Tag Archives: Mathematics

Using Mathematical Functions to Create Stunning Animations

Mathematics is a very powerful tool to create beautiful works of art. By using mathematical formulas, we can create stunning animations. The trigonometric functions (especially sine and cosine) are the most useful mathematical tools to create such animations. Here are a few GIF animations that are made with sine and cosine functions. Each of these animations shows 8,000 moving circles. The sine and cosine functions have been used to determine the center and the radius of the circles.

Credit: Hamid Naderi Yeganeh

Credit: Hamid Naderi Yeganeh

Credit: Hamid Naderi Yeganeh

Credit: Hamid Naderi Yeganeh

Credit: Hamid Naderi Yeganeh

Credit: Hamid Naderi Yeganeh

Credit: Hamid Naderi Yeganeh

Credit: Hamid Naderi Yeganeh

Weapons of math destruction: plane delayed because university professor was writing equations

No matter how bad you are at math, you should be able to recognize an equation when you see it, right? Well, that wasn’t the case for a passenger on the plane from Philadelphia to Ontario. This passenger saw a saw a man “suspiciously” writing down a complicated looking formula on a piece of paper and notified cabin crew.  She then said she was feeling ill, causing the plane to be turned around, and then the man was brought in for questioning. The thing is, the man was Guido Menzio, an Italian-born associate professor in Economics at the University of Pennsylvania, who was simply going through some equations for his upcoming lecture.

It almost sounds too bizarre to be true. Firstly, the passenger (his seat neighbor) thought Menzio looks suspicious, just because he happens to be slightly tanned, dark-haired, bearded and with a foreign accent – things you’d expect from an Italian man after all. Then, she noticed that he was writing “something strange,” cryptic notes in a language she did not understand. But this is where it gets even eerier. She didn’t say anything to the cabin crew, instead preferring to pose ill. The protocol in this case is very strict: the plane must be returned.

But when the plane returned, instead of medical assistance, the woman sought security assistance. Menzio told the Associated Press:

“I thought they were trying to get clues about her illness. Instead, they tell me that the woman was concerned that I was a terrorist because I was writing strange things on a pad of paper.”

His scribbling was actually a differential equation he was preparing for a lecture on Search Theory in Canada, where he was headed. He says he was treated with respect by security, but on Facebook he recalls this bizarre experience:

“The passenger sitting next to me calls the stewardess, passes her a note.” He was then “met by some FBI looking man-in-black”.

“They ask me about my neighbor,” he wrote. “I tell them I noticed nothing strange. They tell me she thought I was a terrorist because I was writing strange things on a pad of paper. I laugh. I bring them back to the plane. I showed them my math.”

After a two-hour questioning, Menzio returned to the plane, but the fact that a system can be so easily perturbed by someone so clueless is disturbing.

If you’re left-handed, you may be a bit better at math

If you’re left-handed, some of the simplest and most mundane things can be an ordeal. Scissors are awful, musical instruments are a drag and house appliances can be quite challenging. But according to a new study, being a leftie is associated with better math skills, at least for teenage boys.

The link between handedness was studied several times in the past. Some studies found no correlation between the two while others suggested lefties are slightly better at math. Now, the largest study of this kind has carried out, focusing on the mathematic performance of 2,300 students in Italy, aged between six and 17.

The study was conducted by University of Liverpool researchers, who asked the Italian students to take a quick, simple test called the Edinburgh Handedness Inventory (which you can take here). The test assesses how right or left handed you are (or if you are ambidextrous). Then, the participants took several math tests, including basic arithmetic and problem-solving. When researchers corrected for age and sex, they found that the handedness actually explained up to 10% of their variation in maths scores.

“This study found there is a moderate, yet significant, correlation between handedness and mathematical skill,” said lead researcher Giovanni Sala. “Moreover, the amount of variance in the maths scores explained by handedness was about 5-10 percent, a surprisingly high percentage for a variable like handedness.”

However, another interesting conclusion was that people who were more to an extreme or the other (very left-handed or very right-handed) had worse results than students who were more comfortable with using both their hands. However, only for male lefties, things were very different – they simply did better than all their counterparts.

“We also found that the degree of handedness and mathematical skills were influenced by age, type of mathematical task and gender. For example, the most lateralized children – that means those who were very one-sided either very left- or very right-handed – tended to underperform compared to the rest of the sample. However, this effect disappeared in male left-handed adolescents, who performed much better than their peers.

The thing is, we still don’t really know how we can interpret these results, but then again handedness itself is pretty much a mystery. We don’t really understand handedness in itself, let alone something as subtle as this type of connection.

“These results must not be considered definitive,” said Sala, “but only a step towards the conception of a new and more comprehensive model of the phenomenon; a model able to account for all the discordant outcomes reported so far.”

 The research was presented at the British Psychological Society 2016 annual conference held in Nottingham in April 2016.

 

All these pentagons are identical. The coloring helps identify the three groups that arrange to form a tilled plane. Image: Casey Mann

Finally, a new pentagon shape that tiles in a plane

Both bathroom decorators and mathematicians have a reason to rejoice (how often does that happen?). Using a computer algorithm, a group of mathematicians at the University of Washington Bothell discovered the 15th kind of pentagon that can tile in a plane. The 14th was discovered in 1985 by mathematician  Rolf Stein, while the previous five before were proven by Majorie Rice, a housewife from San Diego.

All these pentagons are identical. The coloring helps identify the three groups that arrange to form a tilled plane. Image: Casey Mann

All these pentagons are identical. The coloring helps identify the three groups that arrange to form a tiling plane. Image: Casey Mann

You can’t tile a regular pentagon – with all its sides and interior angles equal – but you can tile triangles and squares in innumerable shapes and sizes. As for a convex heptagon or octagon, it was mathematically proven there’s no such shape that can tile in a plane. Tiling pentagons, however, is an open problem, one that’s been fascinating mathematicians for over a century. The first to prove a pentagon could be tiled was  German mathematician Karl Reinhardt who discovered five such shapes that tile in 1918.

