Tag Archives: Mathematics

Scientists find no difference in math ability in the brains of boys and girls

Credit: Pixabay.

There’s a pervasive folk belief that girls are less biologically equipped than boys at math — and this may explain the gender gap in STEM fields. A new study by researchers at Carnegie Mellon University puts such myths to rest, showing no gender difference in brain functionality when performing math.

“We see that children’s brains function similarly regardless of their gender so hopefully we can recalibrate expectations of what children can achieve in mathematics,” said Jessica Cantlon, professor of neuroscience at Carnegie Mellon University and lead author of the new study.

We’re more similar than we are different

The researchers used functional MRI to scan the brains of 104 children, aged 3 to 10, while they watched an educational video covering various early math topics, such as counting and addition. The scans of the boys and girls were compared to evaluate any differences in brain activity. What’s more, the scans were compared to those taken from a group of adults who watched the same videos to examine brain maturity.

Cantlon and colleagues employed a range of statistical methods and comparisons, but none rendered any differences in brain development between boys and girls. According to the researchers, boys and girls were equally engaged with the educational material and exhibited the same brain functions when processing math skills. Lastly, the children’s brain maturity was statistically equivalent to either men or women in the adult group.

“It’s not just that boys and girls are using the math network in the same ways but that similarities were evident across the entire brain,” said Alyssa Kersey, postdoctoral scholar at the Department of Psychology, University of Chicago and first author on the paper. “This is an important reminder that humans are more similar to each other than we are different.”

Besides brain activity, the researchers also examined potential gender differences in mathematical abilities, as measured by standardized tests for 3- to 8-year-old children in a study involving 97 participants. The results showed that math ability was equivalent among boys and girls.

Cantlon says that as children grow up, gender differences in science and math abilities can surface due to the way boys and girls are socialized. She mentions studies showing that most American families encourage young boys to play games that involve spatial cognition. Parents also generally have different expectations from their children in terms of math abilities.

“We need to be cognizant of these origins to ensure we aren’t the ones causing the gender inequities,” Cantlon said.

In the future, the researchers would like to extend their study using a broader array of math skills, such as spatial processing and memory. They would also like to follow children over many years, preferably into adulthood to see how their math abilities and brain functions differ by gender.

The findings appeared in the journal Science of Learning.

This AI invents unique math we’ve never seen before

Credit: Pixabay.

Researchers have taken things to the next level by developing an AI that is basically a mathematical conjecture generator. Conjectures are mathematical statements that are suspected to be true but have not yet been rigorously proven. Any mathematician will tell you that these are their bread and butter, which they use to develop mathematical theorems. Now, we have computers that can feed mathematicians with new conjectures, which they’ll have to prove, and in the process might revolutionize the field.

The AI developed by the team at Technion-Israel Institute of Technology specifically deals with conjectures surrounding another fundamental element of mathematics: constants. In math, constants are key numbers with fixed values that emerge naturally from different mathematical computations and structures.

Take for instance pi, arguably the most important constant in mathematics. It gives the ratio between a circle’s circumference and diameter, which stays the same value for every circle, no matter how large. Other important fundamental constants include Euler’s number and the golden ratio.

Not anyone can make conjectures about such fundamental constants. In fact, this is something typically reserved for geniuses like Newton, Riemann, Gauss, or Srinivasa Ramanujan. The latter was so good at it that Ramanujan is credited for the discovery of thousands of innovative formulas in number theory — and he did so with no formal training, starting from a poor family background.

Srinivasa Ramanujan. Credit: Wikimedia Commons.

In honor of the great mathematician, the researchers named their AI the Ramanujan Machine. Like the late Indian genius, they hope that the AI becomes just as prolific at conjecturing unproven mathematical formulas.

The software has made its own conjectures that independently formulate well-known mathematical constants such as pi, Euler’s number (e),  Apéry’s constant, and the Catalan constant, as well as a couple of original universal constants.

“Our results are impressive because the computer doesn’t care if proving the formula is easy or difficult, and doesn’t base the new results on any prior mathematical knowledge, but only on the numbers in mathematical constants. To a large degree, our algorithms work in the same way as Ramanujan himself, who presented results without proof.”

“It’s important to point out that the algorithm itself is incapable of proving the conjectures it found — at this point, the task is left to be resolved by human mathematicians,” said Assistant Professor Ido Kaminer from the Faculty of Electrical Engineering at the Technion.

For thousands of years of mathematical history, conjectures were reserved for rare genius. This is why we only have a few dozen important formulas discovered in the last hundred years of research. But in a few hours, the Ramanujan Machine “re-discovered” all the formulas for pi discovered by Gauss, which took him a lifetime of work, as well as dozens of new formulas that were unknown to Gauss.

“Similar ideas can in the future lead to the development of mathematical conjectures in all areas of mathematics, and in this way provide a meaningful tool for mathematical research,” wrote the researchers in their study published in Nature.

The researchers launched a website where the public can find algorithmic tools that anyone can use for the advancement of mathematical research.

Solved for 42: long-lasting math problem finally gets its answer

Here’s a problem which sounds simple enough, but isn’t: find three numbers so that the sum of their cubes is equal to 42. A planetary scientist just solved the problem, and the numbers will definitely surprise you.

The original problem was laid out in 1954 at the University of Cambridge, and was very straightforward: find the solutions for x3+y3+z3=k, with k being all the numbers from one to 100. Some of them are pretty obvious. Just take 1, 2, and 3, sum their cubes, and you end up with 1+8+27=36 — so you have an answer to 36. You can be a bit clever and use -1 instead of 1, which leaves you with -1+8+27=34, and you have another solution. After you take out all these easy solutions, you’re left with some weird ones though.

There should be a solution for all numbers, but two proved particularly different to crack: 33 and 42. Thanks to a creative approach (and a week at a world-leading supercomputer), Professor Andrew Booker managed to solve it for 33. Coincidence or not, the only remaining number was 42 — which fans of the Hitchhiker’s Guide to the Galaxy will recognize as “the answer to Everything”, according to a fictional computing machine that worked for 7 million years.

Booker tried to solve the equation for 42, but couldn’t — so he turned to MIT maths professor Andrew Sutherland. Together, the two used a machine that’s surprisingly similar to the fictional one described above. The platform is called Charity Engine: a ‘worldwide computer’ that harnesses idle, unused computing power from over 500,000 home PCs to create a crowd-sourced, super-green platform made entirely from otherwise wasted capacity. It didn’t take 7 million years, but it did take quite a long time. Yet, at the end of it all, the two had their answer.

Are you ready?

The three numbers for which the sum of their cubes is equal to 42 are:

  • X = -80538738812075974
  • Y = 80435758145817515
  • Z = 12602123297335631

These three incredibly large numbers are the solution to the problem — trust us, we did the math.

We checked.

But while doing the math is easy if you have a big calculator, it’s not easy to reach these numbers in the first place. These 17-digit monsters are so big that brute-forcing calculations just won’t work. Instead, researchers used clever algorithms to help finesse the search. But there was no guarantee if or when they would find the solution. Booker, who is based at the University of Bristol’s School of Mathematics, says he’s glad the search is finally over:

“I feel relieved. In this game it’s impossible to be sure that you’ll find something. It’s a bit like trying to predict earthquakes, in that we have only rough probabilities to go by. So, we might find what we’re looking for with a few months of searching, or it might be that the solution isn’t found for another century.”

With ingenuity and sufficient computing power, you can solve almost any problem. Who knows, one day we might even learn the answer to Everthing — and it might be 42.

How simple subtractions can stump even mathematicians — and why that matters for understanding our brains

Math is often regarded as the purest and most elegant form of problem-solving. But a new study claims that our mathematical thinking is often muddled by real-life knowledge. As weird as it sounds, our day-to-day information can get in the way of mathematical calculations — and this can happen to anyone, even experienced mathematicians.

When we learn to solve problems in school, we’re often given real-life scenarios. Jake buys a bunch of melons, then loses some of them, how many melons does he have left? Whether it’s melons, apples, or dividing flowers between vases, we’re taught at an early age to think of math in a practical context. While that approach teaches kids the practical applicability of mathematical calculations, it might also be counterproductive in some situations.

In a new study, researchers report that in some cases, worldly knowledge interferes with mathematical reasoning.

“We argue that such daily-life knowledge interferes with arithmetic word problem solving, to the extent that experts can be led to failure in problems involving trivial mathematical notions,” the  study reads.

They designed twelve problems which they presented to two groups. The main focus was the way in which the problems were presented. They were exactly the same problems, but they were presented in a different way.

“We devised six 5th grade subtraction problems (i.e. for pupils aged 10-11) that could be represented by sets, and six others that could be represented by axes”, begins Emmanuel Sander, an FPSE professor. “But all of them had exactly the same mathematical structure, the same numerical values and the same solution. Only the context was different.”

Half of the problems could be viewed as sets. Whether it’s the number of animals in a pack, the price of a meal in a restaurant or the weight of a stack of books, they all involved elements that can be grouped together in sets. For example:

  • Sarah has 14 animals: cats and dogs. Mehdi has two cats fewer than Sarah, and as many dogs. How many animals does Mehdi have?

The second type of problems (the axes ones) asked participants to calculate things like how long it takes to build a cathedral, to which floor an elevator arrives or how tall a Smurf is. Here’s an example:

  • When Lazy Smurf climbs onto a table, he attains 14 cm. Grumpy Smurf is 2 cm shorter than Lazy Smurf, and he climbs onto the same table. What height does Grumpy Smurf attain?

There’s a mental trick to the way these problems were designed. For instance, you can solve them through a simple subtraction: 14 -2 = 12. But when it comes to sets, the same approach doesn’t work.

Take the animal question with Sarah’s cats and dogs. Instinctively, you’d want to calculate how many cats and dogs Mehdi has — but you can’t. You can solve the problem and calculate how many animals he has, but not how they are divided between cats and dogs. The mathematical structure is identical: it’s the same simple subtraction, 14 – 2 = 12.

