Category Archives: Mathematics

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!

Does education kill the ability to do algebra?

Apparently, the ability to perform basic algebra is innate in humans, mice, fish, and others. Researchers at Johns Hopkins describe some of their findings about algebra and innate abilities in this video,

While the researchers don’t accuse the education system of destroying or damaging one’s ability to perform algebra, I will make the suggestion, the gut level instinct the researchers are describing is educated out of most of us. Here’s more from the March 6, 2014 news item on ScienceDaily describing the research,

Millions of high school and college algebra students are united in a shared agony over solving for x and y, and for those to whom the answers don’t come easily, it gets worse: Most preschoolers and kindergarteners can do some algebra before even entering a math class.

In a just-published study in the journal Developmental Science, lead author and post-doctoral fellow Melissa Kibbe and Lisa Feigenson, associate professor of psychological and brain sciences at Johns Hopkins University’s Krieger School of Arts and Sciences, find that most preschoolers and kindergarteners, or children between 4 and 6, can do basic algebra naturally.

“These very young children, some of whom are just learning to count, and few of whom have even gone to school yet, are doing basic algebra and with little effort,” Kibbe said. “They do it by using what we call their ‘Approximate Number System:’ their gut-level, inborn sense of quantity and number.”

A Johns Hopkins University March 7, 2014 news piece by Latarsha Gatlin describes the research further,

The “Approximate Number System,” or ANS, is also called “number sense,” and describes humans’ and animals’ ability to quickly size up the quantity of objects in their everyday environments. We’re born with this ability, which is probably an evolutionary adaptation to help human and animal ancestors survive in the wild, scientists say.

Previous research has revealed some interesting facts about number sense, including that adolescents with better math abilities also had superior number sense when they were preschoolers, and that number sense peaks at age 35.

Kibbe, who works in Feigenson’s lab, wondered whether preschool-age children could harness that intuitive mathematical ability to solve for a hidden variable. In other words, could they do something akin to basic algebra before they ever received formal classroom mathematics instruction? The answer was “yes,” at least when the algebra problem was acted out by two furry stuffed animals—Gator and Cheetah—using “magic cups” filled with objects like buttons, plastic doll shoes, and pennies.

In the study, children sat down individually with an examiner who introduced them to the two characters, each of which had a cup filled with an unknown quantity of items. Children were told that each character’s cup would “magically” add more items to a pile of objects already sitting on a table. But children were not allowed to see the number of objects in either cup: they only saw the pile before it was added to, and after, so they had to infer approximately how many objects Gator’s cup and Cheetah’s cup contained.

At the end, the examiner pretended that she had mixed up the cups, and asked the children—after showing them what was in one of the cups—to help her figure out whose cup it was. The majority of the children knew whose cup it was, a finding that revealed for the researchers that the pint-sized participants had been solving for a missing quantity. In essence, this is the same as doing basic algebra.

“What was in the cup was the x and y variable, and children nailed it,” said Feigenson, director of the Johns Hopkins Laboratory for Child Development. “Gator’s cup was the x variable and Cheetah’s cup was the y variable. We found out that young children are very, very good at this. It appears that they are harnessing their gut level number sense to solve this task.”

If this kind of basic algebraic reasoning is so simple and natural for 4, 5, and 6-year-olds, then why it is so difficult for teens and others?

“One possibility is that formal algebra relies on memorized rules and symbols that seem to trip many people up,” Feigenson said. “So one of the exciting future directions for this research is to ask whether telling teachers that children have this gut level ability—long before they master the symbols—might help in encouraging students to harness these skills. Teachers may be able to help children master these kind of computations earlier, and more easily, giving them a wedge into the system.”

While number sense helps children in solving basic algebra, more sophisticated concepts and reasoning are needed to master the complex algebra problems that are taught later in the school age years.

Another finding from the research was that an ANS aptitude does not follow gender lines. Boys and girls answered questions correctly in equal proportions during the experiments, the researchers said. Although other research shows that even young children can be influenced by gender stereotypes about girls’ versus boys’ math prowess, “we see no evidence for gender differences in our work on basic number sense,” Feigenson said.

Parents with numerically challenged kids shouldn’t worry that their child will be bad at math. The psychologists say it’s more important to nurture and support young children’s use of their number sense in solving problems that will later be introduced more formally in school.

