## Archive for the ‘Mathematics’ Category

### How to drill a square hole

August 26, 2014### First woman ever among four awarded 2014 Fields medals

August 13, 2014The Fields medal is the most prestigious award for mathematics and was first awarded in 1936. For 2014, four winners were announced this week and Maryam Mirzahkani, Professor of Mathematics at Stanford, becomes the first woman ever to be awarded a Fields medal.

Though women are generally underrepresented in mathematics – I suspect partly because of a lack of interest and partly because it is not a “politically correct” occupation – there have been many **prominent female mathematicians**. But this is the first time in the almost 80 years since it was established that a woman has won the Fields medal.

The Fields Medal is awarded every four years on the occasion of the International Congress of Mathematicians to recognize outstanding mathematical achievement for existing work and for the promise of future achievement.

The Fields Medal Committee is chosen by the Executive Committee of the International Mathematical Union and is normally chaired by the IMU President. It is asked to choose at least two, with a strong preference for four, Fields Medallists, and to have regard in its choice to representing a diversity of mathematical fields. A candidate’s 40th birthday must not occur before January 1st of the year of the Congress at which the Fields Medals are awarded.

Maryam Mirzakhani, a professor of mathematics at Stanford University in California, was named the first female winner of the Fields Medal – often described as the Nobel prize for mathematics – at a ceremony in Seoul on Wednesday morning.

The prize, worth 15,000 Canadian dollars, is awarded to exceptional talents under the age of 40 once every four years by the International Mathematical Union. Between two and four prizes are announced each time.

Three other researchers were named Fields Medal winners at the same ceremony in South Korea. They included Martin Hairer, a 38-year-old Austrian based at Warwick University in the UK; Manjul Bhargava, a 40-year old Canadian-American at Princeton University in the US and Artur Avila, 35, a Brazilian-French researcher at the Institute of Mathematics of Jussieu in Paris.

There have been 55 Fields medallists since the prize was first awarded in 1936, including this year’s winners. The Russian mathematician Grigori Perelman refused the prize in 2006 for his proof of the Poincaré conjecture.

The citations for the four winners:

### Mystical threes and magic scaling number of the Efimov State

June 3, 2014The number three has long been attributed with mystical and divine properties.

Time and Life itself is a matter of threes. Birth, life and death. The past, the present and the future. Third time lucky. Three wishes. The Holy Trinity. Three daughters. The Good, the Bad and the Ugly. The three primary colours. A Troika. Brahma,Vishnu, Shiva. The Creator, the Preserver, the Destroyer. Three monkeys. Three wise men. Three Kings.

Three has its place in Physics as well. Pascal’s triangle and the Golden Number and the Fibonacci series. A theoretical prediction that fundamental particles in sets of three give rise to stable arrangements of infinitely scaleable, nesting sets has now been shown to be real – the **Efimov State**.

Wired:More than 40 years after a Soviet nuclear physicist proposed an outlandish theory that trios of particles can arrange themselves in an infinite nesting-doll configuration, experimentalists have reported strong evidence that this bizarre state of matter is real.

n 1970, Vitaly Efimov was manipulating the equations of quantum mechanics in an attempt to calculate the behavior of sets of three particles, such as the protons and neutrons that populate atomic nuclei, when he discovered a law that pertained not only to nuclear ingredients but also, under the right conditions, to any trio of particles in nature.

While most forces act between pairs, such as the north and south poles of a magnet or a planet and its sun, Efimov identified an effect that requires three components to spring into action. Together, the components form a state of matter similar toBorromean rings, an ancient symbol of three interconnected circles in which no two are directly linked. The so-called Efimov “trimer” could consist of a trio of protons, a triatomic molecule or any other set of three particles, as long as their properties were tuned to the right values. And in a surprising flourish, this hypothetical state of matter exhibited an unheard-of feature: the ability to range in size from practically infinitesimal to infinite.

Efimov had shown that when three particles come together, a special confluence of their forces creates the Borromean rings effect: Though one is not enough, the effects of two particles can conspire to bind a third. The nesting-doll feature — called discrete scale invariance — arose from a symmetry in the equation describing the forces between three particles. If the particles satisfied the equation when spaced a certain distance apart, then the same particles spaced 22.7 times farther apart were also a solution. This number, called a “scaling factor,” emerged from the mathematics as inexplicably as pi, the ratio between a circle’s circumference and diameter.

Now it seems 3 different research teams have shown the existence of Efimov nesting.

“With just one example, it’s very difficult to tell if it’s a Russian nesting doll,” said Cheng Chin, a professor of physics at the University of Chicago who was part of Grimm’s group in 2006. The ultimate proof would be an observation of consecutive Efimov trimers, each enlarged by a factor of 22.7. “That initiated a new race” to prove the theory, Chin said.