Scientists had a hunch that these answers would be a bit more difficult to answer, despite their identical mathematical structure. The context, they argue, makes it somewhat harder to process. Some problems were more difficult than others, but they all followed the same line

But even they weren’t expecting the results to be this striking.

In the non-expert adult group, 82% answered correctly for the axis problems, compared to only 47% for the problems involving sets. Surprisingly, in over more than half of the time (53%), respondents thought that there was no solution to the statement, which the team interprets as reflective of their inability to detach themselves from the elements of the problem.

Even expert mathematicians sometimes struggled with this. A total of 95% answered correctly for the axis problems, but that rate that dropped to only 76% for the sets problems. In other words, 1 out of 4 times, the experts thought there was no solution “even though it was of primary school level,” the study reads.

“We even showed that the participants who found the solution to the set problems were still influenced by their set-based outlook, because they were slower to solve these problems than the axis problems”, continues Hippolyte Gros, a researcher in UNIGE’s Faculty of Psychology and Educational Sciences and one of the study authors.

While the sample size was relatively small and the study design has significant limitations, the results are still intriguing. They seem to suggest that even in mathematical thinking, we are highly dependent on context. Even those who have the capacity to address the problems can suffer from these cognitive biases and be tricked into not finding the answer to a simple problem.

This isn’t the first study to suggest that our mathematical or scientific reasoning can be aided or hindered by semantic context. Given the wide scale at which these findings can make a difference in the education system, it seems there is a need to better understand the full impact of the semantical context.

The study has been published in Psychonomic Bulletin & Review.

New mathematical model describes the growth pattern of plant leaves

Japanese researchers have described one of nature’s most ubiquitous patterns: a model which accurately describes how leaves grow on plants.

“We developed the new model to explain one peculiar leaf arrangement pattern. But in fact, it more accurately reflects not only the nature of one specific plant, but the range of diversity of almost all leaf arrangement patterns observed in nature,” said Associate Professor Munetaka Sugiyama from the University of Tokyo’s Koishikawa Botanical Garden.

Plant leaves have fascinated mankind since time immemorial. Some, like the sunflower, grow in a remarkably ordered geometry. Others seem to be much more chaotic, not subject to any apparent rule. There’s even a special name to describe the growth pattern of leaves on a plant: phyllotaxis.

Understandably, mathematicians have also been fascinated with these patterns. Leonardo da Vinci made observations of the spiral arrangements of plants, although he did not leave behind too many comments. Later on, naturalists noted that the spiral phyllotaxis of plants was sometimes clockwise and otherwise anti-clockwise, but it seemed to follow the so-called mathematical golden ratio.

It became clear that many plants follow a mathematical distribution, but no one was able to find a universal law to describe leaf growth.

In 1996, though, researchers got really close. Douady and Couder developed an algorithm that could account for many, but not all leaf arrangement patterns. This became known as the DC2 equation, and to this day, it is used to infer different variables of plant physiology.

Now, Japanese researchers believe they have found an even better rule, which can account for all the patterns in plants.

Unruly exceptions

They started out from a group of plants called “orixate”, from the species Orixa japonica, a shrub native to Japan, China, and the Korean peninsula. Orixate plants are part of an unruly group that doesn’t obey the DC2 equation. The angles between O. Japonica leaves are 180 degrees, 90 degrees, 180 degrees, 270 degrees, and then the next leaf resets the pattern to 180 degrees.

Leaves on an O. japonica branch (upper left) and a schematic diagram of orixate phyllotaxis (right). The orixate pattern displays a peculiar four-cycle change of the angle between leaves (180 degrees to 90 degrees to 180 degrees to 270 degrees). A scanning electron microscope image (center and bottom left) shows the winter bud of Orixa japonica, where leaves first begin to grow. Primordial leaves are labeled sequentially with the oldest leaf as P8 and the youngest leaf as P1. Image credits: Takaaki Yonekura, Akitoshi Iwamoto, and Munetaka Sugiyama.

At least four other unrelated plants groups follow a similar pattern. Sugiyama and colleagues wanted to see if they could find another equation to describe these plants, starting from the fundamental genetic and cellular machinery shared by all plants. The reason they took this approach is that if four separate groups all evolved this pattern, then it seems likely that there’s an underlying reason for it. Having it randomly pop up 4 times is just too unlikely.

They started from the two main shortcomings of the DC2 equation:

  1. No matter what parameters you put into it, some leaf arrangements are just not accounted for.
  2. The Fibonacci spiral leaf arrangement pattern is the most common spiral pattern observed in nature but is only modestly more common than other spiral patterns calculated by the DC2 equation.

To address these, the team focused on one key assumption of the equation: that leaves emit a constant signal to inhibit the growth of other nearby leaves. This makes sense because the plant would want some balance, and there is some research suggesting that this signal is propagated through a hormone called auxin, although the exact mechanism is not yet clear.

Sugiyama did away with the assumption that this signal was constant.

“We changed this one fundamental assumption – inhibitory power is not constant, but in fact changes with age. We tested both increasing and decreasing inhibitory power with greater age and saw that the peculiar orixate pattern was calculated when older leaves had a stronger inhibitory effect,” said Sugiyama. In other words, the older a leaf is, the less likely it is for new leaves to grow in its direct vicinity.

The resulting equation was not only capable of explaining the growth pattern of orixate plants but fit much better with the pattern observed in all plants, researchers claim.

“Our research has the potential to truly understand beautiful patterns in nature,” said Sugiyama.

The most common leaf arrangement patterns are distichous (regular 180 degrees, bamboo), Fibonacci spiral (regular 137.5 degrees, the succulent Graptopetalum paraguayense), decussate (regular 90 degrees, the herb basil), and tricussate (regular 60 degrees, Nerium oleander sometimes known as dogbane). Image credits: Takaaki Yonekura, Akitoshi Iwamoto, and Munetaka Sugiyama.

However, there are still some shortcomings with this model — while it did account for most of the exceptions, it didn’t account for all the exceptions.

“There are other very unusual leaf arrangement patterns that are still not explained by our new formula. We are now trying to design a new concept that can explain all known patterns of leaf arrangement, not just almost all patterns,” said Sugiyama.

It remains to be seen if biologists or other researchers working with this equation will confirm its results and incorporate it into their work. For now, the relationship between mathematics and botany seems to have gotten even deeper.

Journal Reference: Takaaki Yonekura, Akitoshi Iwamoto, Hironori Fujita, Munetaka Sugiyama. 2019. Mathematical model studies of the comprehensive generation of major and minor phyllotactic patterns in plants with a predominant focus on orixate phyllotaxis. PLOS Computational Biology. DOI: 10.1371/journal.pcbi.1007044.

Karen Uhlenbeck officially receives the Abel Prize — the first woman ever to do so

It’s not common for mathematicians to make the front page, but this is exactly what happened today — in Oslo, at least. The Norwegian capital was teeming with excitement for the Abel Prize ceremony, one of the highest distinctions in mathematics. This year, the award was won by Karen Uhlenbeck who, aside from her pioneering contributions in the fields of gauge theory and geometric partial differential equations, also had a fundamental impact on analysis, geometry, and mathematical physics. In the packed aula at the University of Oslo, she received the well-deserved prize from none other than the King of Norway himself. For a brief moment, the world got a glimpse inside the life of one of the world’s most brilliant mathematicians.

Uhlenbeck receiving the prize.

Esoteric, wonderful, and very useful

Uhlenbeck’s work, like that of any eminent mathematicians, seems esoteric for all but a small group of mathematicians. As it’s so often the case in modern science, overspecialization makes it difficult for researchers to spread their wings in other fields.

This is, in part, what makes Uhlenbeck’s work so special, even among other researchers: her work stretches across multiple fields with remarkable ease. For instance, Uhlenbeck provided remarkable insights on “minimal surfaces” — something commonly referred to as “bubble theory”. This work has given theoretical physicists the tools to describe complex surfaces such as soap bubbles, which are challenging to address mathematically. Uhlenbeck tackled the great puzzles posed by such surfaces, as well as their interaction with physical forces such as electromagnetism and nuclear forces. She also published remarkable work on gauge theory, a class of quantum field theories used, among others, to describe elementary particles and their interactions. Gauge theory is one of the building blocks of the Standard Model and is also used to consider the overall shape of the universe. There too, she excelled.

This proved to be a common theme for Uhlenbeck who, despite substantial adversity, achieved remarkable success.

She was the first woman in 58 years to give a plenary lecture at the International Congress of Mathematicians, after Emmy Noether in 1932 — such a shocking figure reveals just how difficult it is for women to succeed in such a male-dominated field. Now, she’s the first woman to ever receive the Abel Prize, arguably the highest honor in the field. Yet for all her amazing achievements, there’s a very gracious and down-to-earth air about her.

“I think I’m a very lucky mathematician,” she says. She was inspired by the Sputnik, the first satellite sent into space, and the feminist movement. But at the end of the day, luck can only go so far. “I took advantage of my luck,” she continues. “If people are lucky, they often don’t realize it, and if you’re lucky you don’t notice it and you don’t do anything — it’s a wasted opportunity.”

Thankfully, she did notice.

Different colors

Uhlenbeck is not your typical mathematician — she herself will say that much. At some point during the 70s, she attended a university interview wearing running shoes. Of course, that wouldn’t turn too many heads, but at the time, it was shocking. This tiny event in such a long career is very telling: it’s the mark of someone who just wants to do math.

Image credits: IAS.

It wasn’t always like this. She was never supposed to be a mathematician; it just sort of happened along the way. She wanted to study astronomy — but not by looking at the stars, she says. Rather, by using numbers to understand the universe.

“As a girl, I was not designated for a particular profession. During my 2nd year of high school, the Soviets launched Sputnik and the US realized that it wasn’t producing enough mathematicians and scientists and I benefitted from this. It was such a specific goal that women and minorities were specifically included in these programs. I was much encouraged by my professors,” she recalls.

