Category Archives: Mathematics

Epic Scottish poetry and social network science

It’s been a while since I’ve run a social network story here and this research into a 250-year controversy piqued my interest anew. From an Oct. 20, 2016 Coventry University (UK) press release (also on EurekAlert) Note: A link has been removed,

The social networks behind one of the most famous literary controversies of all time have been uncovered using modern networks science.

Since James Macpherson published what he claimed were translations of ancient Scottish Gaelic poetry by a third-century bard named Ossian, scholars have questioned the authenticity of the works and whether they were misappropriated from Irish mythology or, as heralded at the time, authored by a Scottish equivalent to Homer.

Now, in a joint study by Coventry University, the National University of Ireland, Galway and the University of Oxford, published today in the journal Advances in Complex Systems, researchers have revealed the structures of the social networks underlying the Ossian’s works and their similarities to Irish mythology.

The researchers mapped the characters at the heart of the works and the relationships between them to compare the social networks found in the Scottish epics with classical Greek literature and Irish mythology.

The study revealed that the networks in the Scottish poems bore no resemblance to epics by Homer, but strongly resembled those in mythological stories from Ireland.

The Ossianic poems are considered to be some of the most important literary works ever to have emerged from Britain or Ireland, given their influence over the Romantic period in literature and the arts. Figures from Brahms to Wordsworth reacted enthusiastically; Napoleon took a copy on his military campaigns and US President Thomas Jefferson believed that Ossian was the greatest poet to have ever existed.

The poems launched the romantic portrayal of the Scottish Highlands which persists, in many forms, to the present day and inspired Romantic nationalism all across Europe.

Professor Ralph Kenna, a statistical physicist based at Coventry University, said:

By working together, it shows how science can open up new avenues of research in the humanities. The opposite also applies, as social structures discovered in Ossian inspire new questions in mathematics.”

Dr Justin Tonra, a digital humanities expert from the National University of Ireland, Galway said:

From a humanities point of view, while it cannot fully resolve the debate about Ossian, this scientific analysis does reveal an insightful statistical picture: close similarity to the Irish texts which Macpherson explicitly rejected, and distance from the Greek sources which he sought to emulate.”

A statistical physicist, eh? I find that specialty quite an unexpected addition to the team stretching my ideas about social networks in new directions.

Getting back to the research, the scientists have supplied this image to illustrate their work,

Caption: In the social network underlying the Ossianic epic, the 325 nodes represent characters appearing in the narratives and the 748 links represent interactions between them. Credit: Coventry University

Caption: In the social network underlying the Ossianic epic, the 325 nodes represent characters appearing in the narratives and the 748 links represent interactions between them. Credit: Coventry University

Here’s a link to and a citation for the paper,

A networks-science investigation into the epic poems of Ossian by Joseph Yose, Ralph Kenna, Pádraig MacCarron, Thierry Platini, Justin Tonra.  Complex Syst. DOI: http://dx.doi.org/10.1142/S0219525916500089 Published: 21 October 2016

This paper is behind a paywall.

Complex networks to provide ‘grand unified theory’

Trying to mesh classical physics and quantum physics together in one theory which accounts for behaviour on the macro and quantum scales has occupied scientists for decades and it seems that mathematicians have discovered a clue so solving the mystery. A Sept. 13, 2015 news item on Nanotechnology Now describes the findings,

Mathematicians investigating one of science’s great questions — how to unite the physics of the very big with that of the very small — have discovered that when the understanding of complex networks such as the brain or the Internet is applied to geometry the results match up with quantum behavior.

A Sept. 9, 2015 Queen Mary University of London press release, which originated the news item, describes the collaboration between Queen Mary and Karlsruhe Institute of Technology mathematicians,

The findings, published today (Thursday) in Scientific Reports, by researchers from Queen Mary University of London and Karlsruhe Institute of Technology, could explain one of the great problems in modern physics.