“We find links at all ages between the precision of people’s Approximate Number System and their formal math ability,” Feigenson said. “But this does not necessarily mean that children with poorer precision grow up to be bad at math. For example, children with poorer number sense may need to rely on other strategies, besides their gut sense of number, to solve math problems. But this is an area where much future research is needed.”

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

Young children ‘solve for x’ using the Approximate Number System by Melissa M. Kibbe and Lisa Feigenson. Article first published online: 3 MAR 2014 DOI: 10.1111/desc.12177

© 2014 John Wiley & Sons Ltd

This paper is behind a paywall.

Mathematics of Planet Earth lives on past 2013

A Université de Montréal (Québec, Canada) Dec. 11, 2013 news release (also on EurekAlert) proclaims a new life for a worldwide mathematics initiative (Note: I have added paragraph breaks for this formerly single paragraph excerpt),

Although you might not know it, mathematics is able to shed light on many of the issues facing Planet Earth – from the structure of the core of our planet to the understanding of biodiversity, from finding ways to advance cutting edge solar technology to better understanding the Earth’s climate system, and from earthquakes and tsunamis to the spread of infectious diseases – and so mathematicians around the world have decided to launch an international project, Mathematics of Planet Earth (MPE), to demonstrate how their field of expertise contributes directly to our well being.

Mathematics of Planet Earth is growing out of a year-long initiative that was the brainchild of Christiane Rousseau, professor of mathematics at Université de Montréal and vice-president of the International Mathematics Union.

Beginning in 2014, the program will continue under the same name with the same objectives: identify fundamental research questions about Planet Earth and reach out to the general public. As Prof Rousseau observed, “Mathematics of Planet Earth has been a great start. But identifying the research problems is not enough. Mathematics moves slowly, the planetary problems are very challenging, and we cannot expect great results in just one year.” “The International Mathematical Union enthusiastically supports the continuation of Mathematics of Planet Earth. The success of this initiative attests to the foundational role of the mathematical sciences and interdisciplinary partnerships in research into global challenges, increasingly valued by society,” says Ingrid Daubechies, President of the International Mathematical Union.

How did I not hear about this project before now? Well, it’s better to get there late then never get to the party at all. From the news release,

Under the patronage of UNESCO, the MPE initiative brought together over 100 scientific societies, universities, research institutes, and foundations from around the world to research fundamental questions about Planet Earth, nurture a better understanding of global issues, and help inform the public about the essential mathematics of the challenges facing our planet. “The Mathematics of Planet Earth (MPE) initiative resonates strongly with UNESCO’s work to promote the sciences and science education, especially through our International Basic Sciences Programme. Math advances fundamental research and plays an important role in our daily life. More than ever we need to develop relevant learning materials and to spark in every student, especially girls, a sense of joy in the wondrous universe of mathematics and the immense potential unleashed by this discipline. In this spirit, we commend this initiative and fully endorse the proposal to continue this programme beyond 2013,” said Irena Bokova, Director-General of UNESCO.

It’s not about preaching to the converted. “The curriculum material developed for Mathematics of Planet Earth provides schools and educators a free-of-charge wealth of material for and will be used for many years to come. The initiative has presented the public, schools and the media with challenging applications of mathematics, with significant answers to questions like ‘What is mathematics useful for?’” said Mary Lou Zeeman, MPE coordinator for Education. “Mathematics of Planet Earth wonderfully contributed to diffuse an informed culture of environment and helps to get a common mathematical toolkit necessary to deal the dramatic challenges faced today by our planet,” said Ferdinando Arzarello, President of the International Commission of Mathematical Instruction (ICMI).

It’s not only the mathematicians and mathematics pedagogues who’ve gotten excited about this initiative,

MPE2013 has drawn the attention of other disciplines as well. Among its partners are the American Geophysical Union, the International Association for Mathematical Geosciences, and the International Union of Geodesy and Geophysics (IUGG). The research on planetary issues is interdisciplinary, and collaboration and networking are essential for progress. “Great mathematicians understood the importance of research into planet Earth many centuries ago,” said Alik Ismail-Zadeh, a mathematical geophysicist and the Secretary General of the IUGG. “Pierre Fermat studied the weight of the Earth; Carl Friedrich Gauss contributed to the development of geomagnetism and together with Friedrich Wilhelm Bessel made significant contribution to geodesy; Andrei Tikhonov developed regularization techniques intensively used in studies of inverse problems in many areas of geophysics. Mathematics of Planet Earth 2013 highlighted again the importance of international multidisciplinary cooperation and stimulated mathematicians and geoscientists to work together to uncover Earth’s mysteries.”