Despite describing adversity from a male-dominated system which was reluctant to change, Uhlenbeck stubbornly refuses to claim any special merits. Give women equal chances and you will get equal results, she says.

“Everyone makes a big deal that I’m the first woman to win the Abel prize”, she says as she scribbles a theory that took several years to develop on a blackboard. “I was just ahead of the pack,” she adds.

“All you have to do is look at history — I was in the first generation of women for which the opportunities were available.”

Yet despite her firm position, it’s hard not to notice that she is special. At 76 years old, she is still a brilliant mathematician who likes to study the world and take on complicated problems from the comfort of her favorite chair. There’s also something distinctively whimsical about her way of seeing things.

“I write my journal in different colors,” she says, and then giggles. “Just because I like colors.”

A love of mathematics

Mathematics is often strange, especially modern mathematics. It’s easy to see how it lies at the core of virtually all the science we have developed, and yet, it’s hard to know where solving a math problem will take you. Sometimes, it will take years or decades before there is a use for it. For some problems, there may never be a practical application — but that’s completely fine, says Uhlenbeck.

“The role of mathematics is long-term. It doesn’t solve a problem, it creates a language which is used to later solve problems. You would expect things to have consequences years in advance but at the time, it can be really esoteric. Think of Einstein’s Theory of Relativity. It’s very mathematical, and you needed to find good examples to test this theory. It took a few years before there were studies to see that light curves… and now it’s used in your cell phones — but that’s 100 years later!”

At the end of the day, math is much like philosophy — its usefulness is much about teaching people how to think. It’s a fundamental endeavor, even without being practical. Uhlenbeck hopes that people can learn from mathematics, understanding that in modern society, we need farsighted thinking — the kind of thinking we have with mathematics.

“[I’d like] to convince people to think in the long run — stop people from thinking about what’s happening now and think more long-term; and also value ideas for their own sake! Human beings have benefitted tremendously from philosophy, art, music — and none of this is practical, it’s all part of human endeavor. If I sit down and think about what is really fundamental about human endeavor, I think mathematics is part of it.”

So what about her? After such a long and fruitful career, has she had enough of mathematics?

“Pure mathematics is an amazing subject, and I feel very privileged,” she says.

Her eyes sparkle with the unmistakable glow of someone who will forever enjoy their work, and you don’t need too many words (or numbers) to express that.

“I still like to do math,” she ends.

 

Who is Karen Uhlenbeck — the First Female Recipient of the Abel Prize

For her “pioneering achievements in geometric partial differential equations, gauge theory, and integrable systems and for her “fundamental impact” on analysis, geometry, and physics, Karen Keskulla Uhlenbeck has received the 2019 Abel Prize — probably the most prestigious award a mathematician can ever claim, widely regarded as the “Nobel Prize” of math.

Uhlenbeck, a staunch supporter of women in science and math, was the first woman to have ever been awarded the distinction.

A mathematician’s trajectory

Image credits: IAS.

In 1990, in Kyoto, Japan, Karen Uhlenbeck gave a plenary lecture at the International Congress of Mathematician (ICM). She was only the second woman to ever do so, after Emmy Noether in 1932. By the time she addressed the ICM, Uhlbeck was already a well-established name, and one of the most prominent mathematicians in the world.

Although she did receive awards and accolades throughout her career, many of her colleagues felt like her recognition should have been far greater — and she was barely able to have a career in mathematics at all.

As a child, she was always curious about everything. She enjoyed reading everything she could, regularly shutting herself away so she could read, often late at night, sometimes even during her school classes.

Her love affair with mathematics happened rather spontaneously. She had wanted to do physics, but math provided her with more satisfaction, and it also meant that she didn’t have to do any lab work — which she resented greatly. Even as her career progressed, she enjoyed the peace of solitary work.

A brilliant and promising mind coupled with a great desire to succeed in research seemed to put her on a trajectory for success. But things weren’t nearly as simple.

The two body problem — and nepotism

In science, it’s not uncommon for both members of a couple to be active researchers, either in similar or different fields. This makes a lot of sense — that’s how the life of a scientist often goes about. But this also leads to some very practical problems when it comes to finding a job. Finding a research job is hard enough, but finding two in the same city is a very difficult challenge. Among scientists, this is often informally referred to as the “two-body problem.”

[panel style=”panel-default” title=”The two-body problem” footer=””]Academics tend to move a lot chasing job offers — and many of them have life partners also working in academia. It’s estimated that than 70% of academic faculty have a working partner, while more than 30% have an academic partner.

This leads to substantial difficulties in finding two jobs at the same university or within a reasonable commuting distance from each other. This forces many couples to either split up or one of the partners to give up on his or her career. This is called the two-body problem, in allusion to the insolvable three-body problem in classical mechanics.[/panel]

Uhlenbeck experienced this firsthand as she tried to find work along with her then-husband, Olke. She was turned down several times under so-called nepotism regulation: an alleged ban on both husband and wife working at the same institution. Many, including herself, felt this was just a pretext to avoid hiring women as part of the faculty. Decades later, Uhlenbeck tried raising this issue, the universities simply denied it.

“I was told, when looking for jobs after my year at MIT and two years at Berkeley, that people did not hire women, that women were supposed to go home and have babies. So the places interested in my husband – MIT, Stanford, and Princeton – were not interested in hiring me. I remember that I was told that there were nepotism rules and that they could not hire me for this reason, although when I called them on this issue years later they did not remember saying these things.”

Thankfully, she was able to find a generous scholarship at Brandeis University, which would prove to be a turning point for her. It was there that she completed her PhD with Richard Palais as her adviser. Palais was trying to find a link between analysis (a generalization of calculus) and topology and geometry (the structure of shapes). He was also supportive of her, providing a very fruitful collaboration for Uhlenbeck.

Uhlenbeck’s work has applications in surfaces like soap bubbles.

She then moved on to MIT, Berkeley, and the University of Illinois at Urbana-Champaign, where both she and her then-husband were professors. She felt underappreciated there, a mere “faculty wife”. So she kept moving on, reaching the University of Chicago. It wasn’t until Chicago that she received the support of other faculty — there were other female professors there, who gave her valuable guidance. She finally felt like she belonged.

She had already made a name for herself and received the praise of colleagues, but in Chicago, Uhlenbeck was able to make a true mark, establishing herself as one of the most outstanding mathematicians of her generation. Her interests were broad and varied, spanning over fields such as nonlinear partial differential equations, differential geometry, gauge theory, and quantum field theory.

Ultimately, she became a household name in top mathematics but she was a rather unusual presence — not just because a woman, which was and still is relatively rare in top scientific circles, but because of her way of thinking. Mark Haskins, a mathematician at the University of Bath, UK, who was one of Uhlenbeck’s doctoral students, described her as having a non-linear and non-intuitive way of thinking. She would receive a question from someone and give them a baffling answer.

“Your immediate reaction was that Karen had misheard you, because she had answered a different question,” Haskins says. But weeks or months later, you’d realize “you haven’t asked the right question — and the answer to the right question was what Karen said.”

She definitely represents a very specific way of thinking about math, and about things in general. She is one of those mathematicians who appear to have “an innate sense of what should be true”, even if they cannot always explain why, Haskins recalls. The inquisitive mind of the young little girl is still very much present in mature Uhlenbeck.

“If I really understanding something, I’m bored” Haskins remembers her saying.

She also remained stunningly humble in her success. Haskins remembers that even as a professor, Uhlenbeck regularly attended other seminars, even junior seminars — something which for most mathematicians, is simply not a consideration.

Bubbles and the Abel Prize

Instantons are extremely important to physicians, and some have been used to describe the fundamental properties and evolution of the universe, such as the Hawking-Turok Instanton, depicted here. Image credits: Cambridge University.

In a phone interview for the Abel Prize, Uhlenbeck said she was “a bit overwhelmed,” adding that “I hope I can hold myself together for this.” The reasons why her work warranted the Abel Prize are worthy of a separate article — or rather, a book. Indeed, a book she co-authored called Instantons and Four-Manifolds has become a staple for mathematicians in the field.

Instantons were of great interest to Uhlenbeck — these are solutions to equations of movement which define motion as a function of time. They are also useful in quantum mechanics, to describe the probability for a quantum mechanical particle tunneling through a potential barrier.

Perhaps her most important work was on so-called ‘minimal surfaces’. Soap bubbles, beautifully spherical objects are capable of forming minimal surfaces in an instant. They do so in order to minimize the wall tension and energy, by pulling the bubble into the smallest possible shape — known for centuries to be a sphere. But computing these minimal surfaces can be a daunting task, and in more complex situations, the theory simply broke down.

For instance, if instead of blowing the bubble we would dip a deformed wire loop into the soap bubble solution, it would form a disc, with its boundary given by the wire loop and of minimal area, explains Arne Sletsjøe, assistant professor of Mathematics in the University of Oslo. In this situation, soap bubbles might still form the area of minimal surface instantly, but computing and predicting it is a very different matter. Uhlenbeck attacked this problem from a mathematical perspective and expanded our understanding of such surfaces.

A helicoid minimal surface formed by a soap film on a helical frame. Image credits: Blinking Spirit / Wikipedia.

Uhlenbeck also made her mark in gauge theory, which is used in powerful theories in physics. A key idea of Einstein’s work, something still regarded as the Holy Grail of physics, is that laws of physics should be the same in all frames of references — and this is also the general idea of a gauge theory: finding connections that compare measurements taken at different points in space and looking for quantities that don’t change. Working at the interface of mathematics and physics has been a point of focal interest for Uhlenbeck.

Of course, this doesn’t even begin to scratch the surface of Uhlenbeck’s fundamental contribution. She is a founder of geometrical analysis, and her work stretched to cover other fields such as integrable systems and harmonic mappings. Just a few months ago, Uhlenbeck worked with Penny Smith to submit another paper on gauge field equations.

The woman who could

Despite some progress, women in math and science still face great adversity.

“We all thought that once the legal barriers were down, women and minorities would walk through the doors of academia and take their rightful place,” Uhenbeck once said.

