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I met pi in school. You probably met pi that way too. It is that number used to calculate the circumference of a circle. Pi is shown symbolically as:


Pi is the ratio of the circumference of a circle to its diameter. It is an “irrational number” which means its exact value is inherently unknowable.

Using computers, we have calculated billions of digits of pi, starting with 3.14159265358979323…   –  but no recognizable pattern emerges. So strange. The digits of pi continue to infinity. Does anyone really understand infinity?

Ancient mathematicians did not like irrationality because it didn’t work with the concept of an omniscient God.

Recently I read about another pi connection which is also strange. In 1996, the UK earth scientist Hans-Henrik Stølum published a paper announcing that pi explains the seemingly chaotic paths of rivers in a mathematically predictable pattern.

This is called a river’s sinuosity. By dividing the river’s actual meandering length by the length of the direct line drawn from source to sea.

Of course, some rivers flow pretty straight from source to mouth , so they have small meandering ratios. Some rivers wander all over the place and have high meandering ratios.

But the average meandering ratio of rivers seems to be pi. Good old 3.14.

Albert Einstein used fluid dynamics and chaos theory to show that rivers tend to bend into loops.

If a river has a curve that will generate faster currents on the outer side of the curve. Those currents will cause erosion and so a sharper bend. That will eventually make the loop tighten. I have read that then chaos will eventually cause the river to double back on itself and form a loop in the other direction.

I did some more research on this river connection and found that this claim may not be accurate.

Someone put up a website at one point to crowdsource river data. The site at seems to be dead now. People could put in the coordinates of the mouth and the source of a river, and the length of the river (from Google Maps and Wikipedia probably) to calculate the sinuosity of a river. That study looked at 258 rivers and found an average sinuosity of an un-Pi-like 1.94.

Hmmm. Maybe it is another mathematical constant, like the golden ratio (phi) which we often find in nature. That value is 1.618. Nope.

What about if you look at pi/phi? You get 1.94. Okay, that’s a strange “coincidence.”  Or something more than coincidence?

I need to be careful with all this, because I saw the film titled Pi. I saw this science fiction film when it was released in 1998. It is a difficult film to label. It is surrealist, psychological, thriller, that delves into religion, mysticism, the relationship of the universe to mathematics and number theory. It was written and directed by Darren Aronofsky in his directorial debut.

I read it as a cautionary tale. It is about a genius oddball mathematician, Max, who has been working for a decade trying to decode the numerical pattern beneath ordered chaos. The ordered chaos he studies is the stock market.

Max’s belief that there is some mathematical “code” underlying everything compares in my mind with Einstein trying to find that theory that explains it all. That quest frustrated Einstein through the end of his life.

Beware of that quest.

Pie and Pi

A graphic I found by doing a Creative Commons search – but it was made by a friend. Coincidence or…? via

I came across a book at the library this past week quite by coincidence. Well, maybe..

The book is Fluke: The Math and Myth of Coincidence. Don’t be frightened by it being written by a mathematician, Joseph Mazur. It is about the seemingly improbable, surprising moments in our lives that seem to be coincidences. Maybe you attribute those events to serendipity. Or Fate. Look at some of the synonyms for coincidence: correspondence, agreement, accord, concurrence, consistency, conformity, fluke, harmony, compatibility. Do you attribute these kinds of events to coincidence or something else?

Others have said that “extremely improbable events are commonplace.” In 1866, the British mathematician Augustus De Morgan wrote, “Whatever can happen will happen if we make trials enough.”

What are the odds of being hit by lightning  once? More than once?  Roy Sullivan, a park ranger in Virginia who spent a lot of time outside in all kinds of weather was struck 7 times.

Enter the mathematical concepts of probability. This was one of those things that actually interested me in that rare interesting math class I was required to take.

Have you heard of the birthday paradox? What is the lowest number of people who must be in the same room to make it likely that at least two people will have the same birth day and month? Answer: 23. With 30 people in the room, the probability of a shared birthday is about 0.7 (or 70 percent).