The entry foyer of the  Mathematical Association of America's building was designed using the congruent pentagons discovered by Marjorie Rice.

The entry foyer of the Mathematical Association of America’s building was designed using the congruent pentagons discovered by Marjorie Rice.

For almost thirty years there was no tiling pentagon reported, but now using the power of computing Casey Mann, Jennifer McLoud and David Von Derau of the University of Washington Bothell have finally found a new one.

“The problem of classifying convex pentagons that tile the plane is a beautiful mathematical problem that is simple enough to state so that children can understand it, yet the solution to the problem has eluded us for over 100 years,” said Casey for the Guardian. “The problem also has a rich history, connecting back to the 18th of David Hilbert’s famous 23 problems.”

Detailed view of the new tiling pentagon. Image: Casey Mann

Detailed view of the new tiling pentagon. Image: Casey Mann

To find the tiling pentagon, the researchers basically used brute force to search a large, but finite set of possibilities. Eventually they got lucky, but are there more? It’s a simple, yet challenging problem at the same time. After all, it took a 30 years dry spell.

Of course, there are practical uses to finding tiling surfaces, from biochemistry to structural design.

“Many structures that we see in nature, from crystals to viruses, are comprised of building blocks that are forced by geometry and other dynamics to fit together to form the larger scale structure,” he added.

“I am too cautious to make predictions about whether or not more pentagon types will be found, but we have found no evidence preventing more from being found and are hopeful that we will see a few more. As we continue our computerized enumerations, we also hope to gather enough data to start making specific predictions that can be tested.”

All the pentagonal tilings discovered thus far. Ed Pegg/Wikipedia

All the pentagonal tilings discovered thus far. Ed Pegg/Wikipedia

Love equation

Is there really a mathematical formula that predicts happy relationships?

In a recent TED talk,  Hannah Fry outlines a mathematical formula that predicts long-lasting relationships. In her recent book,  The Mathematics of Love, she discusses the findings of psychologist John Gottman who studied hundreds of couples over many years to find out what sets apart the happy couples from the miserable. Gottman than enlisted the help of a mathematician who correlated all the data the psychologists gathered and came up with an empirical formula that seems to predict if a couple will be happy together.

Love equation

Photo: Alamy

Mathematicians, maybe under an compulsion to reduce things – even human feelings – to their abstract essence, have long tried to come with a mathematical formula for love. It sounds silly, but some have actually tried with mixed reports. Enlightened analyst and geometer Alexis Clairaut may have been the first mathematician to propose a love equation in the form of the Archimedean spiral.

Left: portrait of Clairaut. Right: his love equation (1745). V is the fixed fulcrum and the i's and I's moving clockwise represent the motion of its endpoint. Credit: Mathematics Without Apologies, Michael Harris

Left: portrait of Clairaut. Right: his love equation (1745). V is the fixed fulcrum and the i’s and I’s moving clockwise represent the motion of its endpoint. Credit: Mathematics Without Apologies, Michael Harris

“We seek the curve described by the endpoint of a body, initially vertical and pointing downward, that subsequently changes in length and position,”  Clairaut writes in the IXXth century.

As the years passed, “serious” mathematicians had moved on to more tangible things, discounting any idea of a love equation as absurd. It made its way into fiction though. In Jan Patocky’s late enlightenment novel, “The Saragossa Manuscript”, we meet the so-called geometer Don Pedro Velasquez – a prototype character that mockingly or not describes the absent-minded mathematician. At one point, the lead heroine of the novel, Rebecca, asks Don Velasquez if “this movement we call love, can it be calculated?” The context of the conversation was Rebecca’s bewilderment in the face of a paradox: a man’s love diminishes with intimacy, while the woman’s increases. Velasquez, ever the keen mathematician, jumps to her aid. “I have found a very elegant proof for all problems of this kind: let X….” Hilariously, the author doesn’t let Velasquez present his proof.

Don Pedro Velasquez' Intermediate Love Theorem. Black curve measures the decline of the man's love and the gray curve the increase of the woman's. The point P is the sweet spot - the intersection predicted by Velasquez.  Credit: Mathematics Without Apologies, Michael Harris

Don Pedro Velasquez’ Intermediate Love Theorem. Black curve measures the decline of the man’s love and the gray curve the increase of the woman’s. The point P is the sweet spot – the intersection predicted by Velasquez. Credit: Mathematics Without Apologies, Michael Harris

 

More recently, in cinema, Berkeley math professor Edward Frenkel explores the same theme in the 26-minute-long “Rites of Love and Math”, a movie he wrote, directed and played the leading role. The film has only two characters, a man in the throes of an existential dilemma and the woman he loves.  In the movie, Frenkel found the mathematical formula of love. But then he realizes that others could use his formula to cause harm — and that he must die to safeguard the world. He saves the formula by etching it into his lover’s body.

Love is a complicated equation. It’s been given mathematicians, of all people, headaches for centuries. But while scientists have yet to uncover the love equation – which in all likelihood can’t exist – the same can’t be said about relationships. Empirically, at least, there seems to be a formula that predicts if a couple will be in a long-lasting, happy relationship.

“In relationships where both partners consider themselves as happy, bad behaviour is dismissed as unusual,” Fry says.

“In negative relationships, however, the situation is reversed,” writes Fry. “Bad behaviour is considered the norm.”

Gottman along with mathematician James Murray found a pattern that predicted when these trends of negativity spiral out of control. When a threshold is reached, the equation predicts a high probability of the relationship ending.

relationship equation

Though framed as “husband” and “wife”, it can be used to describe any couple, heterosexual or otherwise.

The most important factor that influences the well being of a couple is the mutual influence the husband and wife have on each other. The researchers found that when a husband behaves positively to his wife, like saying something nice about her or their life together, the wife will respond positively as well. In the same vein, negativity spirals into more negativity.

“The most successful relationships are the ones with a really low negativity threshold,” writes Fry. “In those relationships, couples allow each other to complain, and work together to constantly repair the tiny issues between them. In such a case, couples don’t bottle up their feelings, and little things don’t end up being blown completely out of proportion.”