Uhlenbeck spoke how she felt she was the first generation of female mathematicians who could truly succeed in the corrosive culture of mathematical publishing — and scientific publishing in general. But Caroline Series, the president of the London Mathematical Society, said she would have said the same thing — despite being 10 years younger than Uhlenbeck; and if you would ask younger female mathematicians, you almost invariably get the same result: they feel like they barely get through.

“I think if I were five years older, I wouldn’t have made it. It’s better now, but I worry a lot about the other minorities. There are many women and minorities capable of being great mathematicians,” Uhenbeck said after receiving the prize.

Needless to say, Uhlenbeck did manage to break through, but many others might not.

In 2007, when Uhlenbeck was awarded the American Mathematical Society’s Steele Prize for Seminal Contribution to Research in 2007, she pointed a finger at the mathematical community, blaming it for the small number of leading women mathematicians. She summed it up by saying that “changing the culture is a momentous task in comparison to the other minor accomplishments I have mentioned”.

Uhlenbeck has inspired a generation of mathematicians, and as she wrote in 1996, she is painfully aware of this. It’s a hard job because “what you really need to do is show students how imperfect people can be and still succeed. … I may be a wonderful mathematician and famous because of it, but I’m also very human.” For all her immense talent and hard work, Uhlenbeck is, certainly, very much human.

Karen Keskulla Uhlenbeck is a founder of modern geometric analysis. Her work has opened the door for some of the most dramatic advances in mathematics (and physics) over the past 40 years. She has come a long way from the girl who would secretly read in courses, and in addition to having a stellar career in mathematics, she is also a relentless advocate of women’s’ and minorities’ rights in science.

Her research has already inspired a generation of mathematicians, and she now hopes that her winning the Abel Prize will inspire even more girls to pick up mathematics and science. We can only hope for the same thing.

Time travel Anime inspires solution to puzzling math problem

Sometimes, the internet is extremely weird and beautiful at the same time — and this is a perfect example.

The problem

Formally, the math problem can be expressed thusly:

“What is the shortest string containing all permutations of a set of n elements?”

In a more “common language”, the problem sounds something like this:

“Say you want to watch a series with n episodes. You want to watch all the episodes in every combination possible. Overlapping is allowed, but the sequence must be continuous. For instance:

  • for a series with 2 episodes, 1-2-1 is a solution, because it contains both possible combinations (1-2 and 2-1);
  • for a series with 3 episodes, 1-2-1-3 is NOT a solution, because it does not contain the sequence 1-2-3. The solution is 1-2-3-1-2-1-3-2-1, as it contains all possible sequences (1-2-3, 1-3-2, 2-3-1, 2-1-3, 3-1-2, 3-2-1).

What is the least number of episodes you have to watch?

Anime maths

The problem is surprisingly complex and has remained as a rather obscure mathematical puzzle since 1993, when a demonstration was attempted and subsequently proven incomplete. Recently, Robin Houston, a mathematician and computer scientist, tweeted about finding what seems like a solution to this problem, in the unlikeliest of places: a board on an anime — Haruhi, a 2006 anime based on a series of Japanese light novels.

The reason why Haruhi became linked to this problem is also unusual. The series contains a lot of time travel and is overall very difficult to follow, as the chronology becomes very confusing. To make matters even funkier, when the series went to DVD, the episodes were rearranged, making viewers feel like they were watching a different chronology. Essentially, you can watch episodes in a number of different orders, which has become something of an obsession among fans.

Scientifically, the situation is extremely unusual. The solution seems to work, Houston points out, but mathematicians seem reluctant to address it formally since it’s not presented in a journal. But things get even more interesting.

Houston did a bit of digging and found that the proof was first submitted to 4chan — one of the darker corners of the internet, where threads are only kept for a limited time, though Houston was able to find a permanent mirror. Furthermore, 4chan is entirely anonymous, meaning we don’t know who submitted it, and it’s nigh impossible to verify the author’s identity.

So this anonymous proof posted on 4chan and reposted to an anime board is currently the most elegant solution to the problem. Another mathematician, Jay Pantone, transcribed the proposed solution into a formal layout — and he says it stands up.

Solution

So what is the solution? Well, for Haruhi’s 14 episodes, you’d need to watch at least 93,884,313,611 episodes to be sure you’ve watched all possible combinations. At most, you’d need to watch 93,924,230,411 episodes. Now, mathematicians are working on a more formal version of a formula. The explanation is quite long and difficult to follow, but you can read:

Renowned mathematician Michael Atiyah claims to have solved the Riemann Hypothesis

Mathematics doesn’t usually make headlines and yet, to say that the announcement from Sir Michael Atiyah caused a stir would be an understatement. The renowned mathematician claimed to have solved the long-standing Riemann Hypothesis, with potentially massive implications for worldwide digital security. Well-aware of the skepticism that would surround his announcement, Atiyah pushed on, announcing that he would present his “simple” proof at the Heidelberg Laureate Forum.

Michael Atiyah, speaking at the International Congress of Mathematicians in 2018.

This is a continuation of a previously published article, which you can read hereFor a brief overview of the Riemann Hypothesis itself, check the bottom of the article.

There’s something inherently romantic about solving a math problem — a sort of man-against-nature kind of endeavor — and although most people seemed skeptical of Atiyah’s announcement, I’d dare say that deep down, we all hoped he would come out a victor. In Heidelberg, Germany, the stage was set.

It’s not just that the hypothesis is inherently challenging to solve — it stood unsolved for almost 160 years — but the fact that Atiyah himself claimed this solution struck people as unusual. It’s not that anyone doubted his ability: having won the two most prestigious awards in math (the Fields Medal and the Abel Prize), he’s one of the most renowned and respected mathematicians alive today. But most people in the field make their big findings in the earlier stretches of their career, before they’re 40. At a ripe 90, Atiyah stands in stark contrast, and he says recent papers tend to get rejected because people doubt his ability due to his old age.

So few people (if any) knew what to expect coming into the presentation, which made things all the more exciting, of course. The presentation itself was nothing if not entertaining — which is something you wouldn’t really expect, although Atiyah prides himself on his ability to explain everything at a fairly simple level.

“Solve the Riemann and you become famous. If you’re famous already, you become infamous,” he quipped in an almost hasty David Attenborough tone.

[panel style=”panel-info” title=”Hypothesis vs Theory” footer=””]Riemann’s hypothesis has been repeatedly observed to be accurate over a wide range of domains. However, it was never fully demonstrated — which is why it’s a hypothesis, rather than a theory. In science, ‘theory’ has a very different (and strong) meaning, as opposed to how ‘theory’ is used in our day to day lives.[/panel]

But as the minutes went by, it became clear that Atiyah wasn’t squeezing any long, compelling proof into the 45-minute presentation. At points, it felt like an introduction to mathematical history, with numerous sidetracks and backtracks. But there was also a sense of anticipation — a sense that the simple slides he was presenting were more like puzzle pieces, inconspicuously falling into place to reveal a much bigger picture.

Atiyah wasn’t even looking to solve the Riemann hypothesis — he was working in physics, trying to derive something called the fine structure constant. But sometimes, he says, when you solve problem A, you might end up solving problem B and not even know about it — this transposition of ideas makes math so great. The Riemann hypothesis was merely a problem B, something that came along almost accidentally.

Then, there it was — the “punchline”, as Atiyah himself referred to it: a relatively simple slide, with only a few lines. This is where all the “meat” of the demonstration is.

Michael Atiyah’s “punchline” for the Riemann Hypothesis.

If you’re not a mathematician (or if you are, but work in a different field), that probably looks like gibberish — and that’s fine, we won’t go into specifics; in all honesty, we couldn’t, even if we wanted to. But here’s the thing: extraordinary claims require extraordinary proof, and extraordinary proof rarely fits into one slide.

There was a sense of bewilderment after the presentation finished. When it was time for the questions, no one stood up and no one raised their hand, and for the next 30 seconds, you could cut the atmosphere with a knife. Timidly, one young man raised his hand — and he spoke the words that were on everyone’s mind: has the Riemann hypothesis been proven?

The comments which could be heard in some sections of the room were unflattering at best, and don’t belong in any scientific environment. But the concerns raised are, in a general sense, valid. Atiyah made several assumptions which permitted his leap. For instance, he worked with something called infinite iterations — something which is extremely dangerous; it’s the mathematical equivalent of treading on thin ice. Atiyah says he wouldn’t trust himself enough to take this step, but he builds on the work of John von Neumann, widely regarded as the foremost mathematician of his time (first part of the 20th century). Empowered by von Neumann’s work, Atiyah felt confident enough to claim the solution to this problem.

There was also an issue regarding the scope of his demonstration — had he really “solved” the problem, or has he merely tackled a particular aspect about its application? Atiyah’s answer to the question from the audience was, like the entire speech, charming, but left a bit more to be desired:

“This is just the first step on a long road. But yes, the first step, the solution to the problem, I proved that,” he stated, right after he said that he thinks he deserves the Millennium Prize for solving the problem.

Naturally, the problem of verification popped up afterward. Mathematicians worldwide would like to poke and prod around this proof, and they will have the chance to do so. This is where things took an unpleasant turn.

Atiyah wants to be believed — of course he does. A mathematician of his caliber should not settle for anything else and communication is, after all, the end game of any study. But it can be quite hard to get your message across when you’re 90 years old.

“I do care who believes it. Mathematics involves two steps: creation and dissemination. If you don’t disseminate your ideas, you don’t get anywhere,” he said. But when you’re 90, it’s extremely difficult to publish, he adds.

“When you’re my age, people don’t really want to publish my papers. You’re too old, they say.”

As a knee-jerk reaction, you’d want to say this can’t possibly be true. Surely, his work is subjected to the same scrutiny as all others, regardless of age. But the inherent bias is hard to deny, even in a room full of Atiyah’s peers. In the aftermath of his presentation, age seemed to be the most common topic of discussion — perhaps even more so than the math.