Eight years later, the competition to observe a series of Efimov states has ended in a photo finish. “What you see is three groups, in three different countries, reporting these multiple Efimov states all within about one month,” said Chin, who led one of the groups. “It’s totally amazing.”

### Half of Newton’s papers were on religion, 10% on alchemy and only 30% on science and math

May 16, 2014Unlike Alfred Nobel who I posted about recently, Isaac Newton left no will when he died in 1727. But he left behind him a mass of papers estimated to run to about 10 million words. But most of the notes he left behind dealt with religious subjects and alchemy and his views were not just politically incorrect but potentially embarrassing if not dangerous to his heirs.

Wired has interviewed Sarah Dry who has just published her book on **“****The Newton Papers****“**.

Wired:He wrote a forensic analysis of the Bible in an effort to decode divine prophecies. He held unorthodox religious views, rejecting the doctrine of the Holy Trinity. After his death, Newton’s heir, John Conduitt, the husband of his half-niece Catherine Barton, feared that one of the fathers of the Enlightenment would be revealed as an obsessive heretic. And so for hundreds of years few people saw his work. It was only in the 1960s that some of Newton’s papers were widely published.

Now of course The Newton Project is putting all of his papers online and they have so far transcribed about 6.4 million words:

The Newton Projectis a non-profit organization dedicated to publishing in full an online edition of all of Sir Isaac Newton’s (1642–1727) writings — whether they were printed or not. The edition presents a full (diplomatic) rendition featuring all the amendments Newton made to his own texts or a more readable (normalised) version. We also make available translations of his most important Latin religious texts.

Although Newton is best known for his theory of universal gravitation and discovery of calculus, his interests were much broader than is usually appreciated. In addition to his celebrated scientific and mathematical writings, Newton also wrote many alchemical and religious texts.

Sarah Dry traces the history of the Newton papers and how they languished over the years. It was not perhaps by conspiracy but there was some clear apprehension that sorting and cataloguing them would be embarrassing because there was so much of a “heretical” nature:

Sarah Dry in Wired:There’s roughly 10 million words that Newton left. Around half of the writing is religious, and there are about 1 million words onalchemical material, most of which is copies of other people’s stuff. There are about 1 million words related to his work as Master of the Mint. And then roughly 3 million related to science and math.

…… one of the messages of the book is that getting too involved in the papers can be hazardous to your health. One of the first editors of the papers said an older man should take up the task, because he’d have less to lose than a younger man.

This is highly technical stuff. The alchemical stuff is technical, the scientific stuff is technical, the religious stuff is technical. I was more interested in the papers and the characters that worked on them. One person was David Brewster, who wrote a biography of Newton during the Victorian Era. He fought long and hard to resuscitate Newton’s reputation. But he was also one of these Victorians that had to tell the truth. So when he published his biography [in 1855], it included much of the heresy and alchemy, despite the fact that Brewster was a good orthodox Protestant.

…. When the papers came to Cambridge in the late 1800s, they were unsorted and chaotic. And the two men given to sorting them were John Couch Adams and George Stokes. Adams was the co-discoverer of Neptune. He famously never wrote anything down. And Stokes was just as great a physicist, but he wrote everything down. He in fact wrote 10,000 letters. So these two guys get the papers, and then they sit on them for 16 years; they basically procrastinate.

When actually confronted with Newton’s paper, they were horrified and dismayed. Here was this great scientific hero. But he also wrote about alchemy and even more about religious matters. Newton spent a long time writing a lot of unfinished treatises. Sometimes he would produce six or seven copies of the same thing. And I think it was disappointing to see your intellectual father copying this stuff over and over. So the way Adams and Stokes dealt with it was to say that, “His power of writing a beautiful hand was evidently a snare to him.” Basically, they said he didn’t like this stuff, he just liked his own writing.

There’s also Grace Babson, who created the largest collection of Newton objects and papers in America. She was married to a man who got rich predicting the crash of 1929. And Roger Babson [her husband] based his market research on Newtonian principles, using the idea that for every action there is an equal an opposite reaction. The market goes up so it must come down. Interestingly, he thought of gravity as an evil scourge.

Clearly people felt that tarnishing Newton’s image was a heresy in itself and they felt that publicising his stranger writings could do such damage to their icon. But the time since his death is critical here. Newton’s image is now immune to such damage. I think that no matter how weird his views may have been about the Bible and prophecies and the occult and alchemy, they cannot – now – detract from his work on maths and physics and motion.

