Archive for the ‘Mathematics’ Category

Mathematical images by Yeganeh

January 11, 2015
Yeganeh bird in flight

Yeganeh bird in flight

Hamid Naderi Yeganeh, “A Bird in Flight” (November 2014)

This image is like a bird in flight. It shows 2000 line segments. For each i=1, 2, 3, … , 2000 the endpoints of the i-th line segment are:
(3(sin(2πi/2000)^3), -cos(8πi/2000))
and
((3/2)(sin(2πi/2000)^3), (-1/2)cos(6πi/2000)).

See his gallery of images here.

Hamid Naderi Yeganeh is a Bachelor student of mathematics at the University of Qom. He won gold medal at the 38th Iranian Mathematical Society’s Competition (2014).

A Generalization of Wallis Product by Mahdi Ahmadinia and Hamid Naderi Yeganeh

PlusMaths writes:

…but it’s actually a collection of points in the plane given by a mathematical formula. To be precise, it’s a subset of the complex plane consisting of points of the form

  \[ \lambda A(t)+(1-\lambda )B(t), \]    

where

  \[ A(t)= 3(\sin (t))^{3}- \frac{3i}{4}\cos (4t) \]    

and

  \[ B(t)= \frac{3}{2}(\sin (t))^{5} - \frac{i}{2}\cos (3t) \]    

for $0\leq t \leq 2\pi $ and $0\leq \lambda \leq 1.$

The image was created by Hamid Naderi Yeganeh.

The certainty of the improbable

January 4, 2015

When you toss a coin, there is complete certainty that an event that is only 50% probable will occur. When you roll a dice there is absolute certainty that a 16.67% probable event will come to pass. It sounds trivial. After all the probability is only to distinguish between outcomes once it is certain that the coin will be tossed or that the dice will be rolled. Probability of an outcome is meaningless if the coin were not tossed or the dice not rolled. But note also that the different outcomes must be pre-defined. If you toss a silver coin in the air the return of a golden coin is not included among the pre-defined, possible outcomes. That a roll of the dice can result in a 9 is not “on the cards”.

Probability or improbability of an event or a causal relationship is meaningless unless the certainty of some more general event or relationship is certain. Moreover as soon as we define the event or relationship to which we allocate a probability, we also define that that event or relationship is permitted. It is certain that tomorrow will be another day. Only because it is certain can we consider the probability – or improbability – of what weather tomorrow might bring.  Suppose we define the weather as being either “good”, “bad” or “indifferent”. We can guess or calculate the probability of tomorrow’s weather exhibiting one of these 3 permitted outcomes. My point is that as soon as we define the improbable we also make it certain that the selected outcomes are all permitted. Then even the most improbable – but permitted – weather outcome will, on some day, occur. Not just permitted – but certain. If the improbable never happens then it is impossible – not improbable.

We use statistics and probabilities of occurrence because we don’t know the mechanisms which govern the outcome. If mechanisms were known in their entirety, we would just calculate the result – not the probability of a particular result. The very mention of a probability is always an admission of ignorance. It means that we cannot tell what makes something probable or improbable and even what we consider improbable will surely occur. An outcome of even very low probability will then – given sufficient total occurrence – certainly occur. The 2011 earthquake and tsunami off the Tōhoku coast was a one-in-a-1,000 year occurrence. The probability of it happening next year remains at one-in-a-thousand. But given another 1,000 years it will (almost) certainly happen again.

One of my concerns is that the use of statistics and probability – say in medical trials – is usually taken to imply knowledge, but it is actually an admission of ignorance. No doubt the use of statistics and probability help to constrain the boundaries of the ignorance, but the bottom line is that even the low probability risks will materialise. The very use of probabilities is always because of a lack of knowledge, because of ignorance.

In the beginning of December I was having a regular medical check-up and I was offered a flu-shot for the winter which I took. But I got to wondering why I did. The influenza vaccine is effective in about 50% of cases (i.e. 50% achieve protection). Around 5% – irrespective of whether they achieve protection or not – suffer some adverse reaction to the shot. Around 0.5% of the 5% (1:4,000 of total vaccinated) suffer a fatal reaction. In our little clinic perhaps 3,000 were vaccinated this winter. About 1,500 would have achieved protection and about 150 must have had some adverse reaction. Most likely one person has or will suffer a fatal reaction. I was just gambling that I would not be that one person. When some new drug is said to have a 1% chance of adverse effects it only means that it will certainly have adverse effects for 1 in a hundred cases. When that one person chooses to take that drug, he may be making the best medical choice possible – but it is the wrong choice.