Currently ideas of gravity, developed by Einstein and Newton, explain how physics operates on a very large scale, but do not work at the sub-atomic level. Conversely, quantum mechanics works on the very small scale but does not explain the interactions of larger objects like stars. Scientists are looking for a so called ‘grand unified theory’ that joins the two, known as quantum gravity.

Several models have been proposed for how different quantum spaces are linked but most assume that the links between quantum spaces are fairly uniform, with little deviation from the average number of links between each space. The new model, which applies ideas from the theory of complex networks, has found that some quantum spaces might actually include hubs, i.e. nodes with significantly more links than others, like a particularly popular Facebook user.

Calculations run with this model show that these spaces are described by well-known quantum Fermi-Dirac, and Bose-Einstein statistics, used in quantum mechanics, indicating that they could be useful to physicists working on quantum gravity.

Dr Ginestra Bianconi, from Queen Mary University of London, and lead author of the paper, said:

“We hope that by applying our understanding of complex networks to one of the fundamental questions in physics we might be able to help explain how discrete quantum spaces emerge.

“What we can see is that space-time at the quantum-scale might be networked in a very similar way to things we are starting to understand very well like biological networks in cells, our brains and online social networks.”

Here’s a link to and a citation for the paper,

Complex Quantum Network Manifolds in Dimension d > 2 are Scale-Free by Ginestra Bianconi & Christoph Rahmede. Scientific Reports 5, Article number: 13979 (2015) doi:10.1038/srep13979 Published online: 10 September 2015

This is an open access paper.

Michelangelo, clinical anatomy, mathematics, the Golden Ratio, and a myth

I would have thought an article about Michelangelo, mathematics, and the Golden Ratio would be in a journal dedicated to the arts or mathematics or possibly both. Not even my tenth guess would  have been Clinical Anatomy. As for the myth, not everyone subscribes to the Golden Ratio theory of beauty.

A July 20, 2015 Wiley Periodicals press release (also on EurekAlert) announces the publication of the research,

New research provides mathematical evidence that Michelangelo used the Golden Ratio of 1.6 when painting The Creation of Adam on the ceiling of the Sistine Chapel. The Golden Ratio is found when you divide a line into two parts so that the longer part divided by the smaller part is equal to the whole length divided by the longer part.

The Golden Ratio has been linked with greater structural efficiency and has puzzled scientists for centuries due to its frequent occurrence in nature–for example in snail shells and flower petals. The Golden Ratio can also be found in a variety of works by architects and designers, in famous musical compositions, and in the creations of many artists.

The findings suggest that the beauty and harmony found in the works of Michelangelo may not be based solely on his anatomical knowledge. He likely knew that anatomical structures incorporating the Golden Ratio offer greater structural efficiency and, therefore, he used it to enhance the aesthetic quality of his works.

“We believe that this discovery will bring a new dimension to the great work of Michelangelo,” said Dr. Deivis de Campos, author of the Clinical Anatomy study.

Here’s a link to and a citation for the paper,

More than a neuroanatomical representation in The Creation of Adam by Michelangelo Buonarroti, a representation of the Golden Ratio by Deivis De Campos, Tais Malysz,  João Antonio Bonatto-Costa, Geraldo Pereira Jotz, Lino Pinto De Oliveira Junior, and Andrea Oxley da Rocha. Clinical Anatomy DOI: 10.1002/ca.22580 Article first published online: 17 JUL 2015

© 2015 Wiley Periodicals, Inc.

This paper is open access.

Golden Ratio myth

One final comment, it seems not everyone is convinced that the Golden Ratio plays an important role in design, art, and architecture according to an April 13, 2015 article by John Brownlee for Fast Company titled: The Golden Ratio: Design’s Biggest Myth,

In the world of art, architecture, and design, the golden ratio has earned a tremendous reputation. Greats like Le Corbusier and Salvador Dalí have used the number in their work. The Parthenon, the Pyramids at Giza, the paintings of Michelangelo, the Mona Lisa, even the Apple logo are all said to incorporate it.