Joseph Mazur knows that we are intrigued when someone wins the lottery four times in a row. How did you react when you learned that Abraham Lincoln had dreams that foreshadowed his own assassination? Creepy?

That statistics course you had to take may have taught you about correlation and causation. People confuse the two. Maybe cavemen believed that waking up caused the sun to appear.  You talk about a friend you haven’t talked to in years and they call you on the phone that day. Correlation does not imply causation. A correlation between two variables does not imply that one causes the other.

Some of Mazur’s examples seem to be “pure coincidence.” You find  your college copy of Moby Dick in a used bookstore in Paris on your first visit to the city? How do we explain the unlikelihood of strangers named Maria and Francisco, seeking each other in a hotel lobby, accidentally meet the wrong Francisco and the wrong Maria, another pair of strangers also looking for each other?

Mazur asserts that if there is any likelihood that something could happen, no matter how small the probability, it is bound to happen to someone at some time.

“What are the odds?” is what you might say in one of these situations. Like a déjà vu experience it might feel like some ripple just went through time, space or your universe.

In the paper, Methods for Studying Coincidences, mathematicians defined a coincidence as a “surprising concurrence of events, perceived as meaningfully related, with no apparent causal connection.”

In The Improbability Principle: Why Coincidences, Miracles, and Rare Events Happen Every Day, David Hand says that principle “tells us that events which we regard as highly improbable occur because we got things wrong. If we can find out where we went wrong, then the improbable will become probable.”

It’s no coincidence that ukuleles are popping up in ads on Facebook and other websites this week for me, because I was searching and looking at them on last weekend.

There’s the joke about two guys in a Dublin pub drinking and discovering a series of amazing coincidences in their lives. Another patron listening is stunned by the coincidences. But the bartender says, “Nah, it’s just the O’Reilly twins have been drinking too much.”

More Reading…

Connecting with Coincidence: The New Science for Using Synchronicity and Serendipity in Your Life

There Are No Accidents: Synchronicity and the Stories of Our Lives

Archimedes palimpsest

How do you read a two-thousand-year-old manuscript that has been erased, cut up, written on and painted over?

The manuscript in question is of great importance to the history of science. It is the Archimedes Palimpsest. This thirteenth century prayer book contains erased texts from earlier centuries including two treatises by Archimedes that can be found nowhere else. Those two are The Method and Stomachion.

Archimedes was a Greek mathematician, physicist, engineer, inventor, and astronomer.Among his advances in physics are the foundations of hydrostatics, statics and an explanation of the principle of the lever. He is credited with designing innovative machines, including siege engines and the screw pump that bears his name. Modern experiments have tested claims that Archimedes designed machines capable of lifting attacking ships out of the water and setting ships on fire using an array of mirrors. Archimedes is generally considered to be the greatest mathematician of antiquity and one of the greatest of all time. He used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, and gave a remarkably accurate approximation of pi.He also defined the spiral bearing his name, formulae for the volumes of surfaces of revolution and an ingenious system for expressing very large numbers.


Archimedes Thoughtful by Fetti (1620)

What is a palimpsest? In this case, a 10th-century scribe in Constantinople (present day Istanbul) copied the Archimedes treatise in the original Greek onto parchment. In the 13th century, a monk erased the Archimedes text, cut the pages along the center fold, rotated the leaves 90 degrees and folded them in half. The parchment was then recycled, together with the parchment of other books, to create a Greek Orthodox prayer book. This process of erasing and reusing parchment is called palimpsesting.

This prayer book is the kind of artifact that we expect to find in museums so that the public can see and learn from them.  Unfortunately (though it will turn out to be fortunate), the manuscript sold at auction to a private collector in 1998. Thankfully, the new owner deposited the manuscript at The Walters Art Museum in Baltimore, Maryland  where it has gone through conservation, imaging and scholarship.

The Archimedes Palimpsest project has revealed in revealing the erased text information about Archimedes and the ancient world. These new texts include speeches by an Athenian orator from the fourth century B.C. called Hyperides, and a third century A.D. commentary on Aristotle’s Categories.