In short, a happy couple will interact more often, bring up issues as they happen and work together to fix them.

“Mathematics leaves us with a positive message for our relationships,” Fry says, “reinforcing the age-old wisdom that you really shouldn’t let the sun go down on your anger.”

But is this all it boils down to when discussing happy relationships? Of course not, but, like Fry says, it gives to show at least just how important managing anger and negativity can be. A happy, long-lasting relationship is based on mutual support, understanding and, least not, love. The latter might never be explained in the form of an equation or variable of some sort.

via Science Alert.

mathematics without apologies

Book Review: ‘Mathematics Without Apologies’

mathematics without apologies

Mathematics Without Apologies
By Michael Harris
Princeton University Press, 464pp | Buy on Amazon

Mathematics is considered a problematic vocation, because, let’s face it, mathematicians can be weird. But that’s mostly because people don’t understand mathematics, let alone mathematicians which can be even more problematic. Why do (pure) mathematicians do what they do? Michael Harris, professor of mathematics at the Université Paris Diderot and Columbia University, offers a personal account of “Mathematics without apologies”.

Moving past common themes like Mathematics is “beautiful” or “elegant”, the book tries to situate the place of mathematics in human culture today. The context is laid with the help of his personal experience as a mathematician, but also the myths and assertions of famous figures in the field like Archimedes or, more recently, Alexander Grothendieck and Robert Langlands.

If you have no idea how mathematicians think or what they obsesses about, you’ll find this book intriguing to say the least. You might also find it terribly hard to read. Harris’ extremely cautionary and precise tone doesn’t make this easy, that’s for sure. But Harris delves into some serious debates like are mathematicians (quants) responsible for the 2008 financial crisis (no)? There’s also some mini-courses on mathematics. Don’t worry – you can understand them with no prior training and they’re all laid down in a humorous tones in his chapter series “how to explain number theory at a dinner party”.

 

The folding limiting equation Britney found: L is the minimum possible length of the material, t is material thickness, and n is the number of folds possible in one direction.

If you fold an A4 sheet of paper 103 times its thickness will roughly be the size of the Universe

folding_paper

Whaaaat? It’s just a matter of math, really. Fold an A4 once and it will be twice as thick, fold it again and it will be four times as thick as it initially was. Turns out, according to Dr Karl Kruszelnicki, if you do this 103 times the sheet’s thickness will be larger than the observable Universe: 93 billion light-years.  To do this, however, involves an exponential increase of the necessary energy to fold the paper, which wasn’t computed.

The current record for the most times a standard A4 has been folded in half is twelve, and was set by Britney Gallivan more than 10 years ago when she was only a high school student. What’s interesting is that mathematicians at the time thought it was impossible to fold it more than seven times using input human input energy, i.e. your hands. Britney wasn’t unusually strong, however, but she was very clever about it.

The first thing she did was to recognize the challenge’s limitations. She then derived the folding limit equation for any given dimension and found single direction folding requires less paper. One interesting discovery was to fold paper an additional time about 4 times as much paper is needed, contrary to the intuition of many that only twice as much paper would be needed because it is twice as thick. In one day Britney was the first person to set the record for folding paper in half 9, 10, 11 or 12 times.

The folding  limiting equation Britney found: L is the minimum possible length of the material, t is material thickness, and n is the number of folds possible in one direction.

The folding limiting equation Britney found: L is the minimum possible length of the material, t is material thickness, and n is the number of folds possible in one direction.

With each fold, more and more energy needs to be inputted.  A paper folded in half 10 times will result in thickness roughly the width of your hand, but if it’s folded 20 times will be 10 km high, which makes it higher than Mount Everest. Let’s fold it a bit more; 42 times will get you on the moon and, as Dr. Kruszelnicki demonstrated on ABC Science, 103 times will render a thickness the size of the known Universe!

It’s a very simple calculation, but it shows just how powerful math can be.

Mathematician may have revolutionized the theory of numbers… but nobody understands his proof

Shinichi Mochizuki of Kyoto University, Japan claims he has proven the ABC conjecture, one of the longest standing mysteries of mathematics. However, even though his 500-page paper was published in 2012, no one has managed to understand it. Mochizuki says his fellow mathematicians are failing to get to grips with his work.

The ABC conjecture is deceptively complicated. It roughly states that three numbers a, b and c, which have no common factor and satisfy a + b = c cannot be too smooth (more detailed explanation here). In number theory, a smooth number is an integer which factors completely into small prime numbers.

The beauty of this theory is that it captures the intuitive sense that triples of numbers which satisfy a linear relation, and which are divisible by high perfect powers, are rare. It seems like a bottomless, still lake – it seems very simple, but the more you dive into it, the deeper it gets. In fact, the theory poses some significant questions about the actual nature of numbers, so truly understanding it would be highly significant for mathematics.

Step in, Mochizuki. In 2012, he published a 500-page paper online that claimed to solve the puzzle. There were two problems however – the first one, that it’s a 500 page paper, and the second (and more important one), that it uses a dense framework of new maths created by him and dubbed “Inter-universal Teichmüller Theory” – a framework that has even experience mathematicians struggling. Inter-universal Teichmüller Theory is basically a whole new way of looking at numbers which he used to prove other conjectures, such as the Szpiro conjecture, and part of the Vojta’s conjecture for the case of hyperbolic curves, as well as Fermat’s Last Theory.

Mochizuki is a highly-respected mathematician and his work is taken seriously – it’s not like no one has tried to understand his work. It’s just that … no one could. Now, he has released a report in which he says that mathematicians working with him studying his theory have yet to find a flaw, but he criticized the community, saying that most researchers are “simply not qualified” to issue a definitive statement on the proof because they don’t truly understand his theory – and haven’t tried hard enough. Minhyong Kim of the University of Oxford is one of the many who are still awaiting the verdict in this case.

“It’s a bit disappointing that no one has come out and said it’s right or wrong,” he says.

