Music has its own Geometry

The connection between music and mathematics has fascinated scholars for centuries. More than 200 years ago Pythagoras reportedly discovered that pleasing musical intervals could be described using simple ratios.

And the so-called musica universalis or "music of the spheres" emerged in the Middle Ages as the philosophical idea that the proportions in the movements of the celestial bodies -- the sun, moon and planets -- could be viewed as a form of music, inaudible but perfectly harmonious.

Now, three music professors – Clifton Callender at Florida State University, Ian Quinn at Yale University and Dmitri Tymoczko at Princeton University -- have devised a new way of analyzing and categorizing music that takes advantage of the deep, complex mathematics they see enmeshed in its very fabric.

Writing in the April 18 issue of Science, the trio has outlined a method called "geometrical music theory" that translates the language of musical theory into that of contemporary geometry. They take sequences of notes, like chords, rhythms and scales, and categorize them so they can be grouped into "families."

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Creativity in mathematics

Mathematicians have always felt a strong creative aspect in their subject, but only in recent years has the flowering of connections between mathematics and the arts made this aspect apparent to the general public. This collection of three articles explores some of the various ways in which art and beauty appear in mathematics.

Mathematics and Mime

In "Envisioning the Invisible", Tim Chartier describes how the performing arts can be used to capture mathematical concepts...In one of Chartier's mime sketches, he gets the audience to visualize the one-dimensional number line as a rope of infinite length.....

Mathematics and Music

How does the brain sometimes fool us when we listen to music, and how have composers used such illusions?....How can math help create new music?...

Mathematics and Visual Art

...The forms emerging from this iterated function system are fractals. By serendipity, the article on music by Don et al employs some of Barnsely's work on fractal images to produce new music. Using Barnsley's Iterated Function Systems formulas, the authors created fractal images of a fern and of Sierpinski's triangle and used these images to create notes for musical compositions...

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Mystery of the golden ratio

The Egyptians supposedly used it to guide the construction the Pyramids. The architecture of ancient Athens is thought to have been based on it. Fictional Harvard symbologist Robert Langdon tried to unravel its mysteries in the novel The Da Vinci Code.

"It" is the golden ratio, a geometric proportion that has been theorized to be the most aesthetically pleasing to the eye and has been the root of countless mysteries over the centuries. Now, a Duke University engineer has found it to be a compelling springboard to unify vision, thought and movement under a single law of nature's design.

Also know the divine proportion, the golden ratio describes a rectangle with a length roughly one and a half times its width. Many artists and architects have fashioned their works around this proportion. For example, the Parthenon in Athens and Leonardo da Vinci's painting Mona Lisa are commonly cited examples of the ratio.

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Math proffessor working on new ways to see through the human body

Thanks to medical imaging techniques such as X-ray CT, ultrasound imaging and MRI, doctors have long been able to see to varying degrees what's going on inside a patient's body, and now a Texas A&M University mathematician is trying to find new and better ways to do so.

The professor, Peter Kuchment, a leading researcher in mathematical techniques for medical imaging, says the research may enhance the process for detecting cancer and many other diseases.

When talking about medical imaging, most people know that physics and computer sciences are involved, but few may be aware that mathematics is indispensable. Indeed, many imaging methods are based on mathematical analysis.

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Have a symmetrical object named after you

WildAboutMath by Sol-I became a big fan of Marcus du Sautoy when I read his books Symmetry, and Music of the Primes. From that video I learned about du Sautoy’s fundraising page for Common Hope:

Common Hope promotes hope and opportunity in Guatemala, partnering with children, families, and communities who want to participate in a process of development to improve their lives through education, health care, and housing.

And I learned how to get my own symmetrical object:

People have stars named after them, craters on the moon, even comets…but how about naming a symmetrical object in hyperspace. For a donation of over $10 you can have a new symmetrical object named after you or a friend. A great birthday present. My new book FINDING MOONSHINE (UK) or SYMMETRY (US) narrates the discovery of these new symmetrical objects that have interesting connections with objects in number theory called elliptic curves. Here is the chance to claim one of these groups and have the group named after you. I have created infinitely many of these groups so they won’t run out!

