Author: IMC

DNA knot as seen under the electron microscope. Javier Arsuaga, CC BY-ND

If you’ve ever seen a picture of a DNA molecule, you probably saw it in its famous B-form: two strands coiling around each other in a right-handed fashion to form a double helix. But did you know that DNA can change its shape?

DNA molecules, which carry the genetic code of an organism, have to be tightly packed to fit inside a cell. However, every few hours, the cell produces a faithful copy of its genome in preparation for cell division. This replication process puts tremendous stress on the DNA and can change its shape in lethal ways.

As a mathematician and a biologist, I am interested in how mathematics can describe the many shapes of DNA, as well as cellular processes like DNA replication. The answers to these questions inspire new mathematics and possibly a better understanding of the molecule of life.

The shape of DNA

To understand the mathematics of the shape of DNA, you need to consider both its geometry and its topology. These are related but distinct concepts.

Geometry describes an object at a particular moment in time – frozen rigid in space, like a sculpture. In the cell, the DNA helix coils upon itself, or “supercoils.” The way DNA folds and coils encodes valuable geometric information that can be crucial to control the way genes are expressed.

Topology describes how an object deforms smoothly, as if made out of clay without making new holes or breaks. For example, imagine a rubber band tumbling around in a whirlpool. As the water swirls, the rubber band twists, stretches and shrinks. All of the shapes adopted by the band as it moves are topologically identical, but geometrically different.

These three objects have very different geometries, but are topologically the same – meaning that the objects can be bent or twisted from one shape into another. Mariel Vazquez, CC BY

Merely copying DNA creates a large number of shape-related problems, but textbook images rarely illustrate this topological conundrum.

During the cell cycle, each chromosome is replicated into two identical copies. In order for that to happen, the DNA helix must unwind, causing stress on the DNA. DNA responds to this stress by supercoiling, just like an old telephone cord. But the cell cannot tolerate too much supercoiling. If DNA contorts too much, the cell will suffer.

Sketch of a right handed DNA double helix (left). The opening of the helix, indicated by a triangle, causes the DNA to supercoil (right). A supercoil occurs when the axis of the helix, indicated in purple, coils upon itself. Mariel Vazquez, CC BY

A DNA molecule can be linear – as in the case of human chromosomes – or circular. Examples of circular DNA molecules include bacterial chromosomes and human mitochondrial DNA. If the DNA molecule is circular, then cellular processes such as replication may tie DNA into knots or links, like rings in a keychain. DNA knots and links can cause cells to malfunction or even die.

Stabilizing DNA

Consider the bacterium E. coli. Its genetic code is found in one single DNA chromosome. In E. coli and other bacteria, the DNA double helix closes into a circle, like a twisted rubber band.

Replication of the E. coli chromosome can happen in as short as 20 minutes in a test tube. But when a circular chromosome is replicated, the process yields two interlinked chromosomes. That is, the new chromosomes form two rings linked through each other. The new chromosomes must unlink before the cell divides into two cells. Otherwise they would either break on the way to their target cell, or one cell would inherit two interlinked copies of one chromosome and the other one would be missing the chromosome altogether.

The cell recruits enzymes to unlink the DNA. Enzymes called topoisomerases and recombinases act as scissors and glue for DNA. They can change the geometry and topology of DNA, thus maintaining a stable genome. In E. coli, topoisomerases work tirelessly during and after replication to maintain healthy levels of supercoiling and to safely unlink the chromosomes.

Replication of a circular DNA molecule. The arrows show the direction of replication (left). The new molecules interlink in this process (right). Mariel Vazquez, CC BY

When topoisomerases don’t work

When topoisomerases don’t work, the cell eventually dies. This makes them good targets for drug design. But cells have different types of topoisomerases and other enzymes such as recombinases that may be able to come to the rescue. For example, we showed that, in E. coli cells where the topoisomerases in charge of unlinking have been disabled, other enzymes called site-specific recombinases can untie replication links.

Both topoisomerases and site-specific recombinases bind double stranded DNA and can change its shape by introducing breaks. Type II topoisomerases introduce a break along the DNA molecule and transport another piece of DNA through the break before resealing it. Site-specific recombinases attach to two sites along the DNA, introduce one cut in each, then reconnect the ends.

My lab uses mathematics and computer simulations to understand how these enzymes unlink DNA molecules. While the local action is well understood on a biochemical level, how exactly enzymes simplify the topology of DNA is still a mystery.

In one of our studies, we focused on E. coli cells where the topoisomerases don’t work. We showed how to untie a replication link in the minimum number of steps.

The unlinking pathway of a 6-crossing replication link. This is the only pathway that simplifies the link at each step. Replication links can have any even number of crossings and similar unlinking pathways. Arsuaga-Vazquez lab, Author provided (no reuse)

In general, there can be many unlinking pathways. We use computer simulations to assign probabilities to each pathway. Our work indicates that, in the case of replication links, the simplest pathway is the one that enzymes most likely take.

Sophisticated mathematical methods can help explain how enzymes unlink DNA. Without mathematical modeling, researchers would be restricted to simplified models suggested by biological experiments.

For more insights like this, visit our website at www.international-maths-challenge.com.
Credit of the article given to Mariel Vazquez

Philosophers cannot agree on whether mathematical objects exist or are pure fictions

Credit: Brook VanDevelder

When I tell someone I am a mathematician, one of the most curious common reactions is: “I really liked math class because everything was either right or wrong. There is no ambiguity or doubt.” I always stutter in response. Math does not have a reputation for being everyone’s favorite subject, and I hesitate to temper anyone’s enthusiasm. But math is full of uncertainties—it just hides them well.

Of course, I understand the point. If your teacher asks whether 7 is a prime number, the answer is definitively “yes.” By definition, a prime number is a whole number greater than 1 that is only divisible by itself and 1, such as 2, 3, 5, 7, 11, 13, and so on. Any math teacher, anywhere in the world, anytime in the past several thousand years, will mark you correct for stating that 7 is prime and incorrect for stating that 7 is not prime. Few other disciplines can achieve such incredible consensus. But if you ask 100 mathematicians what explains the truth of a mathematical statement, you will get 100 different answers. The number 7 might really exist as an abstract object, with primality being a feature of that object. Or it could be part of an elaborate game that mathematicians devised. In other words, mathematicians agree to a remarkable degree on whether a statement is true or false, but they cannot agree on what exactly the statement is about.

One aspect of the controversy is the simple philosophical question: Was mathematics discovered by humans, or did we invent it? Perhaps 7 is an actual object, existing independently of us, and mathematicians are discovering facts about it. Or it might be a figment of our imaginations whose definition and properties are flexible. The act of doing mathematics actually encourages a kind of dual philosophical perspective, where math is treated as both invented and discovered.

This all seems to me a bit like improv theater. Mathematicians invent a setting with a handful of characters, or objects, as well as a few rules of interaction, and watch how the plot unfolds. The actors rapidly develop surprising personalities and relationships, entirely independent of the ones mathematicians intended. Regardless of who directs the play, however, the denouement is always the same. Even in a chaotic system, where the endings can vary wildly, the same initial conditions will always lead to the same end point. It is this inevitability that gives the discipline of math such notable cohesion. Hidden in the wings are difficult questions about the fundamental nature of mathematical objects and the acquisition of mathematical knowledge.

Invention

How do we know whether a mathematical statement is correct or not? In contrast to scientists, who usually try to infer the basic principles of nature from observations, mathematicians start with a collection of objects and rules and then rigorously demonstrate their consequences. The result of this deductive process is called a proof, which often builds from simpler facts to a more complex fact. At first glance, proofs seem to be key to the incredible consensus among mathematicians.