The news release closes with these interesting bits of information,

About Mathematics of Planet Earth

On January 1, 2014, Mathematics of Planet Earth 2013 (MPE2013) will continue as “Mathematics of Planet Earth” (MPE). The objectives remain unchanged – identify fundamental research questions about Planet Earth and reach out to the general public. With support from the U.S. National Science Foundation, MPE will maintain a website where additional educational and outreach materials will be posted. New modules will be developed and added to the MPE Exhibition. Plans for more MPE activities exist in several countries in the form of workshops, summer schools, and even the creation of new graduate programs in Mathematics of Planet Earth.

About Christiane Rousseau

Christian Rousseau is a professor at Université de Montréal’s Department of Mathematics and Statistics, Vice-President of the International Mathematics Union, and a member of the Centre de recherches mathématiques. Professor Rousseau conceived and coordinated Mathematics of Planet Earth 2013.

About Mathematics of Planet Earth 2013′s Achievements

MPE2013 activities have included more than 15 long-term programs at mathematical research institutes all over the world, 60 workshops, dozens of special sessions at society meetings, two major public lecture series, summer and winter schools for graduate students, research experiences for undergraduates, an international competition, and an Open Source MPE Exhibition. In addition, MPE2013 has supported the development of high-quality curriculum materials for all ages and grades available on the MPE2013 Web site.

Encouraging Research

The scientific activities of MPE2013 were directed both to the mathematical sciences community, whose members are encouraged to identify fundamental research questions about Planet Earth and their potential collaborators in other disciplines. The program provides evidence that many issues related to weather, climate, sustainability, public health, natural hazards, and financial and social systems lead to interesting mathematical problems. Several summer and winter schools have offered training opportunities for junior researchers in these areas.

Reaching Out

The outreach activities of MPE2013 were as important as the scientific activities. More than sixty public lectures have been given with audiences on all five continents. Particularly noteworthy were the MPE Simons Public Lectures, now posted on the MPE2013 Web site, which were supported financially by the Simons Foundation. MPE2013 has maintained a speakers bureau, supported the development of curriculum materials, maintained a collection of posters, and produced special issues of mathematical magazines and other educational materials. Many activities took place at schools in several countries. The permanent MPE Open Source Exhibition is now hosted on the website of IMAGINARY and can be used and adapted by schools and museums.

Daily Blog

The dual mission of MPE2013 – stimulating the mathematics research community and reaching out to the general public – is reflected in the Daily Blogs (one in English, the other in French), each of which has featured more than 250 posts on topics ranging from astronomy to uncertainty quantification. The blog gets several hundred hits a day.

You can find out more about MPE 2013 and its future here. (English language version website) or go here for the French language version. For those who prefer to read the news release about the ‘morphing’ MPE in French, go here.

NUSIKIMO: plasma and nanotechnology applications

NUKISIMO's plama and nanotechnology applications? Credit: Shutterstock [downloaded from http://cordis.europa.eu/fetch?CALLER=EN_NEWS&ACTION=D&RCN=36206]

NUKISIMO’s plama and nanotechnology applications? Credit: Shutterstock [downloaded from http://cordis.europa.eu/fetch?CALLER=EN_NEWS&ACTION=D&RCN=36206]

It looks like a jewel, doesn’t it? Unfortunately, there’s no explanation for why this image is offered as an illustration for an Oct. 31, 2013 OORDIS news release (h/t phys.org) about plasma and nanotechnology applications, being worked on as part of the NUSIKIMO (‘Numerical simulations and analysis of kinetic models – applications to plasma physics and nanotechnology’) project,

Plasma is one of the four fundamental states of matter, alongside solid, liquid and gas. Ubiquitous in form, plasma is an ionised gas so energised that electrons have the capacity to break free from their nucleus.