What a great idea! So, I donated and now I’m the proud owner of a symmetry group. This could be the perfect gift for the Math lover who has everything...

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Are primates' brains wired for math?

Like a lot of humans, monkeys might not be able to do calculus. But a new study shows that they can learn and rapidly apply abstract mathematical principles.

Previous work has shown that monkeys and birds can count, but flexible applications of higher mathematic rules, the study authors asserted, "require the highest degree of internal structuring"—one thought largely to be the domain of only humans.

So researchers based at the Institute of Neurobiology at the University of Tubingen in Germany set out to see whether rhesus monkeys could learn and flexibly apply the greater-than and less-than rule. They tested the monkeys with groups of both ordered and random dots, many of which were novel combinations to ensure that the subjects couldn't have simply memorized them. The monkeys were cued into applying either the greater-than or less-than rule by the amount of time that elapsed between being shown the first and second group of dots.

"The monkeys immediately generalized the greater than and less than rules to numerosities that had not been presented previously," the two researchers, Sylvia Bongard and Andreas Nieder, wrote.

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200-year-old Encrypted Letter Finally Deciphered

The mathematician who deciphered the final, encrypted page of a letter sent to President Thomas Jefferson in 1801 will visit the University of Oregon to tell how he did it.

The encrypted page -- a mystery to Jefferson and everyone else -- was solved in 2007 by Smithline, then 36, an expert in code-breaking. He detailed his solution in the American Scientist.

The letter was written by Jefferson's colleague in the American Philosophical Society, Robert Patterson, a math professor at the University of Pennsylvania. The ciphered page was devoid of capital letters or spaces and scrambled in a way that left no readable segments. Preceding pages had described the nature of the code but not the specific key required to unlock this message. The code was unlike any normally used at the time. Patterson predicted it would never be broken.

The solution involved both linguistic intuition and a computer algorithm to find the digital key. While the required 100,000 calculations would be easy on today's computers, Smithline's method could have been done over time in Patterson's day. In his talk, Smithline will tell how he was pulled into the mystery, how he broke the code and what was written on the page.

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Emotional Bunny Says: "What's that? In the wrong hands, this information could have been fatal? Ah. I wouldn't worry about that...."

Simple Math Yields Intricate Visual Patterns

Polynomials, the meat and potatoes of high-school algebra, are foundational to many aspects of quantitative science. But it would take a particularly enthusiastic math teacher to think of these trusty workhorses as beautiful.

As with so many phenomena, however, what is simple and straightforward in a single serving becomes intricately detailed—beautiful, even—in the collective.

On December 5 John Baez, a mathematical physicist at the University of California, Riverside, posted a collection of images of polynomial roots by Dan Christensen, a mathematician at the University of Western Ontario, and Sam Derbyshire, an undergraduate student at the University of Warwick in England.

Polynomials are mathematical expressions that in their prototypical form can be described by the sum or product of one or more variables raised to various powers.

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Do Computers Understand Art?

ScienceDaily (Dec. 25, 2009) — A team of researchers has shown that some mathematical algorithms provide clues about the artistic style of a painting. The composition of colours or certain aesthetic measurements can already be quantified by a computer, but machines are still far from being able to interpret art in the way that people do.

How does one place an artwork in a particular artistic period?

The researchers have shown that certain artificial vision algorithms mean a computer can be programmed to "understand" an image and differentiate between artistic styles based on low-level pictorial information. Human classification strategies, however, include medium and high-level concepts.

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Are Computers Blinding Our Perception?

...Computers have become an extension of us: that is a commonplace now. But in an important way we may be becoming an extension of them, in turn. Computers are digital — that is, they turn everything into numbers; that is their way of seeing. And in the computer age we may be living through the digitization of our minds, even when they are offline: a slow-burning quantification of human affairs that promises or threatens, depending on your outlook, to crowd out other categories of the imagination, other ways of perceiving.

Welcome to the Age of Metrics — or to the End of Instinct. Metrics are everywhere. It is increasingly with them that we decide what to read, what stocks to buy, which poor people to feed, which athletes to recruit, which films and restaurants to try. World Metrics Day was declared for the first time this year.