It’s bullshit. The golden ratio’s aesthetic bona fides are an urban legend, a myth, a design unicorn. Many designers don’t use it, and if they do, they vastly discount its importance. There’s also no science to really back it up. Those who believe the golden ratio is the hidden math behind beauty are falling for a 150-year-old scam.

Fascinating, non?

April 2015 (US) National Math festival; inside story on math tournaments; US tv programme: The Great Math Mystery; and the SET Award (tech women in the movies and on tv)

I have three math items for this posting and one women in technology item, here they are in an almost date order.

X+Y

A British movie titled X+Y provides a fictionalized view of a team member on the British squad competing in an International Mathematics Olympiad.The Guardian’s science blog network hosted a March 11, 2015 review by Adam P. Goucher who also provides an insider’s view (Note: Links have been removed),

As a competition it is brutal and intense.

I speak from experience; I was in the UK team in 2011.

So it was with great expectation that I went to see X+Y, a star-studded British film about the travails of a British IMO hopeful who is struggling against the challenges of romance, Asperger’s and really tough maths.

Obviously, there were a few oversimplifications and departures from reality necessary for a coherent storyline. There were other problems too, but we’ll get to them later.

In order to get chosen for the UK IMO team, you must sit the first round test of the British Mathematical Olympiad (BMO1). About 1200 candidates take this test around the country.

I sat BMO1 on a cold December day at my sixth form, Netherthorpe School in Chesterfield. Apart from the invigilator and me, the room was completely empty, although the surroundings became irrelevant as soon as I was captivated by the problems. The test comprises six questions over the course of three and a half hours. As is the case with all Olympiad problems, there are often many distinct ways to solve them, and correct complete solutions are maximally rewarded irrespective of the elegance or complexity of the proof.

The highest twenty scorers are invited to another training camp at Trinity College, Cambridge, and the top six are selected to represent the UK at an annual competition in Romania.

In Romania, there was much maths, but we also enjoyed a snowball fight against the Italian delegation and sampled the delights of Romanian rum-endowed chocolate. Since I was teetotal at this point in time, the rum content was sufficient to alter my perception in such a way that I decided to attack a problem using Cartesian coordinates (considered by many to be barbaric and masochistic). Luckily my recklessness paid off, enabling me to scrape a much-coveted gold medal by the narrowest of margins.

The connection between the UK and Eastern Europe is rather complicated to explain, being intimately entangled with the history of the IMO. The inaugural Olympiad was held in Romania in 1959, with the competition being only open to countries under the Soviet bloc. A Hungarian mathematician, Béla Bollobás, competed in the first three Olympiads, seizing a perfect score on the third. After his PhD, Bollobás moved to Trinity College, Cambridge, to continue his research, where he fertilised Cambridge with his contributions in probabilistic and extremal combinatorics (becoming a Fellow of the Royal Society in the process). Consequently, there is a close relationship between Hungarian and Cantabrigian mathematics.

Rafe Spall’s character was very convincing, and his eccentricities injected some much-needed humour into the film. Similarly, Asa Butterfield’s portrayal of a “typical mathmo” was realistic. On the other hand, certain characters such as Richard (the team leader) were unnatural and exaggerated. In particular, I was disappointed that all of the competitors were portrayed as being borderline-autistic, when in reality there is a much more diverse mixture of individuals.

X+Y is also a love story, and one based on a true story covered in Morgan Matthews’ earlier work, the documentary Beautiful Young Minds. This followed the 2006 IMO, in China, where one of the members of the UK team fell in love and married the receptionist of the hotel the team were staying at. They have since separated, although his enamourment with China persisted – he switched from studying Mathematics to Chinese Studies.

It is common for relationships to develop during maths Olympiads. Indeed after a member of our team enjoyed a ménage-a-trois at an IMO in the 1980s, the committee increased the security and prohibited boys and girls from entering each others’ rooms.