The thing is that if you want to understand the proof of this theory, you have to first understand the Inter-universal Teichmüller Theory he developed, and that’s no easy feat in itself.

“I sympathise with his sense of frustration but I also sympathise with other people who don’t understand why he’s not doing things in a more standard way,” says Kim. It isn’t really sustainable for Mochizuki to teach people one-on-one, he adds, and any journal would probably require independent reviewers who have not studied under Mochizuki to verify the proof.

I’m not exactly sure exactly why Mochizuki chose to work in this complicated way, but hey, if it works, it works. I think the scientific community has a duty to rise up to the event and figure out if this is right or wrong. Sure it’s complicated, but that’s modern math – you’d expect no less. I also believe we’re not going to get any reply from Mochizuki, seeing as he wrote the following:

“The papers were released in order to make it possible for specialists to study and assess the validity of IUTeich; they were never intended as an announcement to the general public. Moreover, I never envisaged that these papers might elicit reactions consisting of non-mathematical content from non-specialists. I have no intention of responding to such non-mathematical reactions.”

But until someone figures if Mochizuki is right or wrong, his theories remain in a Schrodinger’s cat like state.

“This sense of stubbornness, dignity and pride is a part of what gives him the personality necessary to embark on a project like this,” says Kim. But for now, the proof remains in limbo. “It’s a curious state.”

sheepdog

How one single sheepdog herds a flock of one hundred – mystery solved

sheepdog

Photo: pleasantview.org

If you’ve ever been to farmlands and kept your eyes opened, you might have noticed not only how beautiful the simple life can be, but also how there’s no better illustration for the phrase “man’s best friend” than a shepherd and his sheepdog. The two are linked together by an ancestral bondage, one that transcended mere friendship – both man and dog depend on each other to make ends meet and put food on the table/bowl. How exactly one man and canine are able to herd whole flocks of mindless sheep over a hundred strong has always been boggling and more or less taken for granted. A new study sheds light on the matter. The result: in the future sheep might be guided by robot herders. Heck, why not?

The math of sheep herding

Researchers at Swansea University, UK and  Uppsala University in Sweden built a mathematical model that explains how one single sheepdog can round up herds made of up to 100 sheep. Their conclusion suggests that the dog needs only to follow two simple mathematical rules. One causes a sheepdog to close any gaps it sees between dispersing sheep – in fact this is sort of where the key lies; the dog doesn’t see the sheep per se. The dog doesn’t distinguish the fluffy white balls in front of him as individual sheep and what it notices are only the gaps that form in an otherwise white sea. The other rule results in sheep being driven forward once the gaps are sufficiently closed.

To reach this conclusion, the researchers fitted one dog and 46 sheep with a highly sensitive GPS device. They then fed the data in a computer simulation to devise the model that would explain what prompts the dog to move and how the animal responds.

Lead researcher Dr Andrew King, from Swansea University, said: “If you watch sheepdogs rounding up sheep, the dog weaves back and forth behind the flock in exactly the way that we see in the model.

“We had to think about what the dog could see to develop our model. It basically sees white, fluffy things in front of it. If the dog sees gaps between the sheep, or the gaps are getting bigger, the dog needs to bring them together.”

Colleague Daniel Strombom, a mathematician from Uppsala University in Sweden, added: “At every time step in the model, the dog decides if the herd is cohesive enough or not. If not cohesive, it will make it cohesive, but if it’s already cohesive the dog will push the herd towards the target. Other models don’t appear to be able to herd really big groups – as soon as the number of individuals gets above 50 you start needing multiple shepherds or sheepdogs.”

So, how does society benefit from an efficient herding algorithm? Well, the obvious benefit is that engineers can now build robot sheep dog, which doesn’t sound like that much of an appealing prospect. C’mon, sheepdogs are awesome! A more immediate application would be for, again, robots. Mechanized swarms are becoming smarter and numerous by the year and mitigating their navigation pattern as efficiently as possible is a crucial prerequisite. Humans sometimes need herding, too. When coupled together by the hundreds or thousands, masses of people are no different from flocks of sheep in many respects, so authorities might want to use these findings for better crowd control.

The findings were reported in the  Journal of the Royal Society Interface.

Iranian is the first woman to win prestigious math award

Maryam Mirzakhani, who was born and raised in Iran, has been awarded the highest honour a mathematician can attain: the Fields Medal.

It’s one of those moments which will go down in history – for the first time in almos 80 years, a woman has won the Fields Medal (officially known as the International Medal for Outstanding Discoveries in Mathematics). Maryam Mirzakhani, an Iranian maths professor at Stanford University in California was rumoured to be among the favorites for quite a while, due to her groundbreaking studies, which seem downright esoterical to less mathematical minds.

Born and raised in Iran, Mirzakhani completed a PhD at Harvard in 2004, even though her childhood passion was literature.

“I dreamed of becoming a writer,” she said in an interview for the Clay Mathematics Institute (CMI) in 2008. “I never thought I would pursue mathematics before my last year in high school.”

Nowadays, she works on geometric structures on surfaces and their deformations. She has a particular interest in hyperbolic planes, which can look like the edges of curly kale leaves. As a matter of fact, hyperbolic planes are so strange that they may be easier to crochet than explain. Mirzakhani says that while advanced math is not for everybody, most students don’t give math enough chances. Frances Kirwan at Oxford University, one of Britain’s leading mathematicians, said:

“Maths is a hugely rewarding subject, but sadly many children lose confidence very early and never reap those rewards. It has traditionally been regarded as a male preserve, though women are known to have contributed to its development for centuries – more than 16 centuries if we go back to Hypatia of Alexandria.

This may also motivate more female students to dive even deeper into research and academic careers.

“In recent years around 40% of UK undergraduates studying maths have been women, but that proportion declines very rapidly when we look at the numbers progressing to PhDs and beyond. I hope that this award will inspire lots more girls and young women, in this country and around the world, to believe in their own abilities and aim to be the Fields medallists of the future.”

