Pi Is Very Old

The number pi = 3.14159265358979323846... is arguably the only mathematical topic from very early history that is still being researched today. The Babylonians used the approximation pi ≈ 3. The Egyptian Rhind Papyrus, dated to roughly 1650 BCE, suggests pi = 256/81 = 3.16049.... Early Indian mathematicians believed pi = √10 = 3.162277... Archimedes, in the first mathematically rigorous calculation, employed a clever iterative construction of inscribed and circumscribed polygons to establish that 3 < 10/71 = 3.14084... < pi < 3 1/7 = 3.14285... This amazing work, done without trigonometry or floating point arithmetic, is charmingly described by George Phillips in *Pi: A Source Book* (Entry 4).

Pi in Popular Culture

The number pi, unique among the pantheon of mathematical constants, captures the fascination both of the public and of professional mathematicians. Algebraic constants, such as √2 , are easier to explain and to calculate to high accuracy. The constant e = 2.71828... is pervasive in physics and chemistry and even appears in financial mathematics. Logarithms are ubiquitous in the social sciences. But none of these other constants has ever gained much traction in the popular culture.

In contrast, we see pi at every turn. In an early scene of Ang Lee's 2012 movie adaptation of Yann Martel's award-winning book *The Life of Pi*, the title character, Piscine ("Pi'') Molitor, writes hundreds of digits of the decimal expansion of pi on a blackboard to impress his teachers and schoolmates, who chant along with every digit. (Good scholarship requires us to say that in the book Pi contents himself with drawing a circle of unit diameter.) This has even led to humorous takeoffs, such as a 2013 Scott Hilburn cartoon entitled "Wife of Pi," which depicts a 4 figure seated next to a pi figure, telling their marriage counselor, "He's irrational and he goes on and on."

This attention comes to a head each year with the celebration of "Pi Day" on March 14, when, in the United States, with its taste for placing the day after the month, 3/14 corresponds to the best-known decimal approximation of pi (with 3/14/15 promising a gala event in 2015). Pi Day was originally founded in 1988 as the brainchild of Larry Shaw of San Francisco's Exploratorium (a science museum), which in turn was founded by Frank Oppenheimer, the younger physicist brother of Robert Oppenheimer, after he was blacklisted by the U.S. government during the McCarthy era.

Originally a lighthearted gag where folks walked around the Exploratorium in funny hats with pies and the like, by the turn of the century Pi Day was a major educational event in North American schools, garnering plenty of press -- visit this Google site to see the seasonal interest in the word "pi." In 2009 the U.S. House of Representatives made Pi Day celebrations official by passing a resolution designating March 14 as "National Pi Day" and encouraging "schools and educators to observe the day with appropriate activities that teach students about Pi and engage them about the study of mathematics." This seems to be the first legislation on pi to have been adopted by a government, though in the late 19th century Indiana came embarrassingly close to legislating its value. (See Singmaster's article in *Pi: A Source Book*, Entry 27, or the Pi Day talk by one of us.)

As a striking example, the March 14, 2007, *New York Times* crossword puzzle featured clues where, in numerous locations, a pi character (standing for "PI") must be entered at the intersection of two words. For example, 33 across, "Vice president after Hubert'' (answer: "SPIRO"), intersects with 34 down, "Stove feature'' (answer: "PILOT"). Indeed 28 down, "March 14, to mathematicians,'' was, appropriately enough, "PIDAY," while "PIPPIN" is now a four-letter word.

Pi Mania in Popular Culture

There are many instances of pi in popular culture. Here are just a few:

- On Sept. 12, 2012, five aircraft armed with dot-matrix-style skywriting technology wrote 1,000 digits of pi in the sky above the San Francisco Bay Area as a spectacular and costly piece of "piformance" art.

*Jeopardy!*, in the category "By the Numbers," the clue was "'How I want a drink, alcoholic of course' is often used to memorize this" (answer: "What is pi?" because the number of letters of these words spells the digits of pi).

*The Matrix*the lead character Neo has only 314 seconds to enter the Source.