The once-mysterious formation of tastes is becoming a quantitative science, as services like Netflix and Pandora and StumbleUpon deploy algorithms to predict, and shape, what we like to watch, listen to and read.

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Math Goes Viral in the Classroom With
the West Nile Virus

ScienceDaily (Dec. 11, 2009) — At least a dozen Alberta high-school calculus classrooms were exposed to the West Nile virus recently.

Luckily, however, it wasn't literally the illness. University of Alberta education professor Stephen Norris and mathematics professor Gerda de Vries used the virus as a theoretical tool when they designed materials for use in an advanced high-school math course. The materials allow students to use mathematical concepts learned in their curriculum to determine the disease's reproductive number, which determines the likelihood of a disease spreading.

"This piece was designed to satisfy an optional unit in Math 31 (Calculus), for which there are no materials, so we said, 'let's fill the gap,'" said Norris. "These materials show a real application of mathematics in the biology curriculum for high-school students."

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Are Our Brains Constantly Making Subconscious Calculations?

Our brain is wired to perform calculations that let us judge how far away an object is when we walk or jump around or reach for a container of milk. Although this task may seem easy, it turns out that calculating depth is surprisingly complex.

When we look at an object, our eyes project the three-dimensional structure onto a two-dimensional retina. To see the three dimensions, our brain must reconstruct the three-dimensional world from our two-dimensional retinal images. We have learned to judge depth using a variety of visual cues, some involving just one eye (monocular vision) and others involving both eyes (binocular vision).

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Corresponding Article!

Computers Based on Brains?

Your brain is doing some amazing calculations as you read these words. Not only are your recognising the letters, the upright and top cross of the 'T', but you are also understanding what they mean. Imagine if you could build a computer with the same kinds of skills? Computer Scientists are looking at how our brains work to build better machines. One area where people are far better than current technology is in seeing. Around half your brain is estimated to be involved in processing some type of information from your eyes.

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Baby brains are hard-wired for math

Experiment indicates that infants recognize apparent errors in subtraction.

Next time someone complains about arithmetic being hard, math lovers can defend themselves by saying "even a 6-month-old can do it."

Through monitoring the brains of infants, researchers confirmed that infants as early as 6 months in age can detect mathematical errors, putting to rest a debate that has gone on for over a decade.

A team of scientists from the United States and Israel exposed 24 infants to a videotaped puppet show. They used the puppets for addition and subtraction while observing the reaction of the babies.

For example, they started the show with two dolls. Before the show ended, a doll was removed and then the infant's vision was blocked with a screen. When the screen was taken away, either one doll was left, as expected, or two dolls, which would not be mathematically correct.

The infants looked at the screen longer (8.04 seconds) when the number of dolls was two, which did not agree with the solution of 2 – 1 = 1.

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The 7 Wonders Of The Ancient World: Mathmatician's Idea?

The man who created the idea to have 7 wonders of the world was the Byzantine Mathematician Philon. When not creating amazingly complicated math formulas, he would travel the lands in search for monuments of beauty. He traveled nearly, all the lands that were known at that time. He wrote on a piece of paper “De Septem Orbis Spectaculis”(the seven wonders of the world). Here were his seven. This was around 150 years before the birth of Christ.


Read all 7 here....

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Math Behind the Characteristic Shape of a Long Leaf Revealed

ScienceDaily (Nov. 26, 2009) — Applied mathematicians dissected the morphology of the plantain lily (Hosta lancifolia), a characteristic long leaf with a saddle-like arc midsection and closely packed ripples along the edges. The simple cause of the lily's fan-like shape -- elastic relaxation resulting from bending during differential growth -- was revealed by using an equally simple technique, stretching foam ribbons.

"These blades have rippled edges when they grow in slowly moving water. When they are transplanted to environments that have rapidly moving water, they generate new blades which are much narrower," says Mahadevan. "This example of phenotypic plasticity, or the ability of the algae to change their shape in response to environmental forces, led to a paper co-authored with Koehl and Silk last year that focused primarily on the experimental findings."

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Eureka! A Theater Play About Math?