Sexism is a big problem in science, Atiyah rightly points out, but so is ageism, he adds. At the end of the day, one can only feel that his paper should be judged by its own value — regardless of whether it comes from an Abel Prize winner or a 90-year-old man — as a paper coming from a mathematician, simple as that.

At this point, it’s not clear whether his papers were reviewed “double-blind” or if the reviewers were aware of who submitted the paper. We don’t know if age really was the decisive factor in the initial rejection of the paper, or if it’s simply invalid — but age shouldn’t be a decisive criterion.

Now, at the very least, his paper (see here) will receive attention and scrutiny. The few people who can truly attest to its worth will presumably review it thoroughly, and while it may take a while (math proofs are often difficult to confirm or invalidate), when the dust will settle, we’ll find out whether we finally have a solution for the Riemann Hypothesis or not. Regardless of that outcome, Atiyah made us all think about the largely overlooked problem of ageism in science.

His presentation was charming and entertaining — which in this context, almost feels like a sin. The stage was brilliantly set, but the final outcome is, as of yet, undecided, and the lead actor’s performance not entirely convincing. Has he really identified a solution? We’ll likely know soon enough. But he’s certainly highlighted a new problem:

[panel style=”panel-default” title=”The Riemann hypothesis” footer=””]The Riemann hypothesis starts with prime numbers — rather strange numbers which can’t be divided by other numbers; one of the oddities of prime numbers is that their distribution is irregular — there’s no precise method to predict where the next prime number will occur. This unpredictability has been widely used in developing digital security systems.

While looking at prime numbers, Berhard Reimann, one of the most prolific German mathematicians, realized something interesting: the distribution of these prime numbers isn’t random at all, it’s very similar to a function, called the Riemann Zeta Function, described below.

ζ(s) = 1/1s + 1/2s + 1/3s + 1/4s + …. up to infinity

This is where it starts to get tricky. The variable can take any value, and Riemann’s work gives us an explicit formula for the number of primes in a given interval. This is done in terms of the so-called ‘zeros’ of the function — the values of under which the function becomes 0. You can think of this function as a way to predict the distribution of prime numbers.

Riemann observed this in action, but he was never really able to prove it. This is why this is still a hypothesis, and not a theory (which, in science and math, has a much stricter meaning than in regular talk). It still remains to be seen whether this will still be the case.

[/panel]

Renowned mathematician claims “simple” solution to 160-year-old problem

Retired mathematician Sir Michael Atiyah claims to have demonstrated a simple solution to the Riemann hypothesis, which has remained unsolved for 159 years. He is set to formally announce his explanation– which could have massive implications for digital security — on Monday.

The real part (red) and imaginary part (blue) of the Riemann zeta function.

Over the centuries, mathematicians have approached more and more complex problems. Some problems have remained stubbornly opaque, defying decades and even centuries of attempts to solve them. The most famous (and most exciting ones) are probably the so-called Millennium Prize Problems: seven problems stated by the Clay Mathematics Institute back in 2000, the solution to which will grant the author $1 million — along with ever-lasting fame and respect, of course. Only one has been solved so far, but a senior mathematician claims to have found the solution to another one.

The mathematician in case is Michael Atiyah, and he’s probably not the kind of person you’d expect to solve a Millennium Problem; not because he doesn’t have the capacity to do that — you could hardly find a more prolific and respected mathematician today — but because at 90 years old, he’s seemingly long retired.

Atiyah is aware of the long history of failures surrounding the Riemann hypothesis:

“Nobody believes any proof of the Riemann hypothesis, let alone a proof by someone who’s 90,” he says, but he is still confident in his proof. Atiyah, a laureate of the two most prestigious awards in mathematics (the Fields Medal and the Abel Prize) is set to present his recent work at the Heidelberg Laureate Forum, at an event involving some of the world’s brightest mathematicians.

While no other mathematicians have officially commented on this, Twitter reactions from other mathematicians have been mixed, ranging from excited to very skeptical. This claim is not to be taken lightly, however: Atiyah is renowned and very well respected, and to say that he is credible is an understatement. To make matters even more intriguing, Atiyah claims that his proof is “simple” — though in this context, “simple” means “relatively simple.”

The implications for this goes way beyond a theoretical math problem — much of today’s cryptography and digital security relies on this random distribution. Essentially, all modern cryptography relies on the fact that prime numbers occur sporadically. Atiyah’s solution could raise new challenges for this approach, potentially bringing a paradigm shift in modern cryptography.

All the more reason to keep an eye out for his announcement.

[panel style=”panel-default” title=”The Riemann hypothesis” footer=””]The Riemann hypothesis starts with prime numbers — rather strange numbers which can’t be divided by other numbers; one of the oddities of prime numbers is that their distribution is irregular — there’s no precise method to predict where the next prime number will occur. This unpredictability has been widely used in developing digital security systems.

While looking at prime numbers, Berhard Reimann, one of the most prolific German mathematicians, realized something interesting: the distribution of these prime numbers isn’t random at all, it’s very similar to a function, called the Riemann Zeta Function, described below.

ζ(s) = 1/1s + 1/2s + 1/3s + 1/4s + …. up to infinity

This is where it starts to get tricky. The variable can take any value, and Riemann’s work gives us an explicit formula for the number of primes in a given interval. This is done in terms of the so-called ‘zeros’ of the function — the values of under which the function becomes 0. You can think of this function as a way to predict the distribution of prime numbers.

Riemann observed this in action, but he was never really able to prove it. This is why this is still a hypothesis, and not a theory (which, in science and math, has a much stricter meaning than in regular talk). It remains to be seen whether this will still be the case after Monday or not.

[/panel]

Pi.

What is Pi (π) and what is it good for?

If you have a straight line but want a circle, you’re going to need some Pi.

Pi.

Image via Max Pixel.

I’m talking about the number, not the delicious baked good. It’s usually represented using the lowercase Greek letter for ‘p’, ‘π’, and probably is the best known mathematical constant today. Here’s why:

The root of the circle

Pi is the ratio of a circle’s diameter to its circumference. No matter the size of a circle, its diameter will always be roughly 3.14 times shorter than its circumference — without fail. This ratio, π, is one of the cornerstones upon which modern geometry was built.

Bear in mind that (uppercase) ∏ is not the same as (lowercase) π in mathematics.

For simplicity’s sake, it’s often boiled down to just two digits, 3.14, or the ratio 22/7. In all its glory, however, pi is impossible to wrap your head around. It’s is an irrational number, meaning a fraction simply can’t convey its exact value. Irrational numbers include a value or a component that cannot be measured against ‘normal’ numbers. For context, there’s an infinite number of irrational numbers between 1.1 and 1.100(…)001. They’re the numbers between the numbers.

There is no unit of measurement small enough in rational numbers that can be used to fully express the value of irrational ones. They’re like apples and oranges — both fruits, but very different.

Real numbers.

Apart from the fact that it implies there are numbers which are neither rational or irrational (there aren’t), this Euler diagram does a good job of showcasing the apples/oranges relationship between the two groups.
Image credits Damien Karras.

Because it can’t properly be conveyed through a fraction, it follows that pi also has an infinite string of decimals. Currently, we’ve calculated pi down to roughly 22.4 trillion digits. Well, I say ‘we’, but it was actually our computers that did it.

Truth be told, we don’t actually need that many digits. They’re very nice to have if you’re NASA and people live or die by how accurate your calculations are — but for us laymen, 3.14 generally does the trick. It’s good enough because it’s just about at the limit of how accurately we can measure things around us. We simply don’t need that much precision in day-to-day activity.

Go around the house, pick up anything round, and run a length of string along its circumference. Unwind it and measure it with a ruler. Measure the circle’s diameter with the same ruler, use this value to divide the circumference, and you’ll get roughly 3.14 each and every time. In other words, if you cut some string in several pieces, each equal to the diameter in length, you’d need 3.14 of those strips to cover the circumference.

Because this simplification is so widely-used, we celebrate Pi day on March 14 (3/14) every year.

If you do happen to need a more-detailed value for Pi, here it is up to 100 million decimal places.

What’s it for?

Pi is used in all manner of formulas. For example, it can be used to calculate a circle’s circumference (π times diameter), or its area: A=πr2 — how I keep this formula lodged in my neurons is using the “all pies are square” trick. It’s also used in calculating various elements of the sphere, such as its volume (3/4πr3) or surface area (4πr²).

But it also shows up in a lot of engineering and computational problems. Weirdly enough, pi can be used to obtain the finite sum of an infinite series. For example, if you add up the inverse of all natural squares — 1/12+1/22+1/32+….+1/n2 — you get π2/6.

Most branches of science stumble into pi in their calculations at one point or another. Computer scientists use it to gauge how fast or powerful a computer is, and how reliable its software, by having the device crunch numbers and calculate pi. It’s very useful for determining both circular velocities (how fast something is spinning) as well as voltage across coils and capacitors. Pi can be used to describe the motion of waves on a beach, the way light moves through space, the motion of planets, or to track population dynamics if you’re into statistics.

Another place pi pops up (that you wouldn’t suspect) is in the value of the gravitational constant. This shows how fast an object will accelerate towards the ground as it’s falling. Its most widely-accepted value is 9.8 m/s2. The square root of that value is 3.1305-ish, which is close to the value of pi. That’s actually because the original definition of a meter involved a pendulum that took 1 second to swing either way. Wired has a more comprehensive explanation here.

Pi also underpins modern global positioning systems (GPS) since the Earth is a sphere. So give a little mental thanks to mathematics the next time you’re drunkenly thumbing your phone to hail an Uber.

Who discovered pi?

Domenico Fetti Archimedes.

“Archimedes Thoughtful” by Domenico Fetti, currently at the Gemäldegalerie Alte Meister in Dresden, Germany. Archimedes calculated one of the most accurate values for Pi during the Antiquity.

Pi is not a newcomer to the mathematical stage by any means. We refer to it using the letter ‘π’ from the ancient Greek word ‘περίμετρος’ — perimetros — which means ‘periphery’ or ‘circumference’. It was introduced by William Jones in 1706 and further popularized by Leonhard Euler. The notation was likely adopted in recognition of the efforts of one great ancient mathematician: Archimedes.