Scientists are keen to shed light on the motion of particles in plasma physics, as well as the dynamics of rarefied gas – a gas whose pressure is much lower than atmospheric pressure. How can this be done? An EU-funded team of researchers has come up with a solution.

Prof. Francis Filbet from Université Claude Bernard Lyon 1 in France decided to tackle the question with mathematical and numerical analyses. He received an European Research Council (ERC) Starting Grant worth almost EUR 500 000 for the NUSIKIMO (‘Numerical simulations and analysis of kinetic models – applications to plasma physics and nanotechnology’) project. Prof Filbet and his research team modelled non-stationary collisional plasma with supercomputers, putting regimes and instabilities under the microscope.

One of the challenges researchers undertook was to approximate kinetic models and to develop novel techniques that could make numerical analysis in kinetic theory possible.

To do this, the team is working on adapting averaging lemmas (proven statements used for obtaining proof of other statements) to examine kinetic equations, including the Boltzmann equation. Devised in 1872, the seven-dimensional equation is used to model the behaviour of gases, but solving it has proved problematic as numerical capabilities fail to capture the complexities involved.

The NUSIKIMO team is also examining asymptotic preserving schemes, which can be described as performant procedures able to solve ‘singularly perturbed problems’ – those for which the character of the problem changes intermittently.

Such problems contain small parameters that cannot be approximated by setting the parameter value to zero. For comparison, an approximation for regular perturbation problems can be obtained when small parameters are set to zero.

Asymptotic preserving schemes were established to help scientists deal with singularly perturbed problems. This is especially the case when they are dealing with kinetic models in a diffusive environment.

Prof. Filbet and his team are developing a method to control numerical entropy (classical thermodynamics) production. Being able to control entropy production, which determines the performance of thermal machines, is an important feature for stability analysis – an assessment that helps us understand what happens to a system when it is perturbed. The researchers believe nonlinear equations could therefore be treated with a strategy based on asymptotic preserving schemes.

Applying these equations to plasma physics is one of the NUSIKIMO goals. The team is evaluating energy transport and seeking to determine the efficiency of plasma heating. The researchers are also looking into the measures required to secure fusion conditions through the interaction of intense, short laser pulses, and schemes like inertial confinement fusion or fast ignition.

Another objective is to apply the equations to microelectromechanical systems (MEMS). Prof. Filbet and his team are developing theoretical and numerical methods to investigate gaseous and liquid flows in micro devices. The key element here is the development of numerical methods. The researchers say: using numerical methods, rather than analytical methods, make modelling the three-dimensional flow geometries in MEMS configurations possible.

The project end date is December 2013 but in the meantime, you can get more information about NUSIKIMO here.

Canadian filmmaker Chris Landreth’s Subconscious Password explores the uncanny valley

I gather Chris Landreth’s short animation, Subconscious Password, hasn’t been officially released yet by the National Film Board (NFB) of Canada but there are clips and trailers which hint at some of the filmmaker’s themes. Landreth in a May 23, 2013 guest post for the NFB.ca blog spells out one of them,

Subconscious Password, my latest short film, travels to the inner mind of a fellow named Charles Langford, as he struggles to remember the name of his friend at a party. In his subconscious, he encounters a game show, populated with special guest stars:  archetypes, icons, distant memories, who try to help him find the connection he needs: His friend’s name.

The film is a psychological romp into a person’s inner mind where (I hope) you will see something of your own mind working, thinking, feeling. Even during a mundane act like remembering the name of an acquaintance at a party, someone you only vaguely remember. To me, mundane accomplishments like these are miracles we all experience many times each day.

Landreth also discusses the ‘uncanny valley’ and how he deliberately cast his film into that valley. For anyone who’s unfamiliar with the ‘uncanny valley’ I wrote about it in a Mar. 10, 2011 posting concerning Geminoid robots,

It seems that researchers believe that the ‘uncanny valley’ doesn’t necessarily have to exist forever and at some point, people will accept humanoid robots without hesitation. In the meantime, here’s a diagram of the ‘uncanny valley’,

From the article on Android Science by Masahiro Mori (translated by Karl F. MacDorman and Takashi Minato)

Here’s what Mori (the person who coined the term) had to say about the ‘uncanny valley’ (from Android Science),