The story of how the conservators did this is an interesting technology detective story itself (see video below) and involves using a particle accelerator.

But who amongst us will read a Byzantine prayer book with writings from ancient Greek mathematician Archimedes?  Without question, it is very few of us. But, should we have access to that information and should it be preserved both in its delicate paper form and in digital versions? Yes.

William Noel has spearheaded the conservation of the Archimedes Palimpsest  and helped to reveal in its parchments the hidden writings from the three original previously-unknown texts. They have also put all their images and findings on the Internet, available to anyone for free under a Creative Commons license.

Though most of will not read the codex, some of us will read about the Archimedes Codex  It pleases me in this mixing of ancient and cutting edge technology that people will be reading about the codex on a Kindle and, as the book’s subtitle says, learn how a medieval prayer book reveals the genius of antiquity’s greatest scientist.

William Noel is the Curator of Manuscripts and Rare Books at the Walters Art Museum and luckily he is also into technology, social media and openness and stresses its value even for the oldest, most established academic and cultural institutions. Noel believes passionately that institutions should free their digital data.

Read more about William Noel in the TED Blog Q&A >>

View Noel’s TED Talk

Yesterday was October 10, 2010. That’s 10-10-10.

I’m not a believer in numerology, but that’s unusual. It hasn’t happened since 1910. It must mean something to believers.

It’s one of many systems, traditions or beliefs in a mystical or esoteric relationship between numbers and physical objects or living things.

Well, I say I’m not a believer, but then I think about a few weeks ago when I was on vacation and was playing roulette. There I was playing birthdays, my sons’ jersey numbers, and a few “mystical” numbers I saw as “hot” or “cold.”

In the book Numerology: Or, What Pythagoras Wrought, mathematician Underwood Dudley uses the term to discuss serious use of serious things like the Elliott wave principle of stock market analysis. (Actually, “serious” might be overdoing it when it comes to the ups and downs of market prognosticators.)

Early mathematicians, such as Pythagoras, were into numerology. Even though today we don’t consider it math but as pseudomathematics, I played numerology games with my sons when they were very young in an effort to inoculate them from my own childhood math phobias.

Isopsephy (from two Greek words meaning “equal” and “pebble”) is the Greek word for the practice of adding up the number values of the letters in a word to form a single number.

“Numerology” has 10 letters – like our number system. Coincidence? Yes.

Those early Greeks used pebbles arranged in patterns to learn arithmetic and geometry. A Latin word for “pebbles” is “calculi”, the origin of the word “calculate.”

So, it’s not all that foolish.

Somewhere along the way, astrology and astronomy, alchemy and chemistry got into the mix. Numerology, even today, is often associated with the occult. Some astrologers believe that each number from 0 to 9 is ruled by a celestial body in our solar system.

Think about some common expressions we use. “There’s safety in numbers.” This expression means that being part of a group makes people feel more secure and more confident when taking action.

“Your number is up.” That means that a person is either in serious difficulty and something bad is going to happen, or the time has come when they will die. Why? Because numbers control our fate. Maybe.

Related to that is “It’s in the cards.” What cards? It means that something is very likely to happen. It’s based on the use of tarot cards which some people use to show what will happen in the future. Using cards with numbers (even an ordinary deck of playing cards is a variation.

So what was in the numbers for yesterday 10/10/10? I don’t know. I’m not a numerologist. But we’ll have to wait until 2110 for the next chance.

Of course, there’s always 11/11/11 next year. And you know what THAT means…

“Yet in another way, calculus is fundamentally naive, almost childish in its optimism. Experience teaches us that change can be sudden, discontinuous, and wrenching. Calculus draws its power by refusing to see that. It insists on a world without accidents, where one thing leads logically to another. Give me the initial conditions and the law of motion, and with calculus I can predict the future — or better yet, reconstruct the past. I wish I could do that now.”