But proofs confer only conditional truth, with the truth of the conclusion depending on the truth of the assumptions. This is the problem with the common idea that consensus among mathematicians results from the proof-based structure of arguments. Proofs have core assumptions on which everything else hinges—and many of the philosophically fraught questions about mathematical truth and reality are actually about this starting point. Which raises the question: Where do these foundational objects and ideas come from?

Often the imperative is usefulness. We need numbers, for example, so that we can count (heads of cattle, say) and geometric objects such as rectangles to measure, for example, the areas of fields. Sometimes the reason is aesthetic—how interesting or appealing is the story that results? Altering the initial assumptions will sometimes unlock expansive structures and theories, while precluding others. For example, we could invent a new system of arithmetic where, by fiat, a negative number times a negative number is negative (easing the frustrated explanations of math teachers), but then many of the other, intuitive and desirable properties of the number line would disappear. Mathematicians judge foundational objects (such as negative numbers) and their properties (such as the result of multiplying them together) within the context of a larger, consistent mathematical landscape. Before proving a new theorem, therefore, a mathematician needs to watch the play unfold. Only then can the theorist know what to prove: the inevitable, unvarying conclusion. This gives the process of doing mathematics three stages: invention, discovery and proof.

The characters in the play are almost always constructed out of simpler objects. For example, a circle is defined as all points equidistant from a central point. So its definition relies on the definition of a point, which is a simpler type of object, and the distance between two points, which is a property of those simpler objects. Similarly, multiplication is repeated addition, and exponentiation is repeated multiplication of a number by itself. In consequence, the properties of exponentiation are inherited from the properties of multiplication. Conversely, we can learn about complicated mathematical objects by studying the simpler objects they are defined in terms of. This has led some mathematicians and philosophers to envision math as an inverted pyramid, with many complicated objects and ideas deduced from a narrow base of simple concepts.

In the late 19th and early 20th centuries a group of mathematicians and philosophers began to wonder what holds up this heavy pyramid of mathematics. They worried feverishly that math has no foundations—that nothing was grounding the truth of facts like 1 + 1 = 2. (An obsessive set of characters, several of them struggled with mental illness.) After 50 years of turmoil, the expansive project failed to produce a single, unifying answer that satisfied all the original goals, but it spawned various new branches of mathematics and philosophy.

Some mathematicians hoped to solve the foundational crisis by producing a relatively simple collection of axioms from which all mathematical truths can be derived. The 1930s work of mathematician Kurt Gödel, however, is often interpreted as demonstrating that such a reduction to axioms is impossible. First, Gödel showed that any reasonable candidate system of axioms will be incomplete: mathematical statements exist that the system can neither prove nor disprove. But the most devastating blow came in Gödel’s second theorem about the incompleteness of mathematics. Any foundational system of axioms should be consistent—meaning, free of statements that can be both proved and disproved. (Math would be much less satisfying if we could prove that 7 is prime and 7 is not prime.) Moreover, the system should be able to prove—to mathematically guarantee—its own consistency. Gödel’s second theorem states that this is impossible.

The quest to find the foundations of mathematics did lead to the incredible discovery of a system of basic axioms, known as Zermelo-Fraenkel set theory, from which one can derive most of the interesting and relevant mathematics. Based on sets, or collections of objects, these axioms are not the idealized foundation that some historical mathematicians and philosophers had hoped for, but they are remarkably simple and do undergird the bulk of mathematics.

Throughout the 20th century mathematicians debated whether Zermelo-Fraenkel set theory should be augmented with an additional rule, known as the axiom of choice: If you have infinitely many sets of objects, then you can form a new set by choosing one object from each set. Think of a row of buckets, each containing a collection of balls, and one empty bucket. From each bucket in the row, you can choose one ball and place it in the empty bucket. The axiom of choice would allow you to do this with an infinite row of buckets. Not only does it have intuitive appeal, it is necessary to prove several useful and desirable mathematical statements. But it also implies some strange things, such as the Banach-Tarski paradox, which states that you can break a solid ball into five pieces and reassemble those pieces into two new solid balls, each equal in size to the first. In other words, you can double the ball. Foundational assumptions are judged by the structures they produce, and the axiom of choice implies many important statements but also brings extra baggage. Without the axiom of choice, math seems to be missing crucial facts, though with it, math includes some strange and potentially undesirable statements.

The bulk of modern mathematics uses a standard set of definitions and conventions that have taken shape over time. For example, mathematicians used to regard 1 as a prime number but no longer do. They still argue, however, whether 0 should be considered a natural number (sometimes called the counting numbers, natural numbers are defined as 0,1,2,3… or 1,2,3…, depending on who you ask). Which characters, or inventions, become part of the mathematical canon usually depends on how intriguing the resulting play is—observing which can take years. In this sense, mathematical knowledge is cumulative. Old theories can be neglected, but they are rarely invalidated, as they often are in the natural sciences. Instead mathematicians simply choose to turn their attention to a new set of starting assumptions and explore the theory that unfolds.

Discovery

As noted earlier, mathematicians often define objects and axioms with a particular application in mind. Over and over again, however, these objects surprise them during the second stage of the mathematical process: discovery. Prime numbers, for example, are the building blocks of multiplication, the smallest multiplicative units. A number is prime if it cannot be written as the product of two smaller numbers, and all the nonprime (composite) numbers can be constructed by multiplying a unique set of primes together.

In 1742 mathematician Christian Goldbach hypothesized that every even number greater than 2 is the sum of two primes. If you pick any even number, the so-called Goldbach conjecture predicts that you can find two prime numbers that add up to that even number. If you pick 8, those two primes are 3 and 5; pick 42, and that is 13 + 29. The Goldbach conjecture is surprising because although primes were designed to be multiplied together, it suggests amazing, accidental relations between even numbers and the sums of primes.

An abundance of evidence supports Goldbach’s conjecture. In the 300 years since his original observation, computers have confirmed that it holds for all even numbers smaller than 4 × 10¹⁸. But this evidence is not enough for mathematicians to declare Goldbach’s conjecture correct. No matter how many even numbers a computer checks, there could be a counterexample—an even number that is not the sum of two primes—lurking around the corner.

Imagine that the computer is printing its results. Each time it finds two primes that add up to a specific even number, the computer prints that even number. By now it is a very long list of numbers, which you can present to a friend as a compelling reason to believe the Goldbach conjecture. But your clever friend is always able to think of an even number that is not on the list and asks how you know that the Goldbach conjecture is true for that number. It is impossible for all (infinitely many) even numbers to show up on the list. Only a mathematical proof—a logical argument from basic principles demonstrating that Goldbach’s conjecture is true for every even number—is enough to elevate the conjecture to a theorem or fact. To this day, no one has been able to provide such a proof.

The Goldbach conjecture illustrates a crucial distinction between the discovery stage of mathematics and the proof stage. During the discovery phase, one seeks overwhelming evidence of a mathematical fact—and in empirical science, that is often the end goal. But mathematical facts require a proof.

Patterns and evidence help mathematicians sort through mathematical findings and decide what to prove, but they can also be deceptive. For example, let us build a sequence of numbers: 121, 1211, 12111, 121111, 1211111, and so on. And let us make a conjecture: all the numbers in the sequence are not prime. It is easy to gather evidence for this conjecture. You can see that 121 is not prime, because 121 = 11 × 11. Similarly, 1211, 12111 and 121111 are all not prime. The pattern holds for a while—long enough that you would likely get bored checking—but then it suddenly fails. The 136th element in this sequence (that is, the number 12111…111, where 136 “1”s follow the “2”) is prime.