A rhesus monkey preparing to choose the four and five combination on the panel. (c) PNAS

Monkeys can do math, study proves

rhesus_monkey

Photo: PNAS

It’s long been supposed that monkeys are capable of mental arithmetics, but it was only recently that this was proven for a fact by neuroscientists at the Margaret Livingstone of Harvard Medical School in Boston. The researchers taught three rhesus macaques to identify symbols representing the numbers zero to 25, then when given the choice between two panels, one depicting a number symbol and the other depicting an addition of two other symbols, the monkeys proved they could do math and choose which of the two was bigger. This doesn’t just mean that monkeys are smarter than everyone might have thought; it also raises important questions as to how mammalians brains, including those of us humans, work and engage with our surroundings.

Previously, researchers showed that chimpanzees could add single-digit numbers. The results were nothing short of remarkable, but the study didn’t conclude what process go on in the primate’s brain when this addition was going on. The new study which studied the rhesus monkeys sheds more light on these aspects.

Margaret Livingstone of Harvard Medical School in Boston and colleagues trained three monkeys to associate the Arabic numbers 0 through 9 and 15 select letters with the values zero through 25. To receive food, the monkey had to choose between two boards: one that showed an addition of two symbols and the other only one symbol. If the monkey chose the greater number of the two, it received more tasty food.  Within 4 months, the monkeys had learned how the task worked and were able to effectively add two symbols and compare the sum to a third, single symbol.

So be certain the monkeys were simply memorizing the symbols and all possible combinations (that’s no how arithmetic works, clearly), the researchers introduced an entirely different set of symbols representing the numbers zero to 25 in the form of tetris-like blocks instead of the familiar Arabic numbers and Latin letters. According to the study, all three monkeys were on average capable of choosing the correct answer “well above” 50 percent of the time, which is statistically relevant enough to infer that the rhesus monkeys could actually do the math and not simply rely on chance.

A rhesus monkey preparing to choose the four and five combination on the panel. (c) PNAS

A rhesus monkey preparing to choose the four and five combination on the panel. (c) PNAS

What’s interesting to note is that after the researchers analyzed their findings in greater deal they began to understand why the monkeys weren’t right most of the time with their calculations. Apparently, they tended to underestimate a sum compared with a single symbol when the two were close in value—sometimes choosing, for example, a 13 over the sum of eight and six. Basically, when the monkey was adding two numbers, it paid close attention to the large of the two and then added only a fraction of the lesser number to make up the sum; which obviously came out wrong from the real answer.

This peculiar, since one prevailing theory on how the brain processes number representations is that it underestimates the value of larger numbers in a systematic and unchangeable way. The present findings contradict this idea and may help researchers better understand how human beings process numbers. Also, the findings could also help shed light on dyscalculia (similar to dyslexia, only it involves failing to perform mathematical operations instead of reading – an interesting piece about it worth reading here). It’s not that people with dyscalculia have an intellect comparable with rhesus monkeys – far from it, apart from their disability to perform arithmetic, they’re totally cognitively functioning human beings. Estimating values, the present study suggests, may be key to how addition works.

Results were published in the journal PNAS.

The Formula that Could Destroy Chess Forever

photo credit: Mukumbura

photo credit: Mukumbura

Norway’s Magnus Carlsen and India’s Viswanathan Anand are duking it out in the world championship, apparently, without anyone else contesting their supremacy. But as insanely good as they are, they still can’t stand up to the best computers today. As a matter of fact, there is one theory which, if proven, could pretty much spoil chess forever.

“Everyone agrees that if a computer were given X number of years, it would be able to calculate the ultimate way to win at chess. Or at least the ultimate way of averting a loss,” says Kjetil Haugen. Hagen is vice-rector of Molde University College, a professor of logistics and sports management and an avid game theory enthusiast.

Kjetil Haugen of Molde University . Credits: ,olde University

Kjetil Haugen of Molde University . Credits: ,olde University

The claim is not new – as a matter of fact, it’s 100 years old. In 1913 the German mathematician Ernst Zermelo published what would later be known as the Zermelo theory; one of the things the Zermelo says is that in any game played by 2 players, which involves alternating moves and has a limited amount of moves, a winning strategy exists.

As time passed, the theory found applications in numerous fields and was interpreted in many ways, but so far, fortunately or not, no one was able to find THE winning strategy for chess.

“The main problem hindering the discovery of that formula involves the limits of computer power,” says Haugen.

The number of total moves in chess is so unfathomably large that with the world’s current processing capacity and taking into consideration predictable advancements, this won’t happen in the near future. But even if it would be possible… would such a thing be desirable?

In a way, chess is like a way more complex and complicated version of Tic-Tac-Toe – you move something, then the other player moves, and you have a clear objective. Of course, finding the winning strategy (in which you either win or draw, but can’t lose) in Tic-Tac-Toe is very simple.

“This is why they don’t organise a world championship in tic-tac-toe. Chess is an overgrown version of it. As a game it is structurally very similar, with two players taking alternating moves within a finite strategic space. But in reality they are far apart. The possible moves in chess are also finite, but the difference between them is an enormous order of magnitude.”

Chess is also something which isn’t simplifiable – it’s gonna take raw computer power, not some out of the box thinking. But if the secret formula for determining the outcome of any game of chess were to be found, that would virtually ruin the game forever. Rules would have to be changed and this fascinating, ancient game would be forever changed.

Professor Kjetil Haugen admits that he doesn’t play chess – he prefers to watch football – which for our American fans means ‘soccer’.

“That, too, is a thrilling game involving plenty of mathematics. Quite a lot about soccer has much in common with my work,” he says with a smile.

Cited paper suggesting a ‘ratio for a good life’ exposed as nonsense by amateur psychologist

A 52-year-old, part-time graduate student with no previous training in psychology and little training in math aside from high-school has discredited a very cited paper published in 2005 in American Psychologist. The paper, then written by Barbara Fredrickson and Marcial Losada suggested a mathematical ratio between positivity and happiness, claiming that humans thrive when ratio of positive to negative statements made in an interaction is about 2.9.

psycc

But Nicholas Brown, who is completing a master’s degree in applied positive psychology at the University of East London in England, teamed up with two other colleagues to show that this theory is, basically, nonsense. Their paper will appear in August 15 in… American Psychologist.