Archimedes put a lot of effort into refining the value of pi. He was also the first to use it to calculate the sum of an infinite number of elements over 2,200 years ago, and it’s still in use today.

But he wasn’t the first to realize the importance of pi(e). In his book A History of Pi, professor Petr Beckmann writes that “the Babylonians and the Egyptians (at least) were aware of the existence and significance of the constant π” as far back as 4,000 years ago. They likely only had rough estimations of its exact value (maths was still a new ‘tech’ back then) but they were in the right ballpark.

“The ancient Babylonians calculated the area of a circle by taking 3 times the square of its radius, which gave a value of pi = 3. One Babylonian tablet (ca. 1900–1680 BC) indicates a value of 3.125 for pi, which is a closer approximation,” writes Exploratorium in a look at the history of pi.

They add that ancient Egyptian mathematicians also settled on a quite-ok-for-the-time value of 3.1605, as revealed by the Rhind Papyrus. Chinese and Indian mathematicians also approximated the value of pi down to seven or five digits, respectively, by the 5th century AD.

Further work, most notably that of Archimedes, helped refine this value. He used the Pythagorean Theorem to measure the area of a circle via the areas of inscribed and circumscribed regular polygons. If you slept during math class, that’s the polygon inside the circle and the one that contains the circle, respectively. It was an elegant method, but it did have its limits — since the areas of those two polygons aren’t exactly the same as the surface area of the circle, what Archimedes got was an interval that contained pi. He was aware of this limitation. His calculations revealed that pi must fall between 3 1/7 and 3 10/71 — which is between 3.14285 and 3.14085. Today we know that the five-digit value of pi is 3.14159, so that result isn’t at all bad for a guy without a proper pen to write it down with.

The first method to calculate the exact value of pi came up during the 14th century, with the development of the Madhava-Leibniz series. By the time the 20th-century swang by, pi was known down to about 500 digits.

The most delicious math problem there is: The Incompatible Food Triad

Math and food don’t often pair well together, but three decades ago, geometer George W. Hart learned of a puzzle that still haunts academics to this day. He called it the Incompatible Food Triad.

We’ve been scratching our brains for a while, but we still haven’t been able to find a conclusive answer.

The premise is simple: what three foods (let’s say A, B, and C) go well in pairs of two, but absolutely don’t go well when taken all together? So A and B are good together, A and C are good together, B and C are good together, but A and B and C are bad.

Like many mathematical mysteries, this one sounds quite simple but is, in fact, quite complicated. The premise seems reasonable enough, and with the sheer number of foods and spices out there, you’d think that a group like this must exist, but when you get down to it, you’ll start to see that solutions are scarce at best. Hart says that it’s been 25 years since he first learned of the problem, and still, no conclusive answer has been put forth.

“I learned of The Incompatible Food Triad problem from the philosopher Nuel Belnap when I was a graduate student in the late 1970’s. He mentioned it in discussion while we were at a dinner together. In the intervening years, I have occasionally passed it on it at various dinners to my colleagues and graduate students, always without success. Recently, (at a wonderful dinner in southern Spain with a colleague, two graduate students, and a vast platter of tentacles and mysterious seafood,) I realized it has been twenty-five years with zero progress.”

But this doesn’t mean that attempts haven’t been made.

The most common fallacy is to overlook a pair. A classic example is tea, milk, and lemon,  famously described by Richard Feynman in his autobiographical Surely You’re Joking Mr. Feynman. Tea and milk go well together, tea and lemon also go well, and tea, milk, and lemon certainly don’t go together. But this overlooks milk and lemon, which don’t go well together (even causing curdles).

Another discussed solution is honey, chocolate, and chicken. Honey and chocolate go well together, honey and chicken are a solid pairing, but I draw my line at chocolate and chicken. Also, if you’d say that chocolate and chicken are acceptable, I’d argue that adding honey doesn’t really make it worse. The same thing goes for another proposed solution which seems to have gathered traction: cheese, peanut butter, and jam. Cheese and peanut butter seem like an odd pairing, but if you can get it to work, adding some jam doesn’t seem like it would do any harm. In my view at least, that’s not a true solution.

But I did manage to find two solutions online. The first one is lemon, cocoa, curry. All three pairs are a bit of a stretch, but cocoa and lemon do work together, cocoa curry is a thing, and lemon curry is also a good pairing. But throw them all together, and that’s a definite no-no. The other one is no longer within the food realm and spills over to drinks: gin, tonic, and orange juice. All three pairs work fine, but take them together, and you end up with something that’s bizarre at best. There are cocktails that involve both gin and orange juice, but gin tonic with orange juice, as nice as it might seem, offers a nasty surprise.

The best solution I could find (which was promptly shut down by my colleagues at ZME) was yogurt, honey, and garlic. Surely the three don’t work together, and surely yogurt & honey and yogurt & garlic are great pairings. But does honey and garlic work? I’d say yes, but alas, my colleagues say no.

Perhaps at the end of it all, George Hart’s favorite solution is the best way to go. What’s that you wonder? Well, a shot of tequila, a shot of tequila, and of course, a shot of tequila. Who knew that pairing math and food together can get you so far?

If that isn’t enough of a brain stretcher, then Craig Westerland (at the University of Michigan) proposes a complementary question: Are there three foods which you would eat together, but you wouldn’t eat any pair without the third? 

Longest Sailable Straight Line Path on Earth. Credit: R. CHABUKSWAR ET AL

The longest straight-line path on Earth is a 20,000-miles ocean journey

Five years ago, Reddit user kepleronlyknows posted a map that illustrated the longest straight-line path on Earth without hitting land. The nearly 20,000-mile-long route begins in southern Pakistan and ends on northeastern Russia. Now, a pair of researchers has officially confirmed the proposed path — they calculated both the longest sailable and longest drivable paths on Earth.

Longest Sailable Straight Line Path on Earth. Credit: R. CHABUKSWAR ET AL

Longest Sailable Straight Line Path on Earth. Credit: R. Chabukswar et al.

Credit: R. CHABUKSWAR ET AL.

Credit: R. Chabukswar et al.

Rohan Chabukswar of the United Technologies Research Center in Ireland and Kushal Mukherjee of IBM Research in India wrote an algorithm that calculates the longest straight path on both land and water. Their model processed data from the National Oceanic and Atmospheric Administration’s ETOPO1 Global Relief model of Earth’s surface. The model’s spatial resolution is about 1.8 kilometers, meaning this is the size of the smallest feature on the map. Because the model also captures altimetry data, the researchers could accurately tell which point on the map was on land or water.

For this purpose, the Earth can be regarded as a sphere, so the distance between two points on its surface traces an arc or the segment of a great circle. A great circle always crosses the maximum circumference of the sphere, meaning it will always lie in the same plane as the center of the sphere. The equator is a great circle, for instance.

The most challenging part of the study was optimizing the plotted routes. The model rendered 233,280,000 possible great circles, with each containing 21,600 points on either land or sea — that’s 5 trillion points to consider, in total. Brute-forcing these many data points would take forever, so the pair of researchers used a technique called branch and bound, which treats possible solutions as branches on a tree. The algorithm goes through one branch after another, where each branch is a subset of possible solutions. The bounding part, however, optimizes the search so only a few subsets of all the possible great circles are considered — the most promising ones that are close to the optimal path. With each iteration, the search for the optimal paths is fine-tuned. It took only 10 minutes on a commercially available laptop to complete the model.

Longest Driveable Straight Line Path on Earth.

Longest Driveable Straight Line Path on Earth. Credit: R. Chabukswar et al.

Credit: R. Chabukswar et al.

The search confirmed the Redditor’s original assertion, finding that the longest-straight path on Earth without hitting land measures 32,089.7 kilometers — or about 19,940 miles. A boat journey along this path would start from Sonmiani, Pakistan, taking you along the shores of Madagascar and continental Africa, squeezing in between South America and Antarctica, before heading north-northwest across the Pacific dodging the Alaskan archipelago until finally landing in the Karaginsky District in Russia.

When the algorithm was run with the parameters in reverse, it produced the longest straight path on land. It starts in Jinjiang, Fujian, in China, and ends at Portugal. The 11,241-kilometers-long route — or about 6,985 miles — passes through 15 countries.

The findings were reported last week in the preprint server arXiv.

 

Amateur mathematician Aubrey de Grey, known for his work on anti-aging, solves decades-old problem

Aubrey de Grey has dedicated his life to understanding how we can live longer — or perhaps, forever. But now, he will also be known for something else: advancing a problem that has puzzled mathematicians for years, the Hadwiger-Nelson problem.

A seven-coloring of the plane, and a four-chromatic unit distance graph in the plane provide the upper and lower boundaries for the problem solution. Now, an amateur mathematician has upped the lower limit from 4 to 5. Image via Wikipedia.

Coloring graphs

Back in 1950, Edward Nelson was still a student at the University of Chicago. He had a question, a question that no one had been able to answer for him. It was the kind of apparently simple and yet deceptively complex question that can stump mathematicians for decades. Nelson asked the following: imagine that you have a graph, a group of points connected by lines. All these lines are equal in length, and everything lies within the same plane. Now, if you were to color all the points in such a way that no two connected points have the same color, what would be the smallest number of colors you’d need?

The answer had been narrowed down to one of the numbers 4, 5, 6 or 7, but no progress had been made, and the problem was stagnating. That is until Aubrey de Grey came along.

Aubrey de Grey made his name in longevity studies — but now he’s making headlines with his mathematical work. Image via Wikipedia.

A biologist who once made the bold claim that people alive today will live to the age of 1,000, de Grey likes to relax from his day job by solving math problems. This time, he struck gold. Writing in his aptly named paper “The Chromatic Number of the Plane Is at Least 5,” he demonstrates that a planar unit-distance graph can’t be colored with only four colors, thus showing that you need 5, 6, or 7 colors.