It is tempting to think that modern computers can help with this problem by allowing you to test the conjecture on more numbers in the sequence. But there are examples of mathematical patterns that hold true for the first 10⁴² elements of a sequence and then fail. Even with all the computational power in the world, you would never be able to test that many numbers.

Even so, the discovery stage of the mathematical process is extremely important. It reveals hidden connections such as the Goldbach conjecture. Often two entirely distinct branches of math are intensively studied in isolation before a profound relation between them is uncovered. A relatively simple example is Euler’s identity, e^iπ + 1 = 0, which connects the geometric constant π with the number i, defined algebraically as the square root of –1, via the number e, the base of natural logarithms. These surprising discoveries are part of the beauty and curiosity of math. They seem to point at a deep underlying structure that mathematicians are only beginning to understand.

In this sense, math feels both invented and discovered. The objects of study are precisely defined, but they take on a life of their own, revealing unexpected complexity. The process of mathematics therefore seems to require that mathematical objects be simultaneously viewed as real and invented—as objects with concrete, discoverable properties and as easily manipulable inventions of mind. As philosopher Penelope Maddy writes, however, the duality makes no difference to how mathematicians work, “as long as double-think is acceptable.”

Real or unreal?

Mathematical realism is the philosophical position that seems to hold during the discovery stage: the objects of mathematical study—from circles and prime numbers to matrices and manifolds—are real and exist independently of human minds. Like an astronomer exploring a far-off planet or a paleontologist studying dinosaurs, mathematicians are gathering insights into real entities. To prove that Goldbach’s conjecture is true, for example, is to show that the even numbers and the prime numbers are related in a particular way through addition, just like a paleontologist might show that one type of dinosaur descended from another by showing that their anatomical structures are related.

Realism in its various manifestations, such as Platonism (inspired by the Greek philosopher’s theory of Platonic forms), makes easy sense of mathematics’ universalism and usefulness. A mathematical object has a property, such as 7 being a prime number, in the same way that a dinosaur might have had the property of being able to fly. And a mathematical theorem, such as the fact that the sum of two even numbers is even, is true because even numbers really exist and stand in a particular relation to each other. This explains why people across temporal, geographical and cultural differences generally agree about mathematical facts—they are all referencing the same fixed objects.

But there are some important objections to realism. If mathematical objects really exist, their properties are certainly very peculiar. For one, they are causally inert, meaning they cannot be the cause of anything, so you cannot literally interact with them. This is a problem because we seem to gain knowledge of an object through its impact. Dinosaurs decomposed into bones that paleontologists can see and touch, and a planet can pass in front of a star, blocking its light from our view. But a circle is an abstract object, independent of space and time. The fact that π is the ratio of the circumference to the diameter of a circle is not about a soda can or a doughnut; it refers to an abstract mathematical circle, where distances are exact and the points on the circle are infinitesimally small. Such a perfect circle is causally inert and seemingly inaccessible. So how can we learn facts about it without some type of special sixth sense?

That is the difficulty with realism—it fails to explain how we know facts about abstract mathematical objects. All of which might cause a mathematician to recoil from his or her typically realist stance and latch onto the first step of the mathematical process: invention. By framing mathematics as a purely formal mental exercise or a complete fiction, antirealism easily skirts problems of epistemology.

Formalism, a type of antirealism, is a philosophical position that asserts that mathematics is like a game, and mathematicians are just playing out the rules of the game. Stating that 7 is a prime number is like stating that a knight is the only chess piece that can move in an L shape. Another philosophical position, fictionalism, claims that mathematical objects are fictions. Stating that 7 is a prime number is then like stating that unicorns are white. Mathematics makes sense within its fictional universe but has no real meaning outside of it.

There is an inevitable trade-off. If math is simply made up, how can it be such a necessary part of science? From quantum mechanics to models of ecology, mathematics is an expansive and precise scientific tool. Scientists do not expect particles to move according to chess rules or the crack in a dinner plate to mimic Hansel and Gretel’s path—the burden of scientific description is placed exclusively on mathematics, which distinguishes it from other games or fictions.

In the end, these questions do not affect the practice of mathematics. Mathematicians are free to choose their own interpretations of their profession. In The Mathematical Experience, Philip Davis and Reuben Hersh famously wrote that “the typical working mathematician is a Platonist on weekdays and a formalist on Sundays.” By funneling all disagreements through a precise process—which embraces both invention and discovery—mathematicians are incredibly effective at producing disciplinary consensus.

For more insights like this, visit our website at www.international-maths-challenge.com.

Credit of the article given to Kelsey Houston-Edwards |

“When will I ever use this?” It’s a question math and science teachers hear all the time from their high school students.

Teaching science, technology, engineering and math (STEM) skills is more important than ever, but it’s often difficult for students to understand the practical applications of such fundamental learning and how it will help them down the road.

Classroom activities should be relevant, meaningful and connected to students’ prior knowledge and experiences. Learning must be based on lived experiences within both formal and informal educational settings.

Increasingly, teacher educators are realizing that we must break away from traditional silos of courses, disciplines and formal schooling. Educators must lead by example and provide students with opportunities to explore interdisciplinary approaches to learning.

Creative thinking

The new British Columbia curriculum embraces these principles of learning. In the same spirit, I’m part of a new and unique Bachelor of Education program at Thompson Rivers University where teacher candidates are learning to teach STEM by actively engaging students. The program promotes cross-curricular and interdisciplinary approaches to learning and is tied to the provincial curriculum core competencies of communication, critical and creative thinking.

So how do you teach a subject like math differently in a way that can help students learn through creative thinking and experience, rather than rote memorization?

Let’s take, for example, Pi.

I often ask my teacher candidates: What is π? Many respond “3.14” and, if probed further, explain the meaning by merely stating an equation like A=πr² (where A is the area of a circle and r is the radius of a circle). Or they may tell me C=2πr (where C is the circumference of a circle).

A door handle in the shape of Pi at the National Museum of Mathematics in New York, (AP Photo/Seth Wenig)

Teaching through discovery

I encourage these teacher candidates to think differently and to help students discover mathematical concepts for themselves. What better way to teach students that π is the ratio of a circle’s circumference to its diameter than to have them trace any circle and then measure it with a piece of string?

They will soon learn that regardless of the size of the circle, the ratio of circumference to diameter will always be 22/7, an approximation of π.

Innovative educators can integrate history, geography, math and science lessons by teaching a thematic unit on ancient civilizations.

For example, the Egyptians succeeded in building great pyramids with incredible precision and accuracy. These magnificent architectural accomplishments have stood the test of time, remaining largely intact after centuries — a tribute to their construction.

The ancient Egyptians understood the significance of mathematics through the very beauty and symmetry of nature. They used geometry to solve everyday problems.

Tearing down silos

Increasingly, teacher educators are realizing that we must break away from traditional silos of courses, disciplines and formal schooling — exactly the opposite of the “back to basics” approach suggested by populist politicians like new Ontario Premier Doug Ford.

Students benefit from learning experiences that are meaningful, relevant and well-connected to their own experiences. For that to happen, the people teaching those students must be prepared to take on new attitudes of reflectiveness and inquisitiveness.

What is necessary is to follow in the footsteps of the great thinkers like Galileo and Newton, who questioned our perceptions of reality and sought answers from tactile experiences rather than textbooks or teachers.