But his catalogers have a point. If one part of his work had been debunked or ridiculed soon after his death, it could have damaged his reputation and even the credibility of his work in Physics and Maths. It is common practice now – as it was common practice then – for detractors to attack an opponent’s views on one subject obliquely, by denigrating his views or work in some other field. Wrong thinking in one field – by association – becomes wrong thinking in all fields.

It may have been different if they had TV in those days. For if Newton had lived in today’s world it could well be that his eminence in Physics and Maths would have made him an instant TV pundit on all subjects. We would be suffering the pain of listening him to expound on his other weird and wonderful ideas. As we all must endure when we have to listen to actors pontificating about environmental science or psychiatrists excusing errant behaviour or politicians pretending they understand economics!!

### Seventeen equations that changed the world

March 20, 2014I just came across this summarising Ian Stewart’s book on **17 Equations That Changed The World** at **Business Insider****:**** **

I have used all of these up to Equation 12. I have never used the equations on Relativity or Schrodinger’s equation or those on Chaos or Information theory or the Black-Scholes Equation. But, I wouldn’t disagree with Equations 12 – 17, but considering the amount of time I spent applying it at University and during my working life I would have liked to see Bernoulli’s Equation on the list:

where:

- is the fluid flow speed at a point on a streamline,
- is the acceleration due to gravity,
- is the elevation of the point above a reference plane, with the positive
*z*-direction pointing upward – so in the direction opposite to the gravitational acceleration, - is the pressure at the chosen point, and
- is the density of the fluid at all points in the fluid.

### Visualising the number of digits in the largest known prime number

January 25, 2014Cool!

The largest known prime number is **M57885161**, which has **17,425,170** digits and was first discovered in **2013**.

### Visualising 17,425,170 digits.

**Click here for the deep zoom into the digits of the largest prime**

The biggest prime number ever discovered is 17 million decimal digits long. Its predecessor, discovered in 2008 was 12 million digits long. Those are huge numbers, but there is also a huge gap between them.

In order to be efficient, the algorithms that have been developed to discover large primes will often leave large areas of unexplored territory in the number-space behind them: the “lost primes”.

We’re challenging you to use cloud computing to find one of those lost primes, and help to increase mathematical knowledge.

Most of the big prime discoveries have used many hundreds of thousands of computers over many years – it takes a lot of computing power to calculate a number that is 17 million digits long. This type of computing power was previously out of reach for casual observers. But cloud computing has changed that and we now all have access to a huge amount of computing power.

This challenge gives everyone the chance to discover new prime number by using cloud computing. We really aren’t expecting to get anywhere near close to the largestrimes ever discovered, but we do expect to find many of the lost primes. The challenge will also highlight which architectures and configurations of cloud computing resources work best for this kind of task.

### Meshing Gears

January 12, 2014Another fabulous image by Paul Nylander at **bugman123.com**.

A set of 242 interlocking bevel gears arranged to rotate freely along the surface of a sphere. This sphere is composed of 12 blue gears with 25 teeth each, 30 yellow gears with 30 teeth each, 60 orange gears with 14 teeth each, and 140 small red gears with 12 teeth each. I also found 3 other gear tooth ratios that will work, but this one was my favorite because the small gears emphasize the shape of atruncated rhombic triacontahedron.

### Numeracy and language

December 2, 2013I tend towards considering mathematics a language rather than a science. In fact mathematics is more like a family of languages each with a rigorous grammar. I like this quote:

*R. L. E. Schwarzenberger*, The Language of Geometry, in *A Mathematical Spectrum Miscellany*, Applied Probability Trust, 2000, p. 112:

My own attitude, which I share with many of my colleagues, is simply that mathematics is a language. Like English, or Latin, or Chinese, there are certain concepts for which mathematics is particularly well suited: it would be as foolish to attempt to write a love poem in the language of mathematics as to prove the Fundamental Theorem of Algebra using the English language.

Just as conventional languages enable culture and provide a tool for social communication, the various languages of mathematics, I think, enable science and provide a tool for scientific discourse. I take “science” here to be analaogous to a “culture”. To follow that thought then, just as science is embedded within a “larger” culture, so is mathematics embedded within conventional languages. This embedding shows up as the ability of a language to deal with numeracy and numerical concepts.

And that means then the value judgement of what is “primitive” when applied to language can depend upon the extent to which mathematics and therefore numeracy is embedded within that language.

