects on a touch screen. These arrays contained one, two or three objects and to receive a reward, the pigeon had to peck the arrays in order – the array with one object first, the array with two objects second, the array with three objects third.

Once this basic requirement was learned, the pigeons were presented with different object sets – one set containing arrays with one to three objects, and sets containing up to nine objects.

Having been presented with these novel object sets, the pigeons were once again required to peck the sets in ascending order. Pigeons solved the task successfully, even though they had never been trained with arrays containing more than three items.

A pigeon taking part in the University of Otago experiment.

In fact, they performed on par with rhesus monkeys, demonstrating that both pigeons and monkeys are able to identify and order the numbers from one to nine. This is significant because it shows these complex numerical abilities are not confined to the primates (and that pigeons are smarter than many people think!)

So if non-human animals possess the cardinal and ordinal aspects of numerical competency, that means it’s the symbolic representation of numbers that makes humans unique, right?

As it turns out, no.

It’s been shown that red wood ants (Formica polyctena) can not only count up to several tens (20, 30 etc.), but can also communicate this numerical information to their brethren.

It would seem, therefore, that not even the symbolic representation of numerical information is specific to humans.

Of course, we still have much more to discover and understand within this fascinating field of research. In the meantime, you might want to think twice before dismissing pigeons as “stupid birds”.

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

*Credit for article given to David Guez and Andrea S. Griffin*

Mathematicians don’t need to worry about AI taking over their jobs just yet

You don’t need a human brain to do maths — even artificial intelligence can write airtight proofs of mathematical theorems.

An AI created by a team at Google has proven more than 1200 mathematical theorems. Mathematicians already knew proofs for these particular theorems, but eventually the AI could start working on more difficult problems.

One of the core pillars of maths is the concept of proof. It is an argument based on known statements, assumptions, or rules, that a certain mathematical statement, such as a theorem, is true.

To train their AI, the Google team started with a database of more than 10,000 human-written mathematical proofs, along with the reasoning behind each step known as a tactic. Tactics could include using a known property about numbers, such as the fact that multiplying x by y is the same as multiplying y by x, or applying the chain rule.

Then, they tested the AI on 3225 theorems it hadn’t seen before and it successfully proved 1253 of them. Those that it couldn’t prove were because it had only 41 tactics at its disposal.

To prove each theorem, the AI split them into smaller and smaller components using the list of tactics. Eventually each of the smaller components could be proven with a single tactic, thus proving the larger theorem.

“Most of the proofs we used are relatively short, so they don’t require a lot of long complicated reasoning, but this is a start,” says Christian Szegedy at Google. “Where we want to get to is a system that can prove all the theorems that humans can prove, and maybe even more.”

Tackling harder problems

While this particular algorithm is focused on linear algebra and complex calculus, changing its training set could allow it to do any sort of mathematics, says Szgedy. For now, the AI’s main application is filling in the details of long and arduous proofs with extreme precision.

Mathematicians often make intellectual jumps in their proofs without spelling out the exact tactics used to get from one step to the next, and provers like this could walk through the intermediate work automatically, without requiring a human mathematician to fill in each exact tactic used.

“You get the maximum of precision and correctness all really spelled out, but you don’t have to do the work of filling in the details,” says Jeremy Avigad at Carnegie Mellon University in Pennsylvania. “Maybe offloading some things that we used to do by hand frees us up for looking for new concepts and asking new questions.”

AIs like this could one day even solve maths problems we don’t know how to solve or that are too long and complicated. But that will take a much larger training set, more tactics, and a simpler way to plug the theorems into the computer. “That’s far away, but I think it could happen in our lifetime,” says Szgedy.

“Pretty much anything that you can state and try to prove mathematically, you can put into this system,” says Avigad. “You can distill just about all of mathematics down to very basic rules and assumptions, and these systems implement those rules and assumptions.”