The film was given a general release March 13, 2015 in the UK and is on the festival circuit elsewhere. Whether or not you can get to see the film, I recommend Goucher’s engaging review/memoir.

The Great Math Mystery and the SET award for the Portrayal of a Female in Technology

David Bruggeman in a March 13, 2015 post on his Pasco Phronesis blog describes the upcoming première of a maths installment in the NOVA series presented on the US PBS (Public Broadcasting Service), Note: Links have been removed,

… PBS has announced a new math special.  Mario Livio will host a NOVA special called The Great Math Mystery, premiering April 15.  Livio is an astrophysicist, science and math writer, and fan of science/culture mashups.  The mystery of the title is whether math(s) is invented or was discovered.

You can find out more about The Great Math Mystery here.

David also mentions this,

The Entertainment Industries Council is seeking votes for its first SET Award for Portrayal of a Female in Technology. … Voting on the award is via a Google form, so you will need a Google account to participate.  The nominees appear to be most of the women playing characters with technical jobs in television programs or recent films.  They are:

  • Annedroids on Amazon
  • Arrow: “Felicity Smoak” played by Emily Bett Rickards
  • Bones: “Angela Montenegro” played by Michaela Conlin

Here’s a video describing the competition and the competitors,

More details about the competition are available in David’s March 13, 2015 post or here or here. The deadline for voting is April 6, 2015. Here’s one more link, this one’s to the SET Awards website.

(US) National Math Festival

H/t to David Bruggeman again. This time it’s a Feb. 6, 2015 post on his Pasco Phronesis blog which announces (Note: Links have been removed),

On April 18 [2015], the Smithsonian Institution will host the first National Math Festival in Washington, D.C.  It will be the culmination of a weekend of events in the city to recognize outstanding math research, educators and books.

On April 16 there will be a morning breakfast briefing on Capitol Hill to discuss mathematics education.  It will be followed by a policy seminar in the Library of Congress and an evening gala to support basic research in mathematics and science.

You can find out more about the 2015 National Math Festival here (from the homepage),

On Saturday, April 18th, experience mathematics like never before, when the first-of-its-kind National Math Festival comes to Washington, D.C. As the country’s first national festival dedicated to discovering the delight and power of mathematics, this free and public celebration will feature dozens of activities for every age—from hands-on magic and Houdini-like getaways to lectures with some of the most influential mathematicians of our time.

The National Math Festival is organized by the Mathematical Sciences Research Institute (MSRI) and the Institute for Advanced Study (IAS) in cooperation with the Smithsonian Institution.

There you have it.

A ‘Magic Square’ stamp from Macau

Alex Bellos describes a fascinating interplay between culture, mathematics, and stamps in his Nov. 4, 2014 posting on the Guardian-hosted Alex’s Adventures in Numberland,

 Old-age mutant number tortoise: Macau stamp displays the origin myth of the magic square. Illustration: Macau Post  [downloaded from http://www.theguardian.com/science/alexs-adventures-in-numberland/2014/nov/04/macaus-magic-square-stamps-just-made-philately-even-more-nerdy]

Old-age mutant number tortoise: Macau stamp displays the origin myth of the magic square. Illustration: Macau Post [downloaded from http://www.theguardian.com/science/alexs-adventures-in-numberland/2014/nov/04/macaus-magic-square-stamps-just-made-philately-even-more-nerdy]

According to Chinese legend a turtle like the one above crept out of the Yellow River about 4000 years ago. It looks like it is riddled with spots, or bullet holes. But if you look carefully, the dots on its back represent the digits from 1 to 9 arranged in the following way:

492

357

816

If you add the numbers in each row together, they are all equal to 15. For example 4 + 9 + 2 = 15, and so on.

If you add the columns, they sum to 15 also. For example, 4 + 3 + 8 = 15. And yes, you guessed it, the diagonals do too.