**GeoCurrents examines numeracy embedded within languages**:

According to a recent article by Mike Vuolo in

Slate.com, Pirahã is among “only a few documented cases” of languages that almost completely lack of numbers. Dan Everett, a renowned expert in the Pirahã language, further claims that the lack of numeracy is just one of many linguistic deficiencies of this language, which he relates to gaps in the Pirahã culture. …..The various types of number systems are considered in the

WALS.info articleon Numeral Bases, written by Bernard Comrie. Of the 196 languages in the sample, 88% can handle an infinite set of numerals. To do so, languages use some arithmetic base to construct numeral expressions. According to Comrie, “we live in a decimal world”: two thirds of the world’s languages use base 10 and such languages are spoken “in nearly every part of the world”. English, Russian, and Mandarin are three examples of such languages. …..Around 20% of the world’s languages use either purely vigesimal (or base 20) or a hybrid vigesimal-decimal system. In a purely vigesimal system, the base is consistently 20, yielding the general formula for constructing numerals as

x20 + y. For example, in Diola-Fogny, a Niger-Congo language spoken in Senegal, 51 is expressed asbukanku-gabadiuɲɛndib-əkɔn‘two twenties and eleven’. Other languages with a purely vigesimal system include Arawak spoken in Suriname,Chukchispoken in the Russian Far East, Yimas in Papua New Guinea, and Tamang in Nepal. In a hybrid vigesimal-decimal system, numbers up to 99 use base 20, but the system then shifts to being decimal for the expression of the hundreds, so that one ends up with expressions of the typex100 + y20 + z. A good example of such a system isBasque, where 256 is expressed asberr-eun eta berr-ogei-ta-hama-sei‘two hundred and two-twenty-and-ten-six’. Other hybrid vigesimal-decimal systems are found in Abkhaz in the Caucasus,Burushaskiin northern Pakistan, Fulfulde in West Africa, Jakaltek in Guatemala, and Greenlandic. In a few mostly decimal languages, moreover, a small proportion of the overall numerical system is vigesimal. In French, for example, numerals in the range 80-99 have a vigesimal structure: 97 is thus expressed asquatre-vingt-dix-sept‘four-twenty-ten-seven’. Only five languages in the WALS sample use a base that is neither 10 nor 20. For instance, Ekari, a Trans-New Guinean language spoken in Indonesian Papua uses base of 60, as did the ancient Near Eastern language Sumerian, which has bequeathed to us our system of counting seconds and minutes. Besides Ekari, non-10-non-20-base languages include Embera Chami in Colombia, Ngiti inDemocratic Republic of Congo, Supyire in Mali, and Tommo So in Mali. ……Going back to the various types of counting, some languages use a restricted system that does not effectively go above around 20, and some languages are even more limited, as is the case in Pirahã. The WALS sample contains 20 such languages, all but one of which are spoken in either Australia, highland New Guinea, or Amazonia. The one such language found outside these areas is !Xóõ, a Khoisan language spoken in Botswana. …….

In some societies in the ancient past, numeracy did not contribute significantly to survival as probably with isolated tribes like the Pirahã. But in most human societies, numeracy was of significant benefit especially for cooperation between different bands of humans. I suspect that it was the need for social cooperation which fed the need for communication within a tribe and among tribes, which in turn was the spur to the development of language, perhaps over 100,000 years ago. What instigated the need to count is in the realm of speculation. The need for a calendar would only have developed with the development of agriculture. But the need for counting herds probably came earlier in a semi-nomadic phase. Even earlier than that would have come the need to trade with other hunter gatherer groups and that probably gave rise to counting 50,000 years ago or even earlier. The tribes who learned to trade and developed the ability and concepts of trading were probably the tribes that had the best prospects of surviving while moving from one territory to another. It could be that the ability to trade was an indicator of how far a group could move.

And so I am inclined to think that numeracy in language became a critical factor which 30,000 to 50,000 years ago determined the groups which survived and prospered. It may well be that it is these tribes which developed numbers, and learned to count, and learned to trade that eventually populated most of the globe. It may be a little far-fetched but not impossible that numeracy in language may have been one of the features distinguishing Anatomically Modern Humans from Neanderthals. Even though the Neanderthals had larger brains and that we are all Neanderthals to some extent!

### From Mandelbrot to Mandelbulbs with Chaos in between

October 31, 2013The

is a three-dimensional analogue of theMandelbulb, constructed by Daniel White and Paul Nylander using spherical coordinates. A canonical 3-dimensional Mandelbrot set does not exist, since there is no 3-dimensional analogue of the 2-dimensional space of complex numbers. It is possible to construct Mandelbrot sets in 4 dimensions using quaternions. However, this set does not exhibit detail at all scales like the 2D Mandelbrot set does.Mandelbrot set

Here is my first rendering of an 8th order Mandelbulb set, based on the following generalized variation of Daniel White’s original squarring formula:

{x,y,z}^{n}= r^{n}{cos(θ)cos(φ),sin(θ)cos(φ),-sin(φ)}

A classic Mandelbrot set