All of this happens in a matter of seconds per proof and the only source of error is the translation of the theorem into formal language the computer can understand. Szegedy says that the team is now working on the problem of automatic translation so that it’s easier for mathematicians to interact with the system.

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

*Credit for article given to Leah Crane*

Mathematicians are celebrating a 1000-page proof of the geometric Langlands conjecture, a problem so complicated that even other mathematicians struggle to understand it. Despite that, it is hoped the proof can provide key insights across maths and physics.

The Langlands programme aims to link different areas of mathematics

Mathematicians have proved a key building block of the Langlands programme, sometimes referred to as a “grand unified theory” of maths due to the deep links it proposes between seemingly distant disciplines within the field.

While the proof is the culmination of decades of work by dozens of mathematicians and is being hailed as a dazzling achievement, it is also so obscure and complex that it is “impossible to explain the significance of the result to non-mathematicians”, says Vladimir Drinfeld at the University of Chicago. “To tell the truth, explaining this to mathematicians is also very hard, almost impossible.”

The programme has its origins in a 1967 letter from Robert Langlands to fellow mathematician Andre Weil that proposed the radical idea that two apparently distinct areas of mathematics, number theory and harmonic analysis, were in fact deeply linked. But Langlands couldn’t actually prove this, and was unsure whether he was right. “If you are willing to read it as pure speculation I would appreciate that,” wrote Langlands. “If not — I am sure you have a waste basket handy.”

This mysterious link promised answers to problems that mathematicians were struggling with, says Edward Frenkel at the University of California, Berkeley. “Langlands had an insight that difficult questions in number theory could be formulated as more tractable questions in harmonic analysis,” he says.

In other words, translating a problem from one area of maths to another, via Langlands’s proposed connections, could provide real breakthroughs. Such translation has a long history in maths – for example, Pythagoras’s theorem relating the three sides of a triangle can be proved using geometry, by looking at shapes, or with algebra, by manipulating equations.

As such, proving Langlands’s proposed connections has become the goal for multiple generations of researchers and led to countless discoveries, including the mathematical toolkit used by Andrew Wiles to prove the infamous Fermat’s last theorem. It has also inspired mathematicians to look elsewhere for analogous links that might help. “A lot of people would love to understand the original formulation of the Langlands programme, but it’s hard and we still don’t know how to do it,” says Frenkel.

One analogy that has yielded progress is reformulating Langlands’s idea into one written in the mathematics of geometry, called the geometric Langlands conjecture. However, even this reformulation has baffled mathematicians for decades and was itself considered fiendishly difficult to prove.

Now, Sam Raskin at Yale University and his colleagues claim to have proved the conjecture in a series of five papers that total more than 1000 pages. “It’s really a tremendous amount of work,” says Frenkel.

The conjecture concerns objects that are similar to those in one half of the original Langlands programme, harmonic analysis, which describes how complex structures can be mathematically broken down into their component parts, like picking individual instruments out of an orchestra. But instead of looking at these with harmonic analysis, it uses other mathematical ideas, such as sheaves and moduli stacks, that describe concepts relating to shapes like spheres and doughnuts.

While it wasn’t in the setting that Langlands originally envisioned, it is a sign that his original hunch was correct, says Raskin. “Something I find exciting about the work is it’s a kind of validation of the Langlands programme more broadly.”

“It’s the first time we have a really complete understanding of one corner of the Langlands programme, and that’s inspiring,” says David Ben-Zvi at the University of Texas, who wasn’t involved in the work. “That kind of gives you confidence that we understand what its main issues are. There are a lot of subtleties and bells and whistles and complications that appear, and this is the first place where they’ve all been kind of systematically resolved.”

Proving this conjecture will give confidence to other mathematicians hoping to make inroads on the original Langlands programme, says Ben-Zvi, but it might also attract the attention of theoretical physicists, he says. This is because in 2007, physicists Edward Witten and Anton Kapustin found that the geometric Langlands conjecture appeared to describe an apparent symmetry between certain physical forces or theories, called S-duality.