A grid containing consecutive numbers starting from 1 such that rows, columns and diagonals all add up to the same number is known as a magic square. The 3×3 square on turtle is known in China as the lo shu.

Magic squares have long fascinated soothsayers, herpetologists, mystics, architects, soldiers, artists, mathematicians…and now, stamp collectors. Macau, the former Portuguese colony now a part of China, has just issued a set of magic square stamps that, it claims, not only promotes Chinese culture but also creates a “unique product in the history of philately.”

I encourage you to read the post in its entirety as Bellos follows the magic square through a number of time periods and cultures.

Mathematicians, political scientists, and cake cutting

If you have a sibling, you’ve likely fought at least once over who got the biggest or ‘best’ piece of cake.  (I do and I did.) In any event, it seems that mathematicians and political scientists have been working on a scheme to avoid disputes over cake.

[downloaded from http://link.springer.com/article/10.1007%2Fs00283-013-9442-0#page-1]

A July 16, 2014 Springer news release (also on EurekAlert) describes the quest for fairly sized cake slices and how that might apply to real life issues such as sharing property,

The next time your children quibble about who gets to eat which part of a cake, call in some experts on the art of sharing. Mathematician Julius Barbanel of Union College, and political scientist Steven Brams of New York University, both in the US, published an algorithm in Springer’s The Mathematical Intelligencer by which they show how to optimally share cake between two people efficiently, in equal pieces and in such a way that no one feels robbed.

The cut-and-choose method to share divisible goods has been regarded as fair and envy-free since Biblical times, when Abraham divided land equally, and Lot could choose the part he wanted. But being free of envy is not the only consideration when sharing something. What happens when more than two cuts can be made, or when people prefer different, specific sections of whatever is to be divided? Barbanel and Brams believe that with a giveback procedure it is possible to make a perfect division between two people that is efficient, equitable and void of jealousy.

An objective referee (such as a Mom or a computer) is essential to the plan. The potential cake eaters first tell the referee which parts of the delicacy they value most. In mathematical terms these are called someone’s probability density functions, or pdfs. The referee then marks out the cake at all points were the pdfs of the disgruntled would-be cake eaters cross, and assigns portions. If at this point the two parties receive the same size of cake, the task is over. If not, the giveback process starts.

The party who received the larger part of the cake during the first round must give a part of it back to the other person, starting with those parts in which the ratio of their pdfs is the smallest. This goes on until the parties value their portions equally, and have the same volume of cake to eat. This method only works with a finite number of cuts if the players’ pdfs are straight-lined, or are so-called piecewise linear sections.

The researchers believe the method can be used to share cake and other divisible goods such as land. In the case of beachfront property being co-owned by two developers, for example, it can help to determine who gets what strips of land to build on based on the pieces of land they value most.

“This allocation is not only equitable but also envy-free and efficient – that is, perfect,” says Barbanel.

“This approach focuses on proving the existence of efficient and envy-free divisions, not on providing algorithms to finding them,” emphasizes Brams.

Here’s a link to and a citation for the paper,

Two-Person Cake Cutting: The Optimal Number of Cuts by Julius B. Barbanel and Steven J. Brams. The Mathematical Intelligencer March 2014 DOI 10.1007/s00283-013-9442.

This paper is behind a paywall although there is a free preview available and a special summer discount (30%) on the purchase price until July 31, 2014.

The geometry of graphene at the University of Arkansas (US)

The University of Arkansas (US) has announced the development of a new mathematical framework useful for studying graphene according to a May 5, 2014 news item on Nanowerk,

Scientists studying graphene’s properties are using a new mathematical framework to make extremely accurate characterizations of the two-dimensional material’s shape.

“The properties of two-dimensional materials depend on shape,” said Salvador Barraza-Lopez, an assistant professor of physics at the University of Arkansas. “And this mathematical framework allows you to make extremely accurate characterizations of shape. This framework is a novel tool to understand shape in materials that behave as atom-thin membranes.”