Recently there are many industrial robots, and as we know the robots do not have a face or legs, and just rotate or extend or contract their arms, and they bear no resemblance to human beings. Certainly the policy for designing these kinds of robots is based on functionality. From this standpoint, the robots must perform functions similar to those of human factory workers, but their appearance is not evaluated. If we plot these industrial robots on a graph of familiarity versus appearance, they lie near the origin (see Figure 1 [above]). So they bear little resemblance to a human being, and in general people do not find them to be familiar. But if the designer of a toy robot puts importance on a robot’s appearance rather than its function, the robot will have a somewhat humanlike appearance with a face, two arms, two legs, and a torso. This design lets children enjoy a sense of familiarity with the humanoid toy. So the toy robot is approaching the top of the first peak.

Of course, human beings themselves lie at the final goal of robotics, which is why we make an effort to build humanlike robots. For example, a robot’s arms may be composed of a metal cylinder with many bolts, but to achieve a more humanlike appearance, we paint over the metal in skin tones. These cosmetic efforts cause a resultant increase in our sense of the robot’s familiarity. Some readers may have felt sympathy for handicapped people they have seen who attach a prosthetic arm or leg to replace a missing limb. But recently prosthetic hands have improved greatly, and we cannot distinguish them from real hands at a glance. Some prosthetic hands attempt to simulate veins, muscles, tendons, finger nails, and finger prints, and their color resembles human pigmentation. So maybe the prosthetic arm has achieved a degree of human verisimilitude on par with false teeth. But this kind of prosthetic hand is too real and when we notice it is prosthetic, we have a sense of strangeness. So if we shake the hand, we are surprised by the lack of soft tissue and cold temperature. In this case, there is no longer a sense of familiarity. It is uncanny. In mathematical terms, strangeness can be represented by negative familiarity, so the prosthetic hand is at the bottom of the valley. So in this case, the appearance is quite human like, but the familiarity is negative. This is the uncanny valley.

Landreth discusses the ‘uncanny valley’ in relation to animated characters,

Many of you know what this is. The Uncanny Valley describes a common problem that audiences have with CG-animated characters. Here’s a graph that shows this:

Follow the curvy line from the lower left. If a character is simple (like a stick figure) we have little or no empathy with it. A more complex character, like Snow White or Pixar’s Mr. Incredible, gives us more human-like mannerisms for us to identify with.

But then the Uncanny Valley kicks in. That curvy line changes direction, plunging downwards. This is the pit into which many characters from The Polar Express, Final Fantasy and Mars Needs Moms fall. We stop empathizing with these characters. They are unintentionally disturbing, like moving corpses. This is a big problem with realistic CGI characters: that unshakable perception that they are animated zombies. [zombie emphasis mine]

You’ll notice that the diagram from my posting features a zombie at the very bottom of the curve.

Landreth goes on to compare the ‘land’ in the uncanny valley to real estate,

… The value of land in the Uncanny Valley has plunged to zero. There are no buyers.

Well, except perhaps me.

Some of you know that my films have a certain obsession with visual realism with their human characters. I like doing this. I find value in this realism that goes beyond simply copying what humans look and act like. If used intelligently and with imagination, realism can capture something deeper, something weird and emotional and psychological about our collective experience on this planet. But it has to be honest. That’s hard.

He also explains what he’s hoping to accomplish by inhabiting the uncanny valley,

When making this film, we knew we were going into the Uncanny Valley. We did it because your subconscious processes, and mine, are like this valley. We project our waking world into our subconscious minds. The ‘characters’ in this inner world are realistic approximations of actual people, without actually being real. This is the miracle of how we get by. My protagonist, Charles, has a mixture of both realistic approximations and crazy warped versions of the people and icons in his life. He is indeed a bit off-kilter. But he gets by, like most of us do. As you probably have guessed, both Charles and the Host are self-portraits. I want to be honest in showing you this world. My own Uncanny Valley. You have one too. It’s something to celebrate.

On the that note, here’s a clip from Subconscious Password,

Subconscious Password (Clip) by Chris Landreth, National Film Board of Canada

 I last wrote about Landreth and his work in an April 14, 2010 posting (scroll down about 1/4 of the way) regarding mathematics and the arts. This post features excerpts from an interview with the University of Toronto (Ontario, Canada) mathematician, Karan Singh who worked with Landreth on their award-winning, Ryan.