For more insights like this, visit our website at www.international-maths-challenge.com.
Credit of the article given to Edward R. Howe

Credit: The mathematician Ken Ono in his office at Emory University in Atlanta. Raymond McCrea Jones for Quanta Magazine

The mathematician Ken Ono believes that the story of Srinivasa Ramanujan—mathematical savant and two-time college dropout—holds valuable lessons for how we find and reward hidden genius

For the first 27 years of his life, the mathematician Ken Ono was a screw-up, a disappointment and a failure. At least, that’s how he saw himself. The youngest son of first-generation Japanese immigrants to the United States, Ono grew up under relentless pressure to achieve academically. His parents set an unusually high bar. Ono’s father, an eminent mathematician who accepted an invitation from J. Robert Oppenheimer to join the Institute for Advanced Study in Princeton, N.J., expected his son to follow in his footsteps. Ono’s mother, meanwhile, was a quintessential “tiger parent,” discouraging any interests unrelated to the steady accumulation of scholarly credentials.

This intellectual crucible produced the desired results—Ono studied mathematics and launched a promising academic career—but at great emotional cost. As a teenager, Ono became so desperate to escape his parents’ expectations that he dropped out of high school. He later earned admission to the University of Chicago but had an apathetic attitude toward his studies, preferring to party with his fraternity brothers. He eventually discovered a genuine enthusiasm for mathematics, became a professor, and started a family, but fear of failure still weighed so heavily on Ono that he attempted suicide while attending an academic conference. Only after he joined the Institute for Advanced Study himself did Ono begin to make peace with his upbringing.

Through it all, Ono found inspiration in the story of Srinivasa Ramanujan, a mathematical prodigy born into poverty in late-19th-century colonial India. Ramanujan received very little formal schooling, yet he still produced thousands of independent mathematical results, some of which—like the Ramanujan theta function, which has found applications in string theory—are still intensely studied. But despite his genius, Ramanujan’s achievements didn’t come easily. He struggled to gain acceptance from Western mathematicians and dropped out of university twice before dying of illness at the age of 32.

While Ono, now 48, doesn’t compare himself to Ramanujan in terms of ability, he has built his career in part from Ramanujan’s insights. In 2014, Ono and his collaborators Michael Griffin and Ole Warnaar published a breakthrough result in algebraic number theory that generalized one of Ramanujan’s own results. Ono’s work, which is based on a pair of equations called the Rogers-Ramanujan identities, can be used to easily produce algebraic numbers (such as phi, better known as the “golden ratio”).

More recently, Ono served as an associate producer and mathematical consultant for The Man Who Knew Infinity, a recently released feature film about Ramanujan’s life. And his new memoir, My Search for Ramanujan: How I Learned to Count (co-authored with Amir D. Aczel), draws connections between Ramanujan’s life and Ono’s own circuitous path to mathematical and emotional fulfillment. “I wrote this book to show off my weaknesses, to show off my struggles,” Ono said. “People who are successful in their careers were not always successful from day one.”

Like Ramanujan, who benefited from years of mentorship by the British mathematician G.H. Hardy, Ono credits his own success to serendipitous encounters with teachers who helped his talents flourish. He now spends a great deal of time mentoring his own students at Emory University. Ono has also helped launch the Spirit of Ramanujan Math Talent Initiative, a venture that “strives to find undiscovered mathematicians around the world and match them with advancement opportunities in the field.”

Quanta Magazine spoke with Ono about finding his way as a mathematician and a mentor, and about Ramanujan’s inspiring brand of creativity. An edited and condensed version of the interview follows.

QUANTA MAGAZINE: What was so special about Ramanujan’s approach to doing mathematics?
KEN ONO: First, he was really a poet, not a problem solver. Most professional mathematicians, whether they’re in academia or industry, have problems that they’re aiming to solve. Somebody wants to prove the Riemann hypothesis, and sets out to do it. That’s how we think science should proceed, and in fact almost every scientist should work that way, because in reality science develops through the work of thousands of individuals slowly adding to a body of knowledge. But what you find in Ramanujan’s original notebooks is just formula after formula, and it’s not apparent where he’s going with his ideas. He was someone who could set down the paths of beginnings of important theories without knowing for sure why we would care about them as mathematicians of the future.

He’s credited with compiling thousands of identities—that is, equations that are true regardless of what values the variables take. Why is that important?
It is true that the vast majority of the contents of his notebooks are what you would call identities. Identities that relate continued fractions to other functions, expressions for integrals, expressions for hypergeometric functions, and expressions for objects that we call q-series.

But that would be a literal interpretation of his notebooks. In my opinion, that would be like taking a cookbook by Julia Child, reading the recipes and saying that it’s about assembling chemical compounds into something more complicated. Strictly speaking that would be true, but you would be missing out on what makes delicious recipes so important to us.

Ramanujan’s work came through flights of fancy. If he had been asked to explain why he did his work, he would probably say that he recorded formulas that he found beautiful, and they were beautiful because they revealed some unexpected phenomenon. And they’re important to us today because these special phenomena that Ramanujan identified, over and over again, have ended up becoming prototypes for big mathematical theories in the 20th and 21st centuries.

Here’s an example. In one of his published manuscripts, Ramanujan recorded a lot of elementary-looking results called congruences. In the 1960s, Jean-Pierre Serre, himself a Fields medalist, revisited some of these results, and in them he found glimpses of a theory that he named the theory of Galois representations. This theory of Galois representations is the language that Andrew Wiles used in the 1990s to prove Fermat’s last theorem.

There’s no “theory of Ramanujan,” but he anticipated mathematical structures that would be important to all of these other more contemporary works. He lived 80 years before his time.

How do you approach your own mathematical work—more as an artist, like Ramanujan, or with the aim of solving specific problems, like a scientist?
I’m definitely much more of a scientist. Science proceeds at a much faster rate than when I started in my career in the early 1990s, and I have to stop often to recognize the beauty in it and try not to be so caught up in the more professionalized side of how one does science. The grant getting, the publications, and all of that—I have to admit, I don’t like it.

What compelled you to juxtapose your own story with his?
Well, I almost didn’t write it. There are a lot of very personal things that I’ve never told anyone before. It wasn’t until I started writing this book that I was mature enough as a parent myself to try to understand the circumstances that led my parents to raise us the way they did. And as a professor at Emory, I see all these kids under tremendous pressure—rarely pressure that they understand the origin of. So many of these super-talented kids are just going through the motions, and aren’t passionate about their studies at all, and that’s terrible. I was like that too. I’d given up on ever trying to live up to my parents’ expectations, but somehow because I’ve had Ramanujan as a guardian angel, things have worked out well for me. It makes you a better teacher when you just tell people how hard it was for you.

This book and your story don’t fit the typical “great man of science” narrative.
I think you’ll find that’s much more common than people are willing to admit. I didn’t discover my passion for mathematics until my early 20s—that’s when [my doctoral adviser Basil] Gordon turned me on to mathematics at a time when I didn’t think anything was beautiful. I thought it was all about test scores, grades and trying to do as well as possible without putting in effort. Colleges are full of kids who think that way. How do you beat the system, right? I wasn’t beating the system. The system was beating me, and Gordon turned me around. When I’ve told people the story I’ve discovered that I’m really not alone.

That’s what I see in Ramanujan. He was a two-time college dropout whom my father looked up to as a hero—which made no sense to me when I was 16, because I was told I had to be a child prodigy. I was supposed to do my geometry problems during the summer sitting next to my dad while he did his research. I wasn’t even really allowed to go out and play, and then to just have my father tell me about Ramanujan out of the blue—it was beyond earth-shattering.