“It’s slightly worrying to discover that a leading journal could publish an article with so many obvious errors in it,” Brown says.

In 2005, the paper used Lorenz equations to calculate how positive and negative emotions change over time, and how their ratio can lead to happiness. These equations were developed in 1963 by mathematician Edward Lorenz to model how fluids are influenced by convection. They also occur in models regarding lasers, dynamos, electric circuits and even chemical reactions – really wide range of uses. Fredrickson and Losada used the equations with emotion data from volunteers and they concluded that the ratio of positive to negative emotions should be above 2.9013:1 and below 11.6346:1 – but more towards the lower limit. As people stray from this ‘optimum value’, they tend to become less happy and less productive.

Lorenz_Equations

Brown first read this paper as an assignment for school; he and his two colleagues had trouble dealing with the math used in, but once they got through it, they were shocked by the magnitude of the errors they found.

“We find no theoretical or empirical justification for the use of differential equations drawn from fluid dynamics, a subfield of physics, to describe changes in human emotions over time,” they write.

Basically, they concluded that the equations used by Fredrickson and Losada to calculate the critical positivity ratio had no connection to their emotion data – regardless of the emotions reported by volunteers, they would generate the same, meaningless numbers. This conclusion was then confirmed by Alan Sokal of New York University – a researcher most well known for publishing an intentionally nonsensical paper in a leading peer-reviewed journal of cultural studies, to show how big errors and even nonsense can creep into peer reviewed papers.

“What’s shocking is not just that this piece of pseudomathematical nonsense received 322 scholarly citations and 164,000 web mentions, but that no one criticized it publicly for eight years, not even supposed experts in the field,” Sokal says.

Even psychologist Fredrickson acknowledges that their paper employed “now questionable mathematics.” So what’s there to learn here? First of all, that scientists aren’t perfect; it’s absolutely normal to make mistakes, but nowhere is it as likely as in scientific research. Second of all, you don’t have to be a well reputed scientist to conduct relevant, significant studies. Also, last but certainly not least – the current peer reviewal system employed by most journals could use a brush-up.

Journal references:

N. Brown et al. The complex dynamics of wishful thinking: The critical positivity ratio. American Psychologist. Published online July 15, 2013. doi:10.1037/a0032850.

B. Fredrickson. Updated thinking on positivity ratios. American Psychologist. Published online July 15, 2013. doi:10.1037/a0033584.

Origin of life needs some serious rethinking, researchers argue

Scientists trying to pinpoint the origin of life have been looking at it the wrong way, a new study claims.

A new perspective

Instead of recreating the chemical building blocks that led to the emergence of life 3.7 billion years ago, they argue scientists should use key differences in the way that living creatures store and process information – something which they believe to be the key.

“In trying to explain how life came to exist, people have been fixated on a problem of chemistry, that bringing life into being is like baking a cake, that we have a set of ingredients and instructions to follow,” said study co-author Paul Davies, a theoretical physicist and astrobiologist at Arizona State University. “That approach is failing to capture the essence of what life is about.”

A way to define living creatures is through their two-way flow of information both from the bottom up and the top down in terms of complexity, he explains. For example, bottom up would go from molecules and cells to more complex organisms, while top down would flow the opposite way. This new perspective would significantly alter not only how researchers try to uncover the origin of life, but also how they search for living creatures on other planets.

“Right now, we’re focusing on searching for life that’s identical to us, with the same molecules,” said Chris McKay, an astrobiologist at the NASA Ames Research Center who was not involved in the study. “Their approach potentially lays down a framework that allows us to consider other classes of organic molecules that could be the basis of life.”

The chemical approach

For over 50 years, scientists have been trying to recreate the exact conditions that led to life on our planet. In the famous Miller-Urey experiments reported in 1953, they created a primordial soup that mimicked the chemical conditions of the planet’s early oceans and found that simple aminoacids, the most primitive building blocks of life, emerged.

However, as big as that first step is, scientists haven’t progressed greatly since then in figuring out how these aminoacids eventually transformed into simple beings capable of reproduction. Part of the problem, the team argues, is that there isn’t a good enough definition of life at the moment.

Sara Walker, study co-author and an astrobiologist at Arizona State University, explains:

“Usually the way we identify life on Earth is always by having DNA present in the organism. We don’t have a rigorous mathematical way of identifying it.”

She claims that if we use a chemical definition for life, like say, life has to have DNA, we unnecessarily limit our search for extraterrestrial life, and we could also wrongly include some non-living system – like for example a petri dish of self replicating DNA.

Everything is information

Walker’s team created a relatively simple mathematical model to capture the transition from a nonliving to a living-breathing being – or so they claim. According to them, remember, all life has one property which defines it: information flows two ways.

For example when you place your hand in a fire (which you really shouldn’t do), your body sends the information to the brain, and then the brain tells your body to move the hand away – as fast as possible. Such a flow governs all living creatures, from the smallest bacteria to the blue whale. By contrast, if you put a pen in a fire, or an aminoacid, it would have no reaction whatsoever. The same flow of information happens practically all the time, with your body constantly interacting with the environment.

However, they warn, the mathematical model is still in its infancy and it will take a while before it can be adapted to check if such molecules have emerged on other planets. But it is another approach, one that seems especially viable.

“This is a manifesto,” said Davies. “It’s a call to arms and a way to say we’ve got to reorient and redefine the subject and look at it in a different way.”

Source

Math Headache

Math anxiety is similar to experiencing physical pain, brain study finds

Math HeadacheFor many of us, mathematics comes with a feeling of anxiety, not while actually performing math, but beforehand in anticipation. Why some people dread math is an interesting question that deserves a systematic, scientific answer – some other time, however. Recently, I came about an equally interesting study, that analyzed how the brain perceives the fear of math. The findings showed that the anxiety experienced in anticipation of math is associated with the experience of physical pain.