A May 5, 2014 University of Arkansas news release, which originated the news item, provides more details,

The mathematical framework being used is known as discrete differential geometry, which is the geometry of two-dimensional interlaced structures called meshes. When the nodes of the structure, or mesh points, correspond with atomic positions, discrete differential geometry provides direct information on the potential chemistry and on the electronic properties of two-dimensional materials, Barraza-Lopez said.

The application of discrete differential geometry to understand two-dimensional materials is an original interdisciplinary development, he said.

“Since two-dimensional materials can be easily visualized as meshes, we asked ourselves how these theories would look if you express them directly in terms of the positions of the atoms, bypassing entirely the common continuum approximation,” Barraza-Lopez said. …

Two papers have been produced about this work,

Quantitative Chemistry and the Discrete Geometry of Conformal Atom-Thin Crystals by Alejandro A. Pacheco Sanjuan, Mehrshad Mehboudi, Edmund O. Harriss, Humberto Terrones, and Salvador Barraza-Lopez. ACS Nano, 2014, 8 (2), pp 1136–1146 DOI: 10.1021/nn406532z Publication Date (Web): January 8, 2014

Copyright © 2014 American Chemical Society

Graphene’s morphology and electronic properties from discrete differential geometry by Alejandro A. Pacheco Sanjuan, Zhengfei Wang, Hamed Pour Imani, Mihajlo Vanević, and Salvador Barraza-Lopez. Phys. Rev. B 89, 121403(R) – Published 6 March 2014 DOI: http://dx.doi.org/10.1103/PhysRevB.89.121403

©2014 American Physical Society

Both papers are behind paywalls.

Happy Pi Day! on March 14, 2014

It;’s no surprise that Canada’s Perimeter Institute (PI) is celebrating Pi Day. Before sharing the institute’s latest public outreach effort and for anyone like me who has a shaky understanding  of what exactly Pi is, there’s this explanation excerpted from the Pi Wikipedia essay (Note: Links have been removed),

The number π is a mathematical constant, the ratio of a circle’s circumference to its diameter, approximately equal to 3.14159. It has been represented by the Greek letter “π” since the mid-18th century though it is also sometimes spelled out as “pi” (/paɪ/).

Being an irrational number, π cannot be expressed exactly as a common fraction. Consequently its decimal representation never ends and never settles into a permanent repeating pattern. The digits appear to be randomly distributed although no proof of this has yet been discovered. Also, π is a transcendental number – a number that is not the root of any nonzero polynomial having rational coefficients. This transcendence of π implies that it is impossible to solve the ancient challenge of squaring the circle with a compass and straight-edge.

Fractions such as 22/7 and other rational numbers are commonly used to approximate π.

Someone at the Perimeter Institute has prepared a ‘facts you don’t know about Pi‘ flyer to commemorate the day, which includes these facts and more,

In the 1995 OJ Simpson trial, one witness’ credibility was called into doubt when he misstated the
value of pi. [for anyone not familiar with the trial, O. J. Simpson murder case Wikipedia entry)

Foucault’s Pendulum by Umberto Eco associates the mysterious pendulum in the novel with the intrigue of pi.

In 2005, Lu Chao of China set a world record by memorizing the first 67,890 digits of pi.

In the year 2015, Pi Day will have special significance on 3/14/15 at 9:26:53.58, with the date and time (including 1/100 seconds) representing the first 12 digits of pi.

Over on the Guardian science blogs (Alex’s Adventures in Nunberland blog), Alex Bellos shares Pi artwork in his March 14, 2014 posting, here’s a sample,

Artist: Cristian Vasile

Artist: Cristian Vasile

In this work, Vasile converted pi into base 16. The sixteen segments around the circle represent the 16 digits of this base. He then traced pi for 3600 digits, going from segment to segment based on the value of the digit. A fuller explanation is here and Vasile’s art can be bought here.

Have a happy Pi Day and a good weekend!