If you’d been interested in something conventionally “artistic,” like music, this kind of painful journey toward success would not seem so surprising. Why does it surprise us to hear about a mathematician having the same struggles?
For whatever reason, we live in a culture where we think that the abilities of our best scientists and our best mathematicians are somehow just God-given. That either you have this gift or you don’t, and it’s not related to help, to hard work, to luck. I think that’s part of the reason why, when we try to talk about mathematics to the public, so many people just immediately respond by saying, “Well, I was never very good at math. So I’m not really supposed to understand it or identify with it.” I might have had some mathematical talent passed through my father genetically, but that was by no means enough. You have to be passionate about a subject.

At the same time, I want it to be known that it’s totally OK to fail. In fact, you learn from your mistakes. We learn early on if that you want to be good at playing the violin, you’ve got to practice. If you want to be good at sports, you practice. But for some crazy reason, our culture assumes that if you’re good at math, you were just born with it, and that’s it. But you can be so good at math in so many different ways. I failed my [graduate-school] algebra qualifications! That doesn’t mean I can’t end up being a successful mathematician. But when I tell people I failed at this, nobody believes me.

But Ramanujan seems to be just that: a unique genius who appeared out of nowhere. What does that have to do with a regular person’s life?
You think no one can be like Ramanujan? Well, I disagree. I think we can search the world looking for a mathematical talent, just not by the usual metrics. I want teachers and parents to recognize that when you do see unusual talent, instead of demanding that these people have certain test scores, let’s find a way to help nurture them. Because I think humanity needs it. I think these are the lessons we learn from Ramanujan.

You’re leading the Spirit of Ramanujan Math Talent Initiative. What is this spirit? How do we recognize it?
First of all, it’s the idea that talent is often found in the most unforgiving and unpromising of circumstances. It’s the responsibility of mentors, teachers and parents first to recognize that talent, which is not always easy to do, and then to offer opportunities that nurture that talent.

There are no age limits, and I don’t want this to be a competition where you’re recognized for high test scores. I have no trouble finding people who can get an 800 on the math SAT. That’s easy. Those people don’t need to be identified. They’ve already self-identified. I’m searching for creativity.

That said, the spirit of Ramanujan does not require finding the next Ramanujan. We would be super lucky to do that, but if we make opportunities for 30 talented people around the world who are presently working in an intellectual desert, or are subjected to inelastic educational systems where they’re not allowed to flourish—or if we can provide an opportunity for someone to work with a scientist who could be their G.H. Hardy—then this initiative will be successful.

Do you wish you had been nurtured differently? Do you resent your parents?
I love my parents. We discussed the draft of the book for months last summer. They were very upset with me at first, because it was difficult for them to get past the first 30 pages, but now they embrace it. One reviewer actually saw the book as a love letter to my parents and to my mentors, because they taught me skills I needed.

If you had never joined the Institute for Advanced Study, would you still be struggling to reconcile your own path with your parents’ expectations?
I think I would still be searching for that recognition today if I hadn’t gotten there.

Both my parents will tell you that you only get to live once, so you might as well be the very, very best that you can be at whatever you choose. Which I don’t necessarily agree with, because if everyone lived that way, there would be nothing but a whole bunch of unhappy people in the world. But that’s how they brought us up. They taught me to be competitive. They taught me not to falsely believe I had done well when I hadn’t. They taught me standards, and those are important. But it’s true that if I hadn’t had the opportunity to work at the Institute, I’m not sure I would have been able to write this book. I might still be struggling with these things.

For more insights like this, visit our website at www.international-maths-challenge.com.

Credit of the article given to John Pavlus & Quanta Magazine

Over the past few days, the mathematics world has been abuzz over the news that Sir Michael Atiyah, the famous Fields Medalist and Abel Prize winner, claims to have solved the Riemann hypothesis.

If his proof turns out to be correct, this would be one of the most important mathematical achievements in many years. In fact, this would be one of the biggest results in mathematics, comparable to the proof of Fermat’s Last Theorem from 1994 and the proof of the Poincare Conjecture from 2002.

Besides being one of the great unsolved problems in mathematics and therefore garnishing glory for the person who solves it, the Riemann hypothesis is one of the Clay Mathematics Institute’s “Million Dollar Problems.” A solution would certainly yield a pretty profitable haul: one million dollars.

The Riemann hypothesis has to do with the distribution of the prime numbers, those integers that can be divided only by themselves and one, like 3, 5, 7, 11 and so on. We know from the Greeks that there are infinitely many primes. What we don’t know is how they are distributed within the integers.

The problem originated in estimating the so-called “prime pi” function, an equation to find the number of primes less than a given number. But its modern reformulation, by German mathematician Bernhard Riemann in 1858, has to do with the location of the zeros of what is now known as the Riemann zeta function.

The technical statement of the Riemann hypothesis is “the zeros of the Riemann zeta function which lie in the critical strip must lie on the critical line.” Even understanding that statement involves graduate-level mathematics courses in complex analysis.

Most mathematicians believe that the Riemann hypothesis is indeed true. Calculations so far have not yielded any misbehaving zeros that do not lie in the critical line. However, there are infinitely many of these zeros to check, and so a computer calculation will not verify all that much. Only an abstract proof will do.

If, in fact, the Riemann hypothesis were not true, then mathematicians’ current thinking about the distribution of the prime numbers would be way off, and we would need to seriously rethink the primes.

The Riemann hypothesis has been examined for over a century and a half by some of the greatest names in mathematics and is not the sort of problem that an inexperienced math student can play around with in his or her spare time. Attempts at verifying it involve many very deep tools from complex analysis and are usually very serious ones done by some of the best names in mathematics.

Atiyah gave a lecture in Germany on Sept. 25 in which he presented an outline of his approach to verify the Riemann hypothesis. This outline is often the first announcement of the solution but should not be taken that the problem has been solved – far from it. For mathematicians like me, the “proof is in the pudding,” and there are many steps that need to be taken before the community will pronounce Atiyah’s solution as correct. First, he will have to circulate a manuscript detailing his solution. Then, there is the painstaking task of verifying his proof. This could take quite a lot of time, maybe months or even years.

Is Atiyah’s attempt at the Riemann hypothesis serious? Perhaps. His reputation is stellar, and he is certainly capable enough to pull it off. On the other hand, there have been several other serious attempts at this problem that did not pan out. At some point, Atiyah will need to circulate a manuscript that experts can check with a fine-tooth comb.

For more insights like this, visit our website at www.international-maths-challenge.com.
Credit of the article given to William Ross

Credit: Getty Images

Better living through set-builder notation

As a publicly mathematical person, one of the matters upon which I am called to adjudicate is what I think of as “viral order of operations questions” with a math problem along the lines of “48÷2(9+3) = ?”

In the past, I used to tell people who asked me about one of those questions something like, “I think the correct answer is __, but it’s better to write the expression unambiguously.” But recently I decided I need to put my foot down. I will no longer give an answer to those questions. The only way to win is not to play.

An ambiguous sequence of digits and mathematical operation symbols is not interesting to mathematicians. Most of us learn something about the order of operations fairly early on in our mathematical educations. We might learn PEMDAS or BIDMAS, or “Please excuse my dear aunt Sally.” All of these expressions tell us that we’re supposed to take care of expressions inside parentheses or brackets first, followed by exponential expressions, followed by multiplication and division, then addition and subtraction. It’s good that we have a system, but mathematicians and other people who use mathematical expressions regularly would never write down something like “48÷2(9+3) = ?” because its potential for causing confusion is too great.