Mathematics has only evolved as a distinct branch of study available to the masses a few centuries ago. Actually, only in the past 150 years or so following education reforms that spurred in Europe did mathematics and its concepts become widely introduced to the public. Society has benefited a great deal from wide spread mathematical knowledge, despite most of its students loathed it. It’s rather clear then that the human brain hasn’t evolved a specialized brain structure dedicated to math anxiety and that the same feeling must come from some other brain function associated with this fear.

To find out which part of the brain is responsible for math anxiety, psychologists devised a series of questions, which they named Short Math Anxiety Rating-Scale, or SMARS. The study volunteers, 28 in total, had to answer these question while a MRI brain scan was performed at the same time. The quizzes were math related in part, while others were focused on verbal skills. Their actual performance was of little significance for the researchers, however – what they were looking at was anxiety.

In order to trigger anxiety, the scientists flashed some colours that warned the participants that math question was following (yellow circle). To fight off extraneous signals, the psychologists managed to pinpoint the differences between people experiencing discomfort while performing math and those while anticipating math. This allowed them to correct their results accordingly.

At the end of their study, the researchers found a limited number of brain regions associated with the math anxiety, the strongest signals coming from the bilateral dorso-posterior insula, an area deep in the core of the brain. The researchers showed when anticipating an upcoming math-task, the higher one’s math anxiety, the more one increases activity in this region of the brain. On the contrary, when getting signaled that a word test was coming, participants actually dropped activity in the insula by a significant margin.

This region of the insular has been documented in the previous studies to be associated with the experience of physical pain. Actually, it’s quite possible to induce the experience of pain simply by stimulating the insula.  Interestingly, this relation was not seen during math performance, suggesting that it is not that math itself hurts; rather, the anticipation of math is painful.

Also, the authors of the study recently published in the journal PLoS one have also confirmed that math anxiety is linked with poor math performance when answering the questions. The value the study provides is that it might explain why people with high anxiety and little tolerance for math behave this way, stirring them away from taking math classes or even entire math-related career paths.

Mathematics

Papers riddled with math put some scientists off

You’re not the only that doesn’t like math, it seems. A new study from scientists at Bristol’s School of Biological Sciences found that biologists pay less attention to theories that are dense with mathematical detail.

Mathematics

Mathematics

The scientists involved in the study compared citation data with the number of equations per page in more than 600 evolutionary biology papers in 1998. The results are rather staggering; they found that most maths-heavy papers were referenced ~50% less often than those with little or no maths.

Apparently, for biologists at least, each additional equation per page reduced a paper’s citation success by 28 per cent. Stephen Hawking envisioned math as a detrimental factor to his readership, and pondered absolutely each equation he included in his popular book, “A Brief History of Time”, in fear of reduced sales. He was on to something.

“This is an important issue, because nearly all areas of science rely on close links between mathematical theory and experimental work,” says Dr Tim Fawcett.

“If new theories are presented in a way that is off-putting to other scientists, then no one will perform the crucial experiments needed to test those theories.  This presents a barrier to scientific progress.”

In the light of these results, which frankly most scientists were already aware of, the researchers recommend some course of action which could potentially offer some tangible solutions. The first, and most difficult to apply, is to  improve the math education of science graduates in less technical fields, like biology, for an increased math-literacy.

Andrew Higginson, Dr Fawcett’s co-author and a research associate in the School of Biological Sciences, said that scientists need to think more carefully about how they present the mathematical details of their work.

“The ideal solution is not to hide the maths away, but to add more explanatory text to take the reader carefully through the assumptions and implications of the theory,” he said.

This isn’t an option for most scientific journals, however, which have a strict policy regarding conciseness and publishing space. An alternative solution is to put much of the mathematical details in an appendix, which tends to be published online.

“Our analysis seems to show that for equations put in an appendix there isn’t such an effect,” said Dr Fawcett.

“But there’s a big risk that in doing that you are potentially hiding the maths away, so it’s important to state clearly the assumptions and implications in the main text for everyone to see.

The findings were reported in the journal Proceedings of the National Academy of Sciences.

[image source]

fourier transformation

Fourier transformation optimized algorithm turns fast into superfast

fourier transformationThe Fourier transformation is arguably the most important algorithm in information technology, with immense applications as well  in optics, signal and image processing, pattern recognition etc. Thanks to this remarkable mathematical operation, we’re able to see videos or listen to music on an iPod, as it turns the digital information into readable frequencies. Recently, MIT scientists have managed to come up with an optimized algorithm of the Fast Fourier Transformation, which was already fast enough as one can imagine. The researchers’ results in some instances had a tenfold increase in processing speed.

In simple terms the Fourier transformation turns signals into frequencies. A simple example as far as applications go is how it can turn voltage signals transmitted through a wire to an mp3 player into sounds rendered through a speaker fast and easy. However, it’s been found indispensable in applications ranging from economics, engineering,  sociology and so on.

In the 1960s the Fast Fourier Transformation algorithm was developed, which provided an absolute breakthrough, still the question remained to this day whether it could be optimized even further. The MIT mathematicians devised the new faster than Fast Fourier Transformation by granting importance to frequencies that “weigh” more and overlooking weak signals.

The algorithm takes a digital signal containing a certain number of samples and expresses it as the weighted sum of an equivalent number of frequencies. Some of these frequencies are more important or “heavy” to the signal, and are thus prioritized. The algorithm slices the signal into narrow bandwidths, each slice containing just one heavy frequency. Each slice is then sliced again and so on once even further until low-weighted frequencies and highly-weighted signals are completely isolated from one another.

In “spare” signals, whose Fourier transforms include a relatively small number of heavily weighted frequencies, the new algorithm can output at lightning speed compared to the old one, as low weight signals are cut out completely with absolutely no loss in quality. “In nature, most of the normal signals are sparse,” says Dina Katabi, one of the developers of the new algorithm.

Considering most of the signals in nature are sparse, and the fact that the FFT was already lightning fast, this new improvement from MIT might have extraordinary consequences. Using your smartphone to wirelessly transmit large video files without draining the battery is just one application, out of countless that might benefit from it.