A low risk for the multitude but a certainty for some. The chances of something improbable never happening are virtually zero.

Improbable – but certain.

How to drill a square hole

August 26, 2014
how to drill a square hole (via imgur)

how to drill a square hole (via imgur)

gif image from here

First woman ever among four awarded 2014 Fields medals

August 13, 2014

The 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.

The Guardian:

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, 2014

The number three has long been attributed with mystical and divine properties.

trinityTime 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.

WiredMore 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 to Borromean 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.”

Read the article.

Related: Physicists Prove Surprising Rule of Threes

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

May 16, 2014

Unlike 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.

WiredHe 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 Project is 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 WiredThere’s roughly 10 million words that Newton left. Around half of the writing is religious, and there are about 1 million words on alchemical 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, 2014

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

seventeen equations

seventeen equations

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:

Bernoulli's Equation

where:

v\, is the fluid flow speed at a point on a streamline,
g\, is the acceleration due to gravity,
z\, 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,
p\, is the pressure at the chosen point, and
\rho\, is the density of the fluid at all points in the fluid.

Idiot paper of the day: “Math Anxiety and Exposure to Statistics in Messages About Genetically Modified Foods”

February 28, 2014

Roxanne L. Parrott is the Distinguished Professor of Communication Arts and Sciences at Penn State. Reading about this paper is not going to get me to read the whole paper anytime soon. The study the paper is based on – to my mind – is to the discredit of both PennState and the state of being “Distinguished”.

I am not sure what it is but it is not Science.

Kami J. Silk, Roxanne L. Parrott. Math Anxiety and Exposure to Statistics in Messages About Genetically Modified Foods: Effects of Numeracy, Math Self-Efficacy, and Form of PresentationJournal of Health Communication, 2014; 1 DOI: 10.1080/10810730.2013.837549

From the Abstract:

… To advance theoretical and applied understanding regarding health message processing, the authors consider the role of math anxiety, including the effects of math self-efficacy, numeracy, and form of presenting statistics on math anxiety, and the potential effects for comprehension, yielding, and behavioral intentions. The authors also examine math anxiety in a health risk context through an evaluation of the effects of exposure to a message about genetically modified foods on levels of math anxiety. Participants (N = 323) were randomly assigned to read a message that varied the presentation of statistical evidence about potential risks associated with genetically modified foods. Findings reveal that exposure increased levels of math anxiety, with increases in math anxiety limiting yielding. Moreover, math anxiety impaired comprehension but was mediated by perceivers’ math confidence and skills. Last, math anxiety facilitated behavioral intentions. Participants who received a text-based message with percentages were more likely to yield than participants who received either a bar graph with percentages or a combined form. … 

PennState has put out a Press Release:

The researchers, who reported their findings in the online issue of the Journal of Health Communication, recruited 323 university students for the study. The participants were randomly assigned a message that was altered to contain one of three different ways of presenting the statistics: a text with percentages, bar graph and both text and graphs. The statistics were related to three different messages on genetically modified foods, including the results of an animal study, a Brazil nut study and a food recall announcement.

Wow! The effort involved in getting all of 323 students to participate boggles. And taking Math Anxiety as a critical behavioural factor stretches the bounds of rational thought. Could they find nothing better to do? This study is at the edges of academic misconduct.

“This is the first study that we know of to take math anxiety to a health and risk setting,” said Parrott.

It ought also to be the last such idiot study – but I have no great hopes.

Visualising the number of digits in the largest known prime number

January 25, 2014

Cool!

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

Visualising 17,425,170 digits.

largest prime known

largest prime known

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

THE PRIME CHALLENGE:

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, 2014

Another fabulous image by Paul Nylander at bugman123.com.

image by Paul Nylander 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 a truncated rhombic triacontahedron.


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