He didn’t completely solve the problem, but it’s the first major advancement since right after the problem was introduced, and it does a lot to narrow the solutions. We know the graph can be colored with 7 colors, and we know it can’t be colored with only 4 — now, it’s all about showing if it could be colored using 5 or 6 colors.

An unlikely mathematician

A game of Othello (in progress). Image credits: Paul_012 / Wikipedia.

De Grey’s solution was inspired by a board game called Othello. Othello shares some similarities with the game of Go — on a chess-like board, two players take turns placing white and black pieces, trying to steal each other’s pieces. Decades ago, de Grey was a competitive Othello player, and through this, he met some talented mathematicians who introduced him to graph theory — the mathematical theory of the properties and applications of graphs.

[panel style=”panel-success” title=”The four-color theorem” footer=””]Perhaps the most famous problem of graph theory is the so-called four-color theorem. Consider a map of countries or randomly defined areas. Regardless of the shape and size of these areas, you need only 4 colors to color the map in such a way that no adjacent countries have the same color.

This was first proposed in 1852, when South-African mathematician Francis Guthrie noticed that you only need four colors to color the counties of England. Notably, this was one of the first theorems in history which was proven through the use of computers.[/panel]

De Grey comes back to graph theory now and again.

“Occasionally, when I need a rest from my real job, I’ll think about math,” he said.

When he “thought about math” on Christmas, this specific problem came to his mind and the breakthrough came not long after that.

“I got extraordinarily lucky,” de Grey said. “It’s not every day that somebody comes up with the solution to a 60-year-old problem.”

His work was possible thanks to the Polymath Project: a large international collaboration among mathematicians to solve important and difficult mathematical problems by coordinating communication between mathematicians. Polymath began about 10 years ago when Timothy Gowers, from the University of Cambridge, wanted to find a way to facilitate massive online collaborations in mathematics. Work on Polymath is open access, and anyone can participate and offer their input. Recently, de Grey was also active on a Polymath collaboration that led to significant progress on the twin prime problem (a twin prime is a prime number that is 2 more or 2 less than another prime number — for instance 17 and 19 or 41 and 43). Not every problem is suitable for such a collaboration, but many consider Polymath to be an extremely important component of modern maths.

Aubrey de Grey believes medical technology could one day develop so much that it will allow us to live indefinitely. It’s not clear if this is the case, but one thing’s for sure: through his work on the Hadwiger-Nelson, de Grey has already ensured that, at least in a way, he’s already immortal.

 

 

There are 15 possible ways to cover a floor with pentagonal tiles

The funny thing about math is that it offers answers to questions you didn’t even think of asking. For instance, did you know that there are 15, and only 15 ways of covering a floor with pentagonal tiles?

Tiling problems initially became interesting due to mosaics. Credits: Archeologisch Museum Sousse.

Tiling a plane with a single pattern has fascinated people since the Antiquity, not only for the sake of geometry but also for aesthetic reasons. That’s how mosaics got started, and how they became the way to exhibit opulence and social status. Most mosaics were covered with rectangular or square tiles. It didn’t take long for the ancients to understand that a floor can also be tiled with triangles or hexagons, and that was pretty much the end of it, until much later.

This characteristic of a shape, the ability to cover an endless plane pattern, is called tessellation. In 1918, Karl Reinhardt published a thesis on tessellation in which he carried out an exhaustive search for all the convex forms (all angles smaller than 180°) that can tile a plane without overlapping. He showed that all types of triangles and four-sided shapes can tile a plane, and only three types of hexagons can do so. He also showed that no polygon with seven or more sides could do so.

The only question mark that remained was that of pentagons.

Now, almost a hundred years later, that pressing question has been answered by Michaël Rao of the Laboratoire d’informatique du parallélisme (CNRS/Inria/ENS de Lyon/Université Claude Bernard Lyon 1).

Five sides

So far, we know of 15 types of pentagons which could fill the tile — many described by Reinhardt himself, several identified by other mathematicians, even amateurs. In 2015, the 15th type was described, 30 years after the previous. But there was no definitive answer as to whether others also remained. In a rather witty introductory note, Reinhardt said his thesis didn’t demonstrate that the list is exhaustive “for the excellent reason that a complete proof would require a rather large book.”

The 15 types of pentagonal tiles and their 4 specific types. Credits: Michael Rao, Laboratoire d’informatique du parallélisme (CNRS/Inria/ENS Lyon/Université Claude Bernard Lyon 1)

Rao started with a computer algorithm which generated all possible pentagonal shapes. In his new computer-assisted proof, he used a computer algorithm and found a total of 371 families of pentagons. They were defined by a common rule, such as “side A is equal to side B” or “Angle C and D are equal.”

“For each of the 371 scenarios,” explainedGreg Kuperberg, a professor of mathematics at the University of California, Davis., “his algorithm tries to piece together a tiling by laying down one tile at a time, using only the allowed vertex configurations.

Here’s a visualization of the computer algorithm:

A video of Rao’s computer program running through tiling possibilities and arriving at the 15th type of pentagon tiling.

Out of these, only 19 were convex and could successfully tile a plane. As it turned out, four of these are particular cases of these 15 types, so lo and behold, 15 and only 15 types of pentagons can fill a tile.

Seeking einstein

Thomas Hales, a professor of mathematics at the University of Pittsburgh and a leader in using computer programming to solve problems in geometry, has independently replicated Rao’s solutions. Rao’s study also provides insight into the search of the legendary einstein (no relation to Albert Einstein, the word just means “one rock” in German). The einstein is a hypothetical shape that can only tile the plane nonperiodically, in a never-repeating orientation pattern. “For everybody who works on tiling, this is a kind of holy grail,” Rao said referring to the einstein. He sees this study not as a goal in itself, but rather as a milestone in a much larger quest.

There’s good reason to believe that at least an einstein exists though if it does, it likely has a very complex shape. As you can imagine, this only adds to its allure.

Researchers believe that the einstein exists because it connects to another problem in tiling theory, called the decision problem. Casey Mann, an associate professor of mathematics at the University of Washington who discovered the 15th tessellating pentagon describes the decision problem:

“The question is, if someone hands you a tile, can you come up with a computer algorithm that will take as input that tile and say, ‘Yes, this tiles the plane,’ or, ‘No, it doesn’t?’”

“Most people think there’s too much complexity for such an algorithm to exist,” Mann said.

But Rao plans to move on and set his algorithms on the search for the elusive einstein. Who knew there was so much complexity to tiling, eh?

Journal Reference: Exhaustive search of convex pentagons which tile the plane. Michaël Rao, available on Arxiv.org, arXiv:1708.00274 

Bonus question

Can you figure out why a floor can’t be tiled with heptagonal shapes? It’s a fairly two-liner simple mathematical proof. Hint: the sum of the heptagon’s interior angles is 900 degrees.

First Zero.

“Nothing” changed: ancient Indian text pushes the history of zero back 500 years

Carbon dating of an ancient Indian text might push the history of zero 500 years earlier than thought, a new paper reports. It might seem like nothing, but this humble number made modern mathematics possible.

First Zero.

Close-up image of one of the manuscript’s sheets. I’ve circled the dot, which stands as a placeholder zero, on the bottom line of the text.
This dot would later evolve into the fully-fledged number zero.
Image credits Bodleian Libraries / University of Oxford.

The number was found in an ancient Indian text known as the Bakhshali manuscript, a collection of 70 Sanskrit-covered birch bark leaves delving into the field of mathematics. The manuscript was first discovered by a local farmer near the village of Bakshali, present-day Pakistan, back in 1881, and has been housed at the University of Oxford’s Bodleian library since 1902.

Luckily for us, it seems that researchers can still teach this old work new tricks. A team lead by Marcus du Sautoy, a professor of mathematics at Oxford, carbon dated the text for the first time in history. Their findings show that we were far off the mark in estimating the text’s date of origin — hailing from somewhere between 224AD and 383AD instead of the commonly-assumed 9th century (801-900AD) — with deep implications for the history of mathematics.

Less is more and nothing is everything

This would make the Bakhshali manuscript the oldest known incidence of the number zero, preceding the current-oldest (an inscription on a temple wall in Gwalior, India, etched in the 9th century) by several centuries.

However, as is often the case, this early-version zero had some quite significant differences from the one we’re familiar with today. For starters, it wasn’t donut-shaped: there are hundreds of zeros throughout the Bakhshali manuscript, all of which are denoted using a simple dot. The shape would evolve over the following centuries. It also seems like dot-zero was yet to come into its full powers, and was originally used as a placeholder. It would be used to write more complex numbers but wasn’t a full-fledged number yet. In other words, it’s used as the “0” in 103 implies there are no tens but doesn’t hold any meaning by itself so it can’t be used to denote zero as in the value of nothing.

The Bakhshali manuscript.

The birch bark leaves that make up the Bakhshali manuscript bound together to preserve them.
Image credits Bodleian Libraries / University of Oxford.

Translations of the text suggest it was a sort of training manual for monks or merchants traveling across the Silk Road, as it includes a lot of practical arithmetic exercises and an early version of algebra.

“It seems to be a training manual for Buddhist monks,” du Sautoy says about the manuscript. “There’s a lot of ‘If someone buys this and sells this how much have they got left?’”

Usage-wise, the Indian concept of zero has counterparts in other cultures, such as the ancient Mayans and Babylonians. But only this dot-zero would go on to become a number in its own right, first described by the Indian astronomer and mathematician Brahmagupta sometime in 628 AD.

The concept of “nothing” was revolutionary for numbers, and went on to change mathematics from the ground up. It underpinned the development of calculus, which is strongly entrenched across fields of science and the nightmares of burgeoning engineers everywhere, and made the digital revolution possible.

In many ways, the moment when “nothing” became a number was a turning point in science and technology, marking a transition from dealing in the palpable to dealing with abstract concepts. It’s likely that India’s cultural background in a sense allowed mathematicians there to think in such abstract terms, according to du Sautoy. For example, despite developing sophisticated maths and geometry, the ancient Greeks had no symbol for zero, showing that the concept of a “nothing” number is far from an obvious one.