In general, mathematicians strive to reduce ambiguity when possible. A mathematician would write (48÷2)(9+3) or 48÷(2(9+3)), depending on which one they meant. Viral order of operations problems are unappealing. Just toss in a few more parentheses to clarify your meaning and move on. There are cat pictures to scroll through, for goodness’ sake!

In fact, I think if mathematicians had their way, they would get rid of easily-fixed ambiguous order of operations problems altogether, and I don’t think they’d stop there. The English language often leaves room for ambiguity, and I think mathematical notation could help us make some improvements.

I remember chuckling to myself when I saw the phrase “I like to play board games and read a book while taking a bath.” It conjures up an image of a very exciting game of Monopoly where Baltic Avenue is replaced by the Baltic Sea. Set-builder notation could resolve that titillating ambiguity there while simultaneously freeing me of a tiny shred of joy in this world of woe. Mathematicians use curly brackets to indicate things that are in one set and can be treated as one object. A person who enjoys two separate activities, one of which is playing board games and one of which is reading a book while taking a bath, could write “I like to {play board games} and {read a book while taking a bath}.” A person with a much more interesting bathtime routine than mine could write “I like to {play board games and read a book} while taking a bath.”

Curly brackets take care of written English, but we communicate through speech as well. The late Danish pianist and comedian Victor Borge had a routine about phonetic punctuation. He assigned sounds to some common punctuation marks and inserted them into sentences.

He didn’t include a sound for curly brackets, and I’m not sure the best option. Perhaps a “zzp” sound would work, but I’m open to other suggestions. In the meantime, there is a precedent for air quotes, and perhaps we could add extend that to air parentheses.

When I started thinking about using brackets and air-parentheses in English writing and speech, I wondered if I was just reinventing sentence diagramming. I don’t know how many people learned to diagram a sentence in school, but a sentence diagram is a graphical representation of a sentence that shows how each word and phrase functions in the sentence. The diagram for the sentence “I like to {play board games} and {read a book while taking a bath}” is different from the diagram for “I like to {play board games and read a book} while taking a bath.” It’s been a while since I did any sentence diagramming, so I beg for lenience from any grammar teachers reading, but these were the diagrams I came up with. The placement of the “while” clause is the only difference between the two diagrams.

Two possible diagrams for the sentence “I like to play board games and read a book while taking a bath.” Credit: Evelyn Lamb

Diagramming the sentence does remove the ambiguity, and it gives even more information than the set-builder notation I’m suggesting, but it comes at a cost of both space and effort. It’s much more practical to throw a few brackets into English prose than to draw every sentence as a complex, multi-storied building.

Set-builder brackets are just the tip of the iceberg when it comes to resolving linguistic ambiguity through mathematical notation. After we’ve mastered those, we can consider incorporating union and disjoint union symbols and the distinction between or and xor into our speech and writing. But that will have to wait until after we’ve relaxed with an extra-realistic game of Battleship in the bathtub.

For more insights like this, visit our website at www.international-maths-challenge.com.

Credit of the article given to Evelyn Lamb

The humble honeybee can use symbols to perform basic maths including addition and subtraction, shows new research published today in the journal Science Advances.

Despite having a brain containing less than one million neurons, the honeybee has recently shown it can manage complex problems – like understanding the concept of zero.

Honeybees are a high value model for exploring questions about neuroscience. In our latest study we decided to test if they could learn to perform simple arithmetical operations such as addition and subtraction.

Addition and subtraction operations

As children, we learn that a plus symbol (+) means we have to add two or more quantities, while a minus symbol (-) means we have to subtract quantities from each other.

To solve these problems, we need both long-term and short-term memory. We use working (short-term) memory to manage the numerical values while performing the operation, and we store the rules for adding or subtracting in long-term memory.

Although the ability to perform arithmetic like adding and subtracting is not simple, it is vital in human societies. The Egyptians and Babylonians show evidence of using arithmetic around 2000BCE, which would have been useful – for example – to count live stock and calculate new numbers when cattle were sold off.

This scene depicts a cattle count (copied by the Egyptologist Lepsius). In the middle register we see 835 horned cattle on the left, right behind them are some 220 animals and on the right 2,235 goats. In the bottom register we see 760 donkeys on the left and 974 goats on the right. Wikimedia commons, CC BY

But does the development of arithmetical thinking require a large primate brain, or do other animals face similar problems that enable them to process arithmetic operations? We explored this using the honeybee.

How to train a bee

Honeybees are central place foragers – which means that a forager bee will return to a place if the location provides a good source of food.

We provide bees with a high concentration of sugar water during experiments, so individual bees (all female) continue to return to the experiment to collect nutrition for the hive.

In our setup, when a bee chooses a correct number (see below) she receives a reward of sugar water. If she makes an incorrect choice, she will receive a bitter tasting quinine solution.

We use this method to teach individual bees to learn the task of addition or subtraction over four to seven hours. Each time the bee became full she returned to the hive, then came back to the experiment to continue learning.

Addition and subtraction in bees

Honeybees were individually trained to visit a Y-maze shaped apparatus.

The bee would fly into the entrance of the Y-maze and view an array of elements consisting of between one to five shapes. The shapes (for example: square shapes, but many shape options were employed in actual experiments) would be one of two colours. Blue meant the bee had to perform an addition operation (+ 1). If the shapes were yellow, the bee would have to perform a subtraction operation (- 1).

For the task of either plus or minus one, one side would contain an incorrect answer and the other side would contain the correct answer. The side of stimuli was changed randomly throughout the experiment, so that the bee would not learn to only visit one side of the Y-maze.

After viewing the initial number, each bee would fly through a hole into a decision chamber where it could either choose to fly to the left or right side of the Y-maze depending on operation to which she had been trained for.

The Y-maze apparatus used for training honeybees. Scarlett Howard

At the beginning of the experiment, bees made random choices until they could work out how to solve the problem. Eventually, over 100 learning trials, bees learnt that blue meant +1 while yellow meant -1. Bees could then apply the rules to new numbers.

During testing with a novel number, bees were correct in addition and subtraction of one element 64-72% of the time. The bee’s performance on tests was significantly different than what we would expect if bees were choosing randomly, called chance level performance (50% correct/incorrect)

Thus, our “bee school” within the Y-maze allowed the bees to learn how to use arithmetic operators to add or subtract.

Why is this a complex question for bees?

Numerical operations such as addition and subtraction are complex questions because they require two levels of processing. The first level requires a bee to comprehend the value of numerical attributes. The second level requires the bee to mentally manipulate numerical attributes in working memory.

In addition to these two processes, bees also had to perform the arithmetic operations in working memory – the number “one” to be added or subtracted was not visually present. Rather, the idea of plus one or minus “one” was an abstract concept which bees had to resolve over the course of the training.

Showing that a bee can combine simple arithmetic and symbolic learning has identified numerous areas of research to expand into, such as whether other animals can add and subtract.

For more insights like this, visit our website at www.international-maths-challenge.com.
Credit of the article given to Scarlett Howard, Adrian Dyer, Jair Garcia

Credit: If you know your height, can you predict how big your arm span is? What about the length of your femur? Or the circumference of your head? Try this simple activity and find out how you knowing these simple ratios can even make you a better artist!  George Retseck

A project that measures up

Introduction
Our bodies are amazing! They are full of mysteries and surprising facts such as this one: Did you know that you are about a centimeter taller in the morning, when you have just woken up after hours of lying down, than you are in the evening? You might never have noticed it. These interesting facts only reveal themselves when you look closely, measure and compare. That is what this activity is about: recording, comparing and discovering how the human body measures up!