*Time*noted the similarity to the digits of pi.

*Pi*received a screenplay award at the Sundance Film Festival. When the two of us were sent advance access to its website, we diagnosed it as a fine hoax.

*The Simpsons*had Apu declaring, "I can recite pi to 40,000 places. The last digit is 1." This digit was supplied to the screen writers by one of the two of us.

*Contact*the lead character (played by Jodie Foster in the movie) searches for patterns in the digits of pi, and after her mysterious experience she seeks confirmation in the base-11 digits of pi.

Several more examples are given in the Pi Day talk.

With regard to item 3 above, there are many such "pi-mnemonics" or "piems" (*i.e.*, phrases or verse whose letter count, ignoring punctuation, gives the digits of pi) in the popular press. Another is "Sir, I bear a rhyme excelling / In mystic force and magic spelling / Celestial sprites elucidate / All my own striving can't relate." (See Brian Bolt's book *More Mathematical Activities: A Resource Book for Teachers*, pg. 106.) Some are very long. (See *Pi: A Source Book*, Entry 59.)

Sometimes the attention given to pi is annoying, such as when, on Aug. 14, 2012, the U.S. Census Office announced the population of the country had passed *exactly* 314,159,265. Such precision was, of course, completely unwarranted. Sometimes the attention is breathtakingly pleasurable. See this 2013 video, or the Pi Day talk.

Poems vs. Piems

While piems are fun, they are usually doggerel. To redress this, we include examples of excellent pi poetry and song. Below we present the first stanza of the much-anthologized poem "PI" by Polish poet Wislawa Szymborska (1923-2012), who won the 1996 Nobel Prize for literature, from his published collection *Poems New and Collected*.

The admirable number pi:

three point one four one.

All the following digits are also just a start,

five nine two because it never ends.

It can't be grasped, six five three five, at a glance,

eight nine, by calculation...

Below we present the beginning of the lyrics of "Pi" by the influential British singer-songwriter Kate Bush. The *Observer* review of her 2005 collection *Aerial*, on which the song appears, wrote that it is "a sentimental ode to a mathematician, audacious in both subject matter and treatment. The chorus is the number sung to many, many decimal places." (She sings over 150 digits but errs after 50 places. The correct digits are given in the published lyrics.)

Sweet and gentle sensitive man

With an obsessive nature and deep fascination

For numbers

And a complete infatuation with the calculation

Of Pi

(The full text of these poems is given in this paper by the two of us.)

Graphical Representations of Pi

A fruitful new approach is to display the digits of pi or other constants graphically, cast as a random walk. For example, the first plot below shows a walk based on one million base-4 pseudorandom digits generated by a computer, where at each step the graph moves one unit east, north, west or south, depending on the whether the pseudorandom base-4 digit at that position is 0, 1, 2 or 3. The color indicates the path followed by the walk, colored by a standard hue-saturation-value scheme that produces a rainbow of colors.

The next figure shows a walk on the first 100 billion base-4 digits of pi. This may be viewed dynamically in more detail online at the Gigapan site, where the full-sized image has a resolution of 372,224 x 290,218 pixels (108.03 billion pixels in total). This is one of the largest mathematical images ever produced, and, needless to say, its production was by no means easy. (See this paper for technical details.)

The above walk was on binary digits of pi. Here, for comparison, is a walk on 10 million base-10 digits.

Such techniques are used to study what is arguably one of the oldest unanswered questions of mathematics: Are the digits of pi "random"? (say in the specific sense that each decimal digit occurs, in the limit, one tenth of the time, each pair of digits occurs one one-hundredth of the time, and so on). Sadly, we still don't know the answer to this age-old question (and many others). But with the advent of modern computer technology, maybe the balance is finally tipping in favor of mathematicians. See this technical paper by the two of us, from which the above blog post is condensed and adapted (with permission of the American Mathematical Monthly) for details.

**CORRECTION:** An earlier version of this blog post erroneously stated that the Egyptian Rhind Papyrus approximated pi to 32/18. The actual quotient is 256/81. The post has been updated accordingly.