Bubblicious

Mathematicians love their bubbles according to the May 9, 2013 news release on EurekAlert,

Two University of California, Berkeley, researchers have now described mathematically the successive stages in the complex evolution and disappearance of foamy bubbles, a feat that could help in modeling industrial processes in which liquids mix or in the formation of solid foams such as those used to cushion bicycle helmets.

Applying these equations, they created mesmerizing computer-generated movies showing the slow and sedate disappearance of wobbly foams one burst bubble at a time.

The applied mathematicians, James A. Sethian and Robert I. Saye, will report their results in the May 10 issue of Science. Sethian, a UC Berkeley professor of mathematics, leads the mathematics group at Lawrence Berkeley National Laboratory (LBNL). Saye will graduate from UC Berkeley this May with a PhD in applied mathematics.

The May 9, 2013 University of California Berkeley news release by Robert Sanders, which originated the news release on EurekAlert, describes a serious side to the work,

“This work has application in the mixing of foams, in industrial processes for making metal and plastic foams, and in modeling growing cell clusters,” said Sethian. “These techniques, which rely on solving a set of linked partial differential equations, can be used to track the motion of a large number of interfaces connected together, where the physics and chemistry determine the surface dynamics.”

The problem with describing foams mathematically has been that the evolution of a bubble cluster a few inches across depends on what’s happening in the extremely thin walls of each bubble, which are thinner than a human hair.

“Modeling the vastly different scales in a foam is a challenge, since it is computationally impractical to consider only the smallest space and time scales,” Saye said. “Instead, we developed a scale-separated approach that identifies the important physics taking place in each of the distinct scales, which are then coupled together in a consistent manner.”

Saye and Sethian discovered a way to treat different aspects of the foam with different sets of equations that worked for clusters of hundreds of bubbles. One set of equations described the gravitational draining of liquid from the bubble walls, which thin out until they rupture. Another set of equations dealt with the flow of liquid inside the junctions between the bubble membranes. A third set handled the wobbly rearrangement of bubbles after one pops.

Using a fourth set of equations, the mathematicians solved the physics of a sunset reflected in the bubbles, taking account of thin film interference within the bubble membranes, which can create rainbow hues like an oil slick on wet pavement. Solving the full set of equations of motion took five days using supercomputers at the LBNL’s National Energy Research Scientific Computing Center (NERSC).

The mathematicians next plan to look at manufacturing processes for small-scale new materials.

Here’s a still image from the video the researchers created to demonstrate their work on soap bubble clusters,

A soap bubble cluster shown with physically accurate thin-film interference, which produces rainbow hues like an oil slick on pavement. A beach at sunset is reflected in the bubbles. Courtesy: UC Berkeley

A soap bubble cluster shown with physically accurate thin-film interference, which produces rainbow hues like an oil slick on pavement. A beach at sunset is reflected in the bubbles. Courtesy: UC Berkeley

You can find the full animation here.

Mathematical theorems as spiritual practice

Alex Bellos in an Oct. 16, 2012 article for the UK’s Guardian newspaper discusses a unique practice combining spirituality and mathematics (Note: I have removed a link),

… one of the most intriguing practices in the history of mathematics.

Between the seventeenth and nineteenth centuries, the Japanese used to hang up pictures of maths theorems at their shrines.

Called “sangaku”, the pictures were both religious offerings and public announcements of the latest discoveries.

It’s a little like as if Isaac Newton had decided to hang up his monographs at the local church instead of publishing them in books.

More than 700 sangaku are known to have survived, and the above shape is a detail from the oldest one that exists in its complete form.

Here’s a picture of a sangaku that Bellos took while in Japan to make a documentary on numeracy for BBC Radio 4,

Picture: Alex Bellos

The purpose of a sangaku was threefold: to show off mathematical accomplishment, to thank Buddha and to pray for more mathematical knowledge.

There are more images and details in Bellos article about this intriguing practice. I look forward to hearing more about Bellos’ documentary, Land of the Rising Sums, due to be broadcast Monday, Oct. 29, 2012 on BBC Radio 4 from 11 – 11:30 am GMT.