Background
Did you know that human bodies come in all sizes and forms? When you start measuring them, however, you will find our bodies show surprising similarities—and even more surprisingly, we can express these with mathematical concepts.

For one thing, our bodies are quite symmetrical. When you draw a vertical line down the center of a body, the left and right sides are almost mirror images of each other. Human bodies also show interesting ratios. Ratios compare two quantities, like the size of one part of the body to the size of another part, or to the size of the whole. An example of a human body ratio is a person’s arm span—the distance from the middle fingertip of the left hand to that of the right hand when stretching out both arms horizontally—to their height. This ratio is approximately a one to one ratio, meaning that a person’s arm span is about equal to their height. There are many more human body ratios; some are independent of age, and others change as we grow from a baby to an adult.

Wondering who would be interested in these ratios? Artists are avid users of human body ratios, because it helps them draw realistic-looking figures. They are also used in the medical world; a sizable deviation from a human body ratio can indicate a body that does not develop according to expectations. In this science activity we will examine some human body ratios and, if you like, we can explore how they can help you draw more realistic-looking figures.

Materials

• Yarn
• Scissors
• A hardcover book
• A helper
• Pen and paper (optional)
• Measuring tape (optional)

Preparation

• To compare the length of different parts of your body with your height, we will first create a string the length of your height. Take off your shoes. The easiest way is to lie on the ground with your heels pressing against a wall. Look straight up and have your helper place a hardcover book flat against the top of your head, resting on the ground. Get out from under the book and, together, span the yarn across the floor from the wall to the book, cutting the yarn just where it reaches it. Now you have a piece of yarn that is as long as you are tall. (If lying on the ground is not possible, you can also stand flat on the floor against the wall and have the book rest on top of your head and against the wall.)

Procedure

• First, we examine your arm span to height ratio. Your arm span is the distance between the middle fingertips on each hand when you stretch your arms out as far as they can reach. How do you think your height compares with your arm span? Would it be similar, way longer or way shorter?
• Now stretch your arms out as far as they can reach. Your arms will be parallel to the ground. Hold one end of the piece of yarn you just cut off with the fingertips of your left hand. Let your helper span the yarn toward the tip of your right hand’s middle finger. Is piece long enough, way longer or way too short? What does this tell you about how your arm span compares to your height?
• For most people, their arm span is about equal to their height. Mathematicians say the arm span to height ratio is one to one: your arm span goes once into your height.
• Now let’s explore another ratio: the length of your femur bone to your height. The femur bone is the only bone in your thigh. To measure its length, sit down and span a new piece of yarn over your thigh from the hip joint to the edge of your knee and cut the yarn there.
• Make an estimate. How many times would this piece of yarn go into the piece that is as long as you are tall? Can you find a way to test your estimate?
• There are several ways to compare the length of the two pieces of yarn: You might cut several pieces of the length of your shorter string, lay them end to end next to your longer piece, and count how many you need. Another way is to fold the longer string into equal parts so the length of the folded string equals the length of the shorter string. The number of folds needed is exactly the number of times your shorter string goes into your longer string.
• Did you see that the length of your femur bone goes about four times into your height? You can also say that if you divide your height in four equal pieces, you have the length of your femur bone, or the length of your femur bone is one fourth of your height. Mathematicians call this a one to four ratio.
• Now let’s move on to a ratio that might help you make more realistic drawings: the head to body ratio. How many times would the length of your head fit into your height? Maybe four, six or eight times? To test six times, fold the yarn with length equaling your height into six equal pieces. Have your helper place a book flat on your head and hang the folded string from the side of the book. If the other end of the string is about level with your chin, your height would be about six times the length of your head, or your head to body ratio would be one to six. Which number of folds fits best for you?
• There are many more bodily ratios you can explore: the circumference of your head compared with your height, or the length ratios of your forearm and foot or thumb and hand. Use pieces of yarn to measure, compare and detect these and/or your other bodily ratios.
• Extra: You have explored some ratios in your body and might wonder if these hold for other people as well. Do you think they hold for most people of your age? What about adults or babies? Do you think these ratios hold for them or would some be different? Make a hypothesis, find some volunteers, measure and compare. Was your hypothesis correct?
• Extra: This activity uses pieces of yarn to compare lengths. You can also measure your height, arm span, femur bone, etcetera with measuring tape, round the values and write the ratios as fractions. Can you find a way to simplify these fractions?
• Extra: Draw some stick figures on a sheet of paper. Can you apply some of the bodily ratios you explored (like the arm span to height or the head to body ratio) to the figures?Which ones look most realistic to you?
• Extra: Ratios are all around us. Can you find other places where ratios play an important role? To get you started, think about a recipe and the ratio of the quantity of one ingredient to another. For avid bikers, can you find the ratios that correspond to the different gears on a bike?

Observations and results
You probably found your arm span to height ratio approximately to be one to one whereas the femur to height was approximately one to four. This is expected because on average and over a large age range the human body has an arm span that is roughly equal to its height and a femur bone roughly a quarter of its height.

The head to body ratio is a little more complex as it changes from a ratio of about one to four for a small child to about one to eight for an adult. A five-year-old is likely to have a head to body ratio of about one to six.

It is good to remember these ratios are averages over a large group of people. Individual variations occur; some might even be used to one’s advantage—for example, having exceptionally long arms can be advantageous when playing basketball.

For more insights like this, visit our website at www.international-maths-challenge.com.

Credit of the article given to Science Buddies & Sabine De Brabandere

In mathematics, no researcher works in true isolation. Even those who work alone use the theorems and methods of their colleagues and predecessors to develop new ideas.

But when a known technique is too difficult to use in practice, mathematicians may neglect important – and otherwise solvable – problems.

Recently, I joined several mathematicians on a project to make one such technique easier to use. We produced a computer package to solve a problem called the “S-unit equation,” with the hope that number theorists of all stripes can more easily attack a wide variety of unsolved problems in mathematics.

Diophantine equations

In his text “Arithmetica,” the mathematician Diophantus looked at algebraic equations whose solutions are required to be whole numbers. As it happens, these problems have a great deal to do with both number theory and geometry, and mathematicians have been studying them ever since.

Why add this restriction of only whole-number solutions? Sometimes, the reasons are practical; it doesn’t make sense to raise 13.7 sheep or buy -1.66 cars. Additionally, mathematicians are drawn to these problems, now called Diophantine equations. The allure comes from their surprising difficulty, and their ability to reveal fundamental truths about the nature of mathematics.

In fact, mathematicians are often uninterested in the specific solutions to any particular Diophantine problem. But when mathematicians develop new techniques, their power can be demonstrated by settling previously unsolved Diophantine equations.

Andrew Wiles, right, receives the Wolflskehl award for his solution of Fermat’s Last Theorem. Peter Mueller/REUTERS

Andrew Wiles’ proof of Fermat’s Last Theorem is a famous example. Pierre de Fermat claimed in 1637 – in the margin of a copy of “Arithmetica,” no less – to have solved the Diophantine equation xⁿ + yⁿ = zⁿ, but offered no justification. When Wiles proved it over 300 years later, mathematicians immediately took notice. If Wiles had developed a new idea that could solve Fermat, then what else could that idea do? Number theorists raced to understand Wiles’ methods, generalizing them and finding new consequences.

No single method exists that can solve all Diophantine equations. Instead, mathematicians cultivate various techniques, each suited for certain types of Diophantine problems but not others. So mathematicians classify these problems by their features or complexity, much like biologists might classify species by taxonomy.

