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10 stunning images show the beauty hidden in pi

Saturday -- March 14, 2015, or 3/14/15 -- marks an extremely nerdy holiday. It is the official celebration of π, the magical, mathematical and infinite constant that is the ratio of a circle’s circumference to its diameter.

For any circle you can imagine, if you divide the distance around the circle by the distance across it, you will get pi, or 3.1415926535897932384626433832795028841971693993751058209749

We could keep going, but you get the picture.

Some people will celebrate the holiday by making and eating pies (Washington restaurants are offering specials on everything from pizza to banana cream). Others will run a Pi-K race of 3.14 kilometers. And some data tinkerers are making art that visualize pi’s infinite and random digits.

One of the best known of these data tinkerers is Martin Krzywinski, a scientist who specializes in bioinformatics, or using computer science and statistics to understand biological data. Krzywinski started publishing his pi art in 2013, beginning with this visualization:

Each digit of pi is represented by a dot of a different color: 3 is orange, 1 is red, 4 is yellow, and so on. Krzywinski then folded these colored dots, each of which represents a different digit ("1" or "4"), into a spiral. Going from the center of the circle outward, here is the first 13,689 digits of pi:

Working with Cristian Ilies Vasile, a self-described "artist by accident," Krzywinski also created a series of circular representations of pi, where the numbers are connected across the circle with a chord. The artists start at 3, draw a line to 1, draw a line to 4, and so on, changing the color with each new digit.

The image below follows the same process, except now, when a number is repeated (for example a "1," followed by another "1"), Krzywinski and Vasile place a dot at the outer edge of the circle. The more digits that are repeated, the larger the dot.

The big purple dot near the top of the circle represents a unique point in pi: A place where six 9s occur in a row. Called the Feynman Point, the repetition appears much earlier than probability would dictate.

Below is Krzywinski's new illustration for 2015, a type of diagram that is called a treemap. He first divides the box by drawing "3" lines vertically. Then he divides the first box horizontally by drawing "1" line, the second by drawing "4" lines, and so on. Here, Krzywinski randomly colored the graphic with the primary colors used by members of the De Stijl and Bauhaus art movements in the 1920s, like Piet Mondrian, Paul Klee and Joseph Albers.

So what is the point of all this? Mostly, the works are meant to be beautiful and fun to look at. But beyond that, Krzywinski says the art is meant to awaken emotions about math (hopefully emotions other than dislike and confusion) and start conversations about numbers and randomness.

"All numbers are necessarily interesting," Krzywinski says. "But, to echo the ending of Orwell's 'Animal Farm,' some numbers are more interesting than others. Pi is one of those."

Why is pi so interesting? For one thing, pi describes a perfect circle, and thus is included in any formula that describes a circle or some kind of repetition, from a heart beat to the Earth's orbit around the sun.

For another, pi has the appearance of being random (or, more accurately, "uniformly distributed") -- meaning that, as its digits continue, there is an equal chance of any digit between 0 and 9 appearing. In the first six billion digits of pi, each of the digits 0 through 9 shows up about six hundred million times.

If pi were truly random, that would mean that the number sequence in pi would never repeat itself, and -- because pi is infinite -- it would contain all patterns in existence. Any word that you can think of, when encoded in numbers, would show up in pi, says Kryzwinski. So would the entire works of Shakespeare, all possible misprints and permutations of Shakespeare, and even, if you were patient enough, pi itself. As Cornell mathematician Steven Strogatz writes for The New Yorker, pi is so special in part because it "puts infinity within reach."

Pi looks random: Mathematicians have computed pi out to 10 trillion digits and seen no evident pattern. But what really vexes mathematicians is that no one can definitely say that pi is random -- no one has figured out the mathematical proof. And in another sense, pi is anything but random: After all, the number embodies the order of a perfect circle. "The tension between order and randomness is one of the most tantalizing aspects of pi," writes Strogatz.

This randomness is illustrated in another visualization of pi created by Nadieh Bremer, an astronomer who now does data art and analytics at the blog Visual Cinnamon. Bremer says she was inspired by Krzywinski's work to create a kind of map of pi, in which each digit is assigned both a color and a direction.

Bremer notes that the idea has been around for a while: In the 1888 book “The Logic of Chance,” mathematician John Venn suggested that the digits 0 to 7 in pi represent eight compass directions, and followed the path tracked by those digits. Cristian Vasile also did a random walk using pi digits, mapping the image over the Earth's surface.

Bremer's graphics follow pi as it "walks" out 100, 1,000, 10,000, 100,000 and finally 1 million digits.

Bremer says that her favorite part about the visualization is that the shape of 1,000 digits has no obvious relationship to 10,000 digits, and seeing 10,000 digits gives you no idea how 100,000 digits will look. Bremer describes this as the epitome of randomness: What came before has no influence on what happens next, and there is no evident structure or pattern.

What else can we take away from these graphics? For one, seeing all of the digits of pi visualized in physical space should help bring home another attribute of mathematics. Math isn't just numbers on a page: From physics to architecture, math is the language that we use to describe and construct the physical world around us. That's especially true for pi, which describes a perfect circle that appears everywhere in nature.

Says Kryzwinski, "Pi Day is a great time to take a moment and recognize the extent to which, as a language to describe reality, math has allowed us to write the details of the workings of our universe."

"Thanks to numbers and math, we can build devices that will deliver this text to you. And, as you read it, we know how fast the photons will be traveling and what happens when they hit your retina. The rest is up to you."

Read more stories on data visualization:

-How a dog sees a rainbow, and 12 other images that explain how we see color

-12 fascinating optical illusions show how color can trick the eye

-A data genius computes the ultimate American road trip