When Marshfield math teacher Jean Kelley suggested last year that her friend Spring Sirkin produce a play about mathematics, Sirkin was skeptical.

“I didn’t think math concepts would naturally lend themselves to dramatic situations,’’ said Sirkin, whose Boston-based Chamber Theatre Productions tours the country with plays that support middle and high school instruction.

“We’ve always focused on the literature curriculum,’’ Sirkin said. “But we constantly ask teachers for feedback on what they’re reading and what they need help with. We were looking for something a little different, and when we polled teachers across the country, they overwhelmingly asked for a math play.’’

With Kelley as her consultant, Sirkin put out a call last spring to several play wrights to see what they might come up with. “Eureka!’’ by Shaun Wainwright-Branigan, jumped out at her right away.

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Heads or tails? It all depends on some key - but unknown - variables

Physorg - Everyone knows the flip of a coin is a 50-50 proposition. Only it's not. You can beat the odds. So says a three-person team of Stanford and UC-Santa Cruz researchers. They produced a provocative study that turns conventional wisdom, well, on its head for anyone who has ever settled a minor dispute with a simple coin toss.

It also could have profound implications in America's favorite sport -- pro football -- because the coin flip plays an integral role in deciding games that go into overtime.

But first, here's what the researchers concluded: Using a high-speed camera that photographed people flipping coins, the three researchers determined that a coin is more likely to land facing the same side on which it started. If tails is facing up when the coin is perched on your thumb, it is more likely to land tails up.

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A Fraction of History

- An extensive treatment of fractions appeared around 1600 B.C. in the Rhind Papyrus, which contained the work of Egyptians mathematicians.

The Egyptians did not express fractions as ratios such as 2:5 or 2/5.

They expressed ratios in unit fractions.

Egyptians used their symbols to represent this fraction.
1/3 + 1/15 would be represented as shown below:

Notice that all we care about is the man's feet. Feet pointing toward the direction of writing means add. Otherwise, it means subtract...

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Math Model May Speed Healing of Chronic Wounds

Victims of permanent, sometimes fatal wounds may receive hope from a new mathematical model published by researchers at Ohio State University.

Ischemic wounds, which arise from conditions such as diabetes, obesity, and high blood pressure, heal extremely slowly―if at all―and may result in loss of limb or even death. Inadequate blood supply in the affected area decreases the amount of oxygen and proteins that reaches the lesions, essential components of the healing process.

The model, a system of partial differential equations, uses some data from animal studies, but also includes values the researchers assigned to the various cells and chemicals in wound healing. Chuan Xue, a postdoctoral researcher in Ohio State's Mathematical Biosciences Institute, helped the team calculate numerical coefficients for oxygen concentration, the concentration of growth factors, density of white blood cells, density of fibroblasts, and density of tips and sprouts of new blood vessels.

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For Decades, Puzzling People With Mathematics

For today’s mathematical puzzle, assume that in the year 1956 there was a children’s magazine in New York named after a giant egg, Humpty Dumpty, who purportedly served as its chief editor.

Mr. Dumpty was assisted by a human editor named Martin Gardner, who prepared “activity features” and wrote a monthly short story about the adventures of the child egg, Humpty Dumpty Jr. Another duty of Mr. Gardner’s was to write a monthly poem of moral advice from Humpty Sr. to Humpty Jr.

At that point, Mr. Gardner was 42 and had never taken a math course beyond high school. He had struggled with calculus and considered himself poor at solving basic mathematical puzzles, let alone creating them. But when the publisher of Scientific American asked him if there might be enough material for a monthly column on “recreational mathematics,” a term that sounded even more oxymoronic in 1956 than it does today, Mr. Gardner took a gamble.

He quit his job with Humpty Dumpty.

On Wednesday, Mr. Gardner will celebrate his 95th birthday with the publication of another book — his second book of essays and mathematical puzzles to be published just this year. With more than 70 books to his name, he is the world’s best-known recreational mathematician, and has probably introduced more people to the joys of math than anyone in history.

How is this possible?

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Emotional Bunny says: "OMG! I finished his book before I was 15, and I never knew! I thought for sure he had majored in math or something similar, but alas it appears he majored in philosophy....."