Finer classification

This classification produces specialists, as different number theorists specialize in the techniques related to different families of Diophantine problems, such as elliptic curves, binary forms or Thue-Mahler equations.

Within each family, the finer classification gets customized. Mathematicians develop invariants – certain combinations of the coefficients appearing in the equation – that distinguish different equations in the same family. Computing these invariants for a specific equation is easy. However, the deeper connections to other areas of mathematics involve more ambitious questions, such as: “Are there any elliptic curves with invariant 13?” or “How many binary forms have invariant 27?”

The S-unit equation can be used to solve many of these bigger questions. The S refers to a list of primes, like {2, 3, 7}, related to the particular question. An S-unit is a fraction whose numerator and denominator are formed by multiplying only numbers from the list. So in this case, 3/7 and 14/9 are S-units, but 6/5 is not.

The S-unit equation is deceptively simple to state: Find all pairs of S-units which add to 1. Finding some solutions, like (3/7, 4/7), can be done with pen and paper. But the key word is “all,” and that is what makes the problem difficult, both theoretically and computationally. How can you ever be sure every solution has been found?

In principle, mathematicians have known how to solve the S-unit equation for several years. However, the process is so convoluted that no one could ever actually solve the equation by hand, and few cases have been solved. This is frustrating, because many interesting problems have already been reduced to “just” solving some particular S-unit equation.

The process of solving the S-unit equation is so convoluted that few have attempted to do it by hand. Jat306/shutterstock.com

How the solver works

Circumstances are changing, however. Since 2017, six number theorists across North America, myself included, have been building an S-unit equation solver for the open-source mathematics software SageMath. On March 3, we announced the completion of the project. To illustrate its application, we used the software to solve several open Diophantine problems.

The primary difficulty of the S-unit equation is that while only a handful of solutions will exist, there are infinitely many S-units that could be part of a solution. By combining a celebrated theorem of Alan Baker and a delicate algorithmic technique of Benne de Weger, the solver eliminates most S-units from consideration. Even at this point, there may be billions of S-units – or more – left to check; the program now tries to make the final search as efficient as possible.

This approach to the S-unit equation has been known for over 20 years, but has been used only sparingly, because the computations involved are complicated and time-consuming. Previously, if a mathematician encountered an S-unit equation that she wanted to solve, there was no automated way to solve it. She would have to carefully step through the work of Baker, de Weger and others, then write her own computer program to do the computations. Running the program could take hours, days or even weeks for the computations to finish.

Our hope is that the software will help mathematicians solve important problems in number theory and enhance their understanding of the nature, beauty and effectiveness of mathematics.

For more insights like this, visit our website at www.international-maths-challenge.com.
Credit of the article given to Christopher Rasmussen

Credit: Annalisa Crannell analyzes art with the help of chopsticks and projective geometry. Evelyn Lamb

To fully appreciate perspective art, mathematician Annalisa Crannell says both the artist and the art viewer need to do some math

This article was published in Scientific American’s former blog network and reflects the views of the author, not necessarily those of Scientific American

Annalisa Crannell goes to art museums with chopsticks. She is not unusually hungry or over-prepared; she uses them to figure out how to look at the art.

Crannell, a mathematician at Franklin and Marshal College in Lancaster, Pennsylvania, studies mathematical perspective and applies her work to the world of art. She writes not only about how artists use perspective but also about how viewers can use it to see art in different ways.

In a 2014 Math Horizons article (pdf, also available in The Best Writing on Mathematics 2015, edited by Mircea Pitici), she and coauthors Marc Frantz and Fumiko Futamura take on the case of the mysterious table in 15th- and 16th-century German artist Albrecht Dürer’s famous engraving St. Jerome in His Study.

Credit: St. Jerome in His Study, by Albrecht Dürer. Public domain, via Wikimedia Commons.

This work is an early example of mathematical perspective in art, but some critics have maligned Dürer’s technique. William Mills Ivins Jr., a former curator at the Metropolitan Museum of Art in New York, described it as “the oddest trapezoidal shape” and claimed it wasn’t even level with the floor. Crannell and her coauthors say it’s a matter of perspective. They write, “Surprisingly, the answers to these questions depend not only on what Dürer did 500 years ago, but also on what Ivins did in 1938. And, as we will show, it depends on what you, the reader, do when you look at St. Jerome today.”

Crannell and her coauthors describe how to use straight lines in St. Jerome in His Study to determine exactly where the viewer should stand to see the painting from the perspective Dürer probably intended, and therefore to see the table as a square instead of a trapezoid. The proper viewing location for that particular engraving turns out to be closer to the picture and farther to the right than most people would naturally stand. They write, “The oddness that Ivins saw in the table wasn’t because Dürer was in the wrong, but because Ivins was in the wrong, literally: he was looking from the wrong place!”

In February, I had the pleasure of going to the Brigham Young University Museum of Art with Crannell, and she shared some of her secrets with me. (The BYU Museum of Art does not normally allow photography, so I thank them for graciously making an exception for us.)

In perspective art, lines in the painting that represent parallel lines in the real world—say, train tracks or the opposite sides of a table—intersect on the canvas at so-called vanishing points. These vanishing points are the key to determining the optimal location from which to view a painting.

The most obvious way to find the vanishing points of a painting and thus to determine the optimal viewing location is to place rulers directly on “parallel” lines in the drawing, but shockingly, most museums frown on that practice. That’s where the chopsticks come in.

Standing in front of a piece of art, Crannell closes one eye and holds the chopsticks in front of her so they line up with lines in the artwork that represent parallel lines in the real world. The place the chopsticks appear to intersect is in front of the vanishing point of those lines. For art that has one vanishing point, the viewer should stand directly in front of that point. The viewing distance can be determined by trial and error or by some sneaky geometry with squares.

Credit: A diagram illustrating the orthocenter of a triangle. Each red line is one altitude of the triangle. Image: Public domain-self, via Wikimedia Commons.

For art with two vanishing points, the optimal viewing point is somewhere on the semicircle that connects the two vanishing points. For three vanishing points, determining the optimal vantage point is a bit more involved. It is at the intersection of three hemispheres, each one of which has two of the vanishing points as a diameter. Equivalently, it is somewhere in front of the orthocenter of the triangle whose vertices are the vanishing points. (The orthocenter of a triangle is the intersection point of the three altitidues of the triangle, as illustrated in the diagram on the right.) For a more complete description of how to find viewing points in art, check out Viewpoints: Mathematical Perspective and Fractal Geometry in Art by Crannell and Frantz.

In addition to giving me a way to look eccentric at the museum, Crannell’s technique helps me understand why some paintings seem to leap off the page, and some, even though they basically look realistic, don’t quite pop. In some pieces we looked at, lines that should have represented parallel lines in the real world didn’t end up determining a consistent vanishing point. Looking at any one part of the painting, nothing was clearly wrong, but the overall effect was slightly imperfect. When artists do manage to deploy perfect perspective—and viewers manage to find the correct vantage point—the effect can be startlingly realistic.

As Crannell and her coauthors describe it,  we can see the effect that the master geometer Albrecht Dürer intended. If you view St. Jerome in His Study [from the mathematically determined vantage point], you’ll see that the engraving takes on an amazing realism and depth. The gourd in the picture seems to hover over your head; you feel you could stick your hand in the space under the table; the bench off to the left invites you to come sit down and fluff up the pillows.

If you would like to know how to find those fluffy pillows for yourself, Crannell has information about projective geometry and perspective art on her website.

For more insights like this, visit our website at www.international-maths-challenge.com.

Credit of the article given to Evelyn Lamb |