The most basic example of this in the real world is in electricity and magnetism, which are mirror images of one another and interchangeable in many scenarios, but S-duality was also used by Witten to famously unite five competing string theory models into a single theory called M-theory.

But before anything like that, there is much more work to be done, including helping other mathematicians to actually understand the proof. “Currently, there’s a very small group of people who can really understand all the details here. But that changes the game, that changes the whole expectation and changes what you think is possible,” says Ben-Zvi.

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*Credit for article given to Alex Wilkins*

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.

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

*Credit for article given to Karmela Padavic-Callaghan*

There is a crisis in the education system, and it’s affecting the life chances of many young Australians. The number of secondary teaching graduates with adequate qualifications to teach mathematics is well below what it should be, and children’s education is suffering.

A report completed for the Australian Council of Deans of Science in 2006 documented the problem, but the situation has deteriorated since. The percentage of Year 12 students completing the more advanced mathematics courses continues to decline. This affects mathematics enrolments in the universities and a number no longer offer a major in mathematics, worsening an already inadequate supply of qualified teachers.

Changing qualifications

To exacerbate an already serious problem, the Australian Institute for Teaching and School Leadership (AITSL) currently proposes that graduate entry secondary programs must comprise at least two years of full-time equivalent professional studies in education.

There will be no DipEd pathway, which allows graduates to enter the profession within a year. Forcing them to spend more time in education will lead to increased debt. You couldn’t blame people for changing their mind about becoming a teacher.

I believe the changes in qualifications will lead to a disaster, denying even more young people access to a quality mathematics education that gives them real opportunities in the modern world.

An unequal opportunity

This is a social justice issue because access to a decent mathematics education in Australia is now largely determined by where you live and parental income.

In the past there have been concerns regarding the participation of girls in mathematics and the effect on their careers and life chances.

Australia now seems incapable of responding to a situation where only the privileged have access to well-qualified teachers of mathematics.

The Northern Territory is a prime example. The contraction of mathematics at Charles Darwin University means the NT is now totally dependent on the rest of Australia for its secondary mathematics teachers. And how can talented mathematics students in the NT be encouraged to pursue mathematical careers when it means moving away?

Elsewhere most of regional Australia is largely dependent on mathematics teachers who complete their mathematics in the capital or large regional cities.

Examine the policy

In what is supposed to be a research-driven policy environment, has anyone considered the consequences of the AITSL proposal? And whether this will actually give teachers the skills they need for the positions they subsequently occupy?

In my own case I came to Melbourne with a BSc (Hons) from the University of Adelaide. In the early 1970s I completed a DipEd at La Trobe. The only real cost was some childcare. If I remember correctly the government was so keen to get professional women into the workforce they even helped with the cost of books. Would I have committed to a two-year course? I’m not sure but I had no HECS debt and ongoing employment was just about guaranteed.

My first school had a very high percentage of students from a non-English speaking background. Many of the Year 7s had very poor achievement in mathematics and I turned my attention to finding out what could be done to help them reach a more appropriate standard.

In the course of this I met Associate Professor John Munro who stressed the importance of language in the learning of mathematics. To be a better mathematics teacher, I completed another degree in teaching English as a second language.

Later I coordinated a DipEd program. Many of our better students were of a mature age and struggling with money, family, jobs and a host of other things. They managed for a year. Requiring them to complete two would have seen many of them not enrol in the first place or drop out when it became too much.

Learn on the job

A two-year teaching qualification does not necessarily equip you for the teaching situation you find yourself in. If AITSL wants all teachers to have a second year, let that be achieved in work-related learning over, for example, 5-7 years.

Australia can’t afford to lose a single prospective teacher who is an articulate, well-qualified graduate in mathematics. If the one-year DipEd goes, many will be lost. They have too many options. The new graduates will think about other courses, the career change, mature-age graduates will decide it is all too hard.

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

*Credit for article given to Jan Thomas*