Archive for the ‘Mathematics’ Category

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.

Numeracy and language

December 2, 2013

I 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. SchwarzenbergerThe 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 article on 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 as bukan ku-gaba di uɲɛn di b-əkɔn ‘two twenties and eleven’. Other languages with a purely vigesimal system include Arawak spoken in Suriname, Chukchi spoken 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 type x100 + y20 + z. A good example of such a system is Basque, where 256 is expressed as berr-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, Burushaski in 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 as quatre-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 in Democratic 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. ……. 

Read the whole article. 

Counting monkey?

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, 2013

The Mandelbulb is a three-dimensional analogue of the Mandelbrot set, 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.

From bugman123

an 8th order Mandelbulb set by bugman123

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 = rn{cos(θ)cos(φ),sin(θ)cos(φ),-sin(φ)}

Paul Nylander, bugman123.com

A classic Mandelbrot set

Mandelbrot set – Wikipedia

The mathematics of a pizza bite (by Sheffield University for Pizza Express)

October 19, 2013

English: Picture of an authentic Neapolitan Pi...

It is now crystal clear.  Eugenia Cheng is both a mathematician and a pizza lover.

A median bite from an 11” pizza has 10% more topping than a median bite from the 14” pizza.

On the perfect size for a pizza

cheng-pizza pdf
Eugenia Cheng
School of Mathematics and Statistics, University of Sheffield
E-mail: e.cheng@sheffield.ac.uk
October 14th, 2013
Abstract
We investigate the mathematical relationship between the size of a pizza and its ratio of topping to base in a median bite. We show that for a given recipe, it is not only the overall thickness of the pizza that is affected by its size, but also this topping-to-base ratio.

Acknowledgements: This study was funded by Pizza Express.

The ratio of topping to base in a median bite is given by

Formula for median pizza bite (Cheng)

where

r = radius of pizza (half the diameter) in inches
d = volume of dough (constant)
t = volume of topping (constant)
α = scaling constant for the edge

The IPCC 95% trick: Increase the uncertainty to increase the certainty

October 17, 2013

Increasing the uncertainty in a statement to make the statement more certain to be applicable is an old trick of rhetoric. Every politician knows how to use that in a speech. It is a schoolboy’s natural defense when being hauled up for some wrongdoing. It is especially useful when caught in a lie. It is the technique beloved of defense lawyers in TV dramas. Salesmen are experts at this. It is standard practice in scientific publications when experimental data does not fit the original hypothesis.

Modify the original statement (the lie) to be less certain in the lie, so as to be more certain that the statement could be true. Widen the original hypothesis to encompass the actual data. Increase the spread of the deviating model results to be able to include the real data within the error envelope.

  • “I didn’t say he did it. I said somebody like him could have done it”
  • “Did you start the fight?” >>> “He hit me back first”.
  • “The data do not match your hypothesis” >>> “The data are not inconsistent with the improved hypothesis”
  • “Your market share has reduced” >>> “On the contrary, our market share of those we sell to has increased!” (Note -this is an old one used by salesmen to con “green” managers with reports of a 100% market share!!)

And it is a trick that is not foreign to the IPCC  - “we have a 95% certainty that the less reliable (= improved) models are correct”. Or in the case of the Cook consensus “97% of everybody believes that climate does change”.

A more rigorous treatment of the IPCC trick is carried out by Climate Audit and Roy Spencer among others but this is my simplified explanation for schoolboys and Modern Environ-mentalists.

The IPCC Trick

The IPCC Trick

The real comparison between climate models and global temperatures is below:

Climate Models and Reality

Climate Models and Reality

With the error in climate models increased to infinity, the IPCC could even reach 100% certainty. As it is the IPCC is 95% certain that it is warming – or not!


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