The Calculus of Friendship: What a Teacher and a Student Learned about Life while Corresponding about Math by Steven Strogatz

I heard Steven Strogatz interviewed about his book. Though the book’s title got my attention, the basics of the story got me to listen and then pick up the book. It’s about a mentorship and friendship between Strogatz and his high-school math teacher over 30 years.

They share math – problems in calculus and  chaos theory – and then life events. It moves through their changing roles his from student to professor, and Mr. Joffray from teacher into retirement.

Joffray goes from the top of his form as a teacher to retirement. He competes in high level whitewater kayaking. He loses a son.

Steven goes from high school math whiz to Professor Strogatz. He loses a parent. He enters a doomed marriage.

After many years of teaching and quite a number of students whom I have kept in touch with, I don’t have a story like this one to tell.

The two of them take a long time to move from being connected by calculus, to that time when the connection moves beyond math.

The title, The Calculus of Friendship, isn’t just a clever twist. Calculus explores change and friendships are all about that too.  Calculus was  Isaac Newton’s way of modeling change mathematically.

Strogatz is professor of applied mathematics at Cornell University and got his Ph.D. from Harvard University. His specialty is nonlinear systems. Don’t ask me to explain but let’s say that can take you from synchronized fireflies to small-world networks.

Their letter writing starts when Steven is in his freshman year in college. Eventually, the letters begin to explore the more philosophical similarities between calculus and human relationships.

Calculus (from the Latin, calculus, meaning a small stone used for counting) is the study of change, in the same way that geometry is the study of shape and algebra is the study of operations and their application to solving equations.

Though I am fascinated by math, I was never good at it. It wasn’t mathphobia. I just didn’t have it in me. The book gets into calculus, differential equations, and chaos theory and I get a bit lost. I’m back in a high school classroom. But Strogatz does a pretty good job in using metaphors, images, and stories to make some of the math approach that beauty that mathematicians always talk about finding in things like solutions.

Of course, I’m not going to rush out to read his Nonlinear Dynamics And Chaos: With Applications To Physics, Biology, Chemistry, And Engineering (Studies in nonlinearity) (“An introductory text in nonlinear dynamics and chaos, emphasizing applications in several areas of science, which include vibrations, biological rhythms, insect outbreaks, and genetic control systems.”)

But from the interview I heard, I am tempted to find a copy of his Sync: How Order Emerges From Chaos In the Universe, Nature, and Daily Life in a library and give it a shot.

The mystery of the synchrony of fireflies flashing in sync by the thousands makes me wonder.

I have written here already about our own body clocks synchronizing with night and day and even with one another.

How freaky is it that Christiaan Huygens discovered in 1665 that when he observed two pendulum clocks they would swing in unison when they were within a certain distance of each other.

Spontaneous synchrony makes a footbridge in London undulate erratically as people on it unconsciously adjust their walking pace to the bridge’s swaying.

Is there chaos systems in playing “six degrees of Kevin Bacon” or in our behavior involving fads or mobs? Is there a “herd mentality” amongst stock traders? What about the way we drive in traffic?

James Glieck’s Chaos: Making a New Science and Barabasi’s Linked: How Everything Is Connected to Everything Else and What It Means were books that had a crossover audience from academics to a general audience too.

I don’t think I’ll head back into a math class though (even if there are free courses from MIT online).

If all that math still scares you, you could go down a novel path with Yoko Ogawa’s The Housekeeper and the Professor. The novel is narrated by the Housekeeper and the characters are known only as the Professor and Root. Root, nicknamed by the Professor because the shape of his hair and head remind the Professor of the square root symbol, is Housekeeper’s 10-year-old son.

The math Professor was seriously injured in a car accident and has short-term memory that only lasts for 80 minutes. He can remember his math and the baseball he loves, but forgets the Housekeeper all the time.

The Professor and Root connect through the baseball more than through math and are able to have a friendship in those short periods of memory.

And there is some education process between the housekeeper (who didn’t finish high school) and the professor. I like how the author uses mathematical theories to make everyday connections in their chaotic world.

We all understand chaos theory these days.

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