“Some of these ideas that we take for granted had to be dreamt up. Numbers were there to count things, so if there is nothing there why would you need a number?” du Sautoy adds. “The whole of modern technology is built on the idea of something and nothing.”

“This [number] is coming out of a culture that is quite happy to conceive of the void, to conceive of the infinite. That is exciting to recognise, that culture is important in making big mathematical breakthroughs. The Europeans, even when it was introduced to them, were like ‘Why would we need a number for nothing?’”

“It’s a very abstract leap,” he concludes.

However, the team notes that the manuscript is far from a homogenous body of work — which is why it took so long for scientists to accurately date it in the first place. The pages come from different dates, with up to 500 years’ difference between the oldest and youngest ones, they write. It’s still unknown how they got collected together, du Sautoy says, but hopefully, future research will provide answers.

The Science Museum of London will put the manuscript on display on October 4th as part of a larger exhibition, Illuminating India: 5000 Years of Science and Innovation.

A paper titled “Carbon dating reveals Bakhshali manuscript is centuries older than scholars believed and is formed of multiple leaves nearly 500 years different in age” describing the findings has been made available online by David Howell, Head of Heritage Science at the Bodleian Libraries.

Laws of mathematics don’t apply here, says Australian PM

Australia’s Prime Minister, Malcolm Turnbull, just out-trumped Trump, saying that the laws of Australia override those of mathematics.

No need to bother with maths, it doesn’t offer any votes. Image via Flickr.

At a recent press conference about cryptosecurity, Turnbull just spewed out this gem:

“JOURNALIST: Won’t the laws of mathematics trump the laws of Australia? And aren’t you also forcing everyone to decentralised systems as a result?”

“PRIME MINISTER: The laws of Australia prevail in Australia, I can assure you of that. The laws of mathematics are very commendable but the only law that applies in Australia is the law of Australia.”

We truly are living in a post-truth world, aren’t we? After all, one can only wonder if a theorem might be overturned with enough votes in Parliament.

Context

Turnbull’s comments come in the context of controversial legislation that wants to force social media companies to give the government access to confidential messages whenever there is some suspicion of illegal activity. Apps like Whatsapp prevent any snoopers, be they the government or otherwise. As NewScientist explains, these use uncrackable end-to-end encryption, “jumbling it up in such a way that only the recipient can de-jumble it”.

Basically, this cryptosecurity involves very advanced mathematics that just doesn’t allow anyone to look in — not even Whatsapp itself. There’s simply no way for anyone other than the sender and the recipient to see the message. There’s also no way of weakening this encryption for say, terrorists, without weakening it for everyone else. So everyone would lose their security.

This isn’t a new approach. UK home secretary Amber Rudd has previously called encryption “completely unacceptable” and the UK prime minister Theresa May has sworn to crack down on social media encryption, despite all the indicators showing that this is not the real problem. Basically, you can’t blame terrorism on the internet, and this will just end up being the worst of both worlds. Terrorists, child abusers, and all the other baddies will find secure ways to send their messages and the rest of us will be left vulnerable to peeking eyes — again, be them governmental eyes or other malicious entities. But hey, who knows, with enough votes, maybe we can change reality.

 

Link

Mathematical concepts can be very useful for us to generate beauty. By using the trigonometric functions sine and cosine, we can make an infinite number of stunning symmetrical images. Below you can see seven images and the formulas I used to create them. Each of these shapes is constructed by 7,000 circles.

7,000 Circles (1)

Credit: Hamid Naderi Yeganeh

7,000 Circles (2)

Credit: Hamid Naderi Yeganeh

7,000 Circles (3)

Credit: Hamid Naderi Yeganeh

7,000 Circles (4)

Credit: Hamid Naderi Yeganeh

7,000 Circles (5)

Credit: Hamid Naderi Yeganeh

7,000 Circles (6)

Credit: Hamid Naderi Yeganeh

7,000 Circles (7)

Credit: Hamid Naderi Yeganeh

See more images at: https://mathematics.culturalspot.org

coffee

Mathematicians use their skills to find the perfect cup of coffee

coffee

Credit: Pixabay, Couleur.

Coffee is one of the most popular brews on the planet. In the U.S. alone, 83% of the population drinks coffee amounting to 580 million cups each day. Not everyone, however, might be brewing their coffee the right way. Containing some 18,000 chemical compounds, the flavor and properties of a cup of joe can vary wildly depending on where you source the beans or the brewing process itself. Undeterred by such complexity, two British mathematicians solved equations that describe drip filter coffee machine operation in effort to spot what the perfect brew looks like.

An ideal brew

Kevin Moroney at the University of Limerick and William Lee at the University of Portsmouth used a model that describes flow and extraction in a coffee bed, specifies extraction mechanisms in terms of the coffee grain properties, and compares the model’s performance with empirical results from previous experiments.

“Our overall idea is to have a complete mathematical model of coffee brewing that you could use to design coffee machines, rather like we use a theory of fluid and solid mechanics to design racing cars.” Dr Lee told BBC News.

Because the interactions can be complex, assumptions had to be made to simplify the model. They assumed isothermal conditions (constant temperature), because optimal brewing circumstances require a narrow temperature range of 91-94 degrees Celsius. Another assumption is that coffee bed properties remain homogeneous in any cross section and that water saturates all pores in the coffee bed, eliminating the need to model unsaturated flow. A set of conservation equations on the bed scale monitor the transport of coffee and liquid throughout the coffee bed.

Location of coffee in the bed: The coffee bed consists of (intergranular) pores and grains. The grains consist of (intragranular) pores and solids. The schematic shows the breakdown of this coffee in the grains (intragranular pores are not represented for clarity). Image credit: Kevin M. Moroney

Location of coffee in the bed: The co ffee bed consists of (intergranular) pores and grains. The grains consist of (intragranular) pores and solids. The schematic shows the breakdown of this coff ee in the grains (intragranular pores are not represented for clarity). Image credit: Kevin M. Moroney

These models suggest that brewing good coffee can be boiled down to a simple rule: grind beans more finely if the coffee is too watery or use a coarser grind if it tastes too bitter. But that’s not what was the most important to come out of this study. Because the researchers performed a quantitative analysis, not just qualitative, their model can determine the exact grain size to brew a coffee of some desired properties. If manufacturers take a hint, then in the near future you might be able to buy coffee machines which grind your coffee per your desired taste, oscillating between bitter and watery. Barista-grade coffee coming out of every kitchen — that could really happen!

Another interesting conclusion is that there are two physical processes that influence the brewing.

“There’s a very quick process by which coffee’s extracted from the surface of the grains. And then there’s a slower tail-off where coffee comes out of the interior of the grains,” said Dr Lee.

Next for the researchers is studying how different dripping methods influence brewing. For instance, what’s better: pour hot water dead center or sprinkle it like a shower? Should the machine have a flatbed or should it be conical?

The findings appeared in the SIAM Journal on Applied Mathematics.

Ref: Asymptotic Analysis of the Dominant Mechanisms in the Coffee Extraction Process. SIAM Journal on Applied Mathematics, 76(6), 2196-2217.

math calc

Struggling with math? You might have what scientists call a ‘math disability’

math calc

Credit: Pixabay

In a typical classroom, you’ll find children who are exceptionally good at math while some struggle. Some kids seem to be particularly poor at math, though, showing difficulty even when routinely adding or subtracting even after extensive schooling. A pair of researchers at Georgetown University Medical Center and Stanford University made an extensive review of the current literature and found evidence of a form of ‘math disability’, which is related to dyslexia. Their theory suggests math disability is linked to abnormalities in brain areas supporting procedural memory.

The two researchers identified the basal ganglia and areas in the frontal and spatial lobes as responsible for the inability of some people to process math problems. These brain structures have been previously linked to dyslexia, which makes people struggle with word order and reading.

These brain structures are involved in procedural memory, a specialized learning and memory system that is crucial to automatization and non-conscious skills like driving.

“Given that the development of math skills involves their automatization, it makes sense that the dysfunction of procedural memory could lead to math disability. In fact, aspects of math that tend to be automatized, such as arithmetic, are problematic in children with math disability. Moreover, since these children often also have dyslexia or developmental language disorder, the disorders may share causal mechanisms,” said Michael T. Ullman, PhD, professor of neuroscience at Georgetown, in a statement.

Previously, other groups have suggested that problems in solving seemingly simple math might arise due to deficits in spatial short-term memory, which makes it difficult to keep numbers in mind. But this hypothesis doesn’t explain math disability in terms of underlying brain structures since the disorder must ultimately depend on a brain-related aberration.

Ullman and Tanya M. Evans, who is now a post-doc fellow at Stanford, have come up instead with what they call the “procedural deficit hypothesis.” Their idea is based on current literature that says learning a skill involves both procedural and declarative memory.

“We believe that learning math is likely similar to learning other skills,” Evans says. “For example, declarative memory may first be used to consciously learn how to drive, but then with practice driving gradually becomes automatized in procedural memory. However, for some children with math disability, procedural memory may not be working well, so math skills are not automatized.”

Math, reading or learning a new language depend on both of these learning systems and evidence so far seems to indicate that when procedural memory is impaired, a child may face math disability, dyslexia or developmental language disorder. Declarative memory often compensates but only to an extent.

“We believe that understanding the role of memory systems in these disorders should lead to diagnostic advances and possible targets for interventions,” Ullman said. “In fact, aspects of math that tend to be automatized, such as arithmetic, are problematic in children with math disability. Moreover, since these children often also have dyslexia or developmental language disorder, the disorders may share causal mechanisms.”

That’s not to say, however, you have an underlying brain abnormality if you did poorly in math in school. Maybe you had the bad luck of dealing with a teacher whose technique relied too heavily on rote memorization isolated from meaning, maybe you weren’t exposed to the right curricular materials or just maybe you’re not that bad at math as you think.

Findings appeared in the journal Frontiers in Psychology.