Read more about the research in technical detail at the MIT press release.

 

Sudoku

Mathematician solves sudoku dilema: 17 minimum clues for a solution

One of my favorite past times is filling sudoku puzzles. There’s something about this seemingly simple, yet challenging, dance of digits up and down, left and right that manages to keep me highly entertain though a perfectly balanced mixture of thrill and frustration. If you think you’re good enough to solve any kind of sudoku, be aware that some combinations are impossible to solve.

SudokuGary McGuire, a mathematician of University College Dublin, has pain-stakingly devised an algorithm through which he has scientifically proven that a sudoku puzzle can’t have less than 17 clues, since puzzles with 16 or fewer clues do not have a unique solution. Your typical newspaper sudoku has around 25 clues, on the safe side – just enough not to get bored, while keeping frustration amounted from failing to see the solution at bay.

Gasp! It suddenly hit me that some ZME readers might not know what a sudoku is in the first place. Well, I must make haste before I continue to explain how the game works. Very simple: sudoku involves filling in a 9×9 grid of squares, according to a set of rules (each 3×3 box needs to be filled with each number from 1-9, the same goes for the every column and line of the 9×9 grid). To kick things off and be able to find a solution, the puzzle has some boxes filled out in a particular order – these are your clues, the fewer you have, the harder the game gets.

Back to science. Professor McGuire presented his work at the recent conference in Boston, where his findings where heralded valid and deemed as an important advance in the growing field of Sudoku mathematics, which is more important than one might think. I’ll get to that soon enough.

There have been literary thousands of 16 clues sudoku trials, however it was found that in every instance, there could be found only one solution – but how to demonstrate this? Gary McGuire developed a “hitting set algorithm” to definitively prove the theory. This algorithm looks for what McGuire calls “unavoidable sets”, couples of filled in values in the completed values which when interchanged can result in multiple solutions. By replacing these unavoidable sets values and positions with clue values the computing task at hand becomes a lot less complex, though still very though.

“The approach is reasonable and it’s plausible. I’d say the attitude is one of cautious optimism,” says Jason Rosenhouse, a mathematician at James Madison University in Harrisonburg, Va., and the co-author of a newly released book on the mathematics of Sudoku.

To run this algorithm for solutions, brute force is the only option. Even with the unavoidable sets algorithm in place,  which took two years to tweak, McGuire and his team used about 700 million CPU hours at the Irish Centre for High-End Computing in Dublin, searching through possible grids with the hitting-set algorithm. The work, says McGuire, has implications beyond Sudoku itself.

“Hitting set problems have applications in many areas of science, such as bioinformatics and software testing,” he says.

McGuire’s paper was published in a recent edition of the journal Nature.

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Spacial reasoning gender gap disappears in female-dominant cultures

Currently, only about 30% percent of the total scientific workforce is comprised of female scientists. Thousands of years of cultural discrepancies might be to blame for this, like stereotyping, however in societies where math gender gaps disappears, the gender gap remains in higher education.

In Sweden or Norway, the math gender gap has been bridged, as persons of both sexes manage to score similarly in tests, however even there men seem to show a better spatial reasoning ability. Are men and women hot-wired differently from birth with these terms in mind or are these discrepancies as a result of social engineering? A team of scientists capitalized on a set of perfect natural experiments as part of a recent study published in PNAS looking to answer these questions.

Remarkably, they managed to find two settlements in Northeast India very similar in all the right ways to make the study relevant, but different enough to make a point. Both are very close to one another, both employ an agrarian lifestyle, which renders the same diet and share a very similar DNA , culturally-wise however they’re at opposite poles.

The inhabitants of one of the settlements, the Karbi, are entirely patrilineal: women have no proprietary rights to land and the oldest son in the family inherits everything when the parents die. On the other side of the fence, the Khasi, are matrilineal: men have no rights to own land, and the youngest daughter in the family inherits everything. The researchers couldn’t ask for possibly more from this naturally perfect case study environment.

To test how the two societies scored at spacial reasoning, the scientists introduced the task of solving a simple three-dimensional puzzle that involved four blocks, with portions of a picture on a single face. The subjects would have to identify the correct side of the block, rotate it to the top, and then arrange the pieces to re-form the picture. Whoever could solve the simple task in under 30 seconds was rewarded with the equivalent of a quarter day’s salary – early 1,300 villagers agreed to participate.

In the patrilinial settlement, the Karbi men took 35% less time to perform the task than Karbi women. A very significant different, which almost vanished in the Khasi tribe where no such differences could be encountered in the scores of the two sexes.

Scientists explain that these differences, they claim, are due to cultural differences. Patrilinial men are more likely to receive education, a factor which when taken into account researchers found it accounts for a third of the performance difference. Male ownership of the home also had a large effect; the gender gap is only a third the size in homes that are not owned solely by males.

Other factors like gender competitiveness or inheritance didn’t seem to influence the results too much. As a conclusion to their study, the authors outline that cultural differences might account to spacial reasoning differences, however they disclaim the fact that their work is correlative and should be taken with a grain of salt. The Karbi/Khasi case study only offers a small snapshot of the human diversity spectrum.

PNAS via wired

The Futurama theorem

In case you don’t know, Futurama is a popular science fiction – I highly recommend it, as a matter of fact, but that’s more of a personal preference. What’s interesting about it is that a mathematical theory was created especially for it, or for an episode to be more exact.

The theory refers to a hypothetical scenario where two persons have used a machine to swap their minds, but cannot directly change them back due to an immune response. The theory from the mathematical field of study called group theory says that no matter how many people have swapped minds with one another, they can still get back to the original configuration with two additional individuals who haven’t swapped minds with anyone. The theory was proven by the show’s writer Ken Keeler, who also has a PhD in mathematics.

For the ones of you who like math, I think you should try and demonstrate it for yourself, it’s quite interesting. For those interested about this, but really that savvy, there’s also a demonstration on Wikipedia. Either way, who says cartoons aren’t smart ?