# Pi Day 2016

Once again Pi Day (March 14, or 3/14 in North American notation) has come and gone here, when both professional mathematicians and students in school celebrate this most famous of mathematical numbers. Last year was a particularly memorable Pi Day, since 3/14/15 gets two more digits correct, although some would argue that this year's Pi Day is also memorable, since 3/14/16 is pi rounded to four digits after the decimal point (the actual value is 3.14159265358979323846... ).

Numerous celebrations were scheduled for Pi Day 2016. San Francisco's Exploratorium featured several events, culminating with a "Pi Procession" at 1:59pm Pacific Time (corresponding to 3.14159) and pie served at 2:15pm. The Illinois Science Council sponsored a Pi Day Pi K Fun Run, starting 6:28pm (= 2 x Pi) at any of four different locations in Chicago. The website teachpi.org lists 50 ideas to make Pi Day "entertaining, educational, tasty and fun."

Just before Pi Day, Pizza Hut, in conjunction with the well-known mathematician John Conway of Princeton University, announced that on March 14 it would release three mathematics problems on its Hut Life blog. It did so. The first person to submit correct solutions to one of the three problems had a *chance*(the term is not defined) to receive 3.14 years of free pizza from Pizza Hut. The problems vary in difficulty from high school to Ph.D. level.

In past years, the present authors have celebrated Pi Day with popular articles and more serious technical pieces. For Pi Day 2014, we presented several examples of "piems," i.e., poems rhapsodising on the wonders of pi. We also presented some examples of recent research using high-tech graphical tools to explore the intriguing question of whether and why the digits of pi are "normal" (i.e., its digits are statistically "random" in a particular sense).

For Pi Day 2015, we summarised recent computations of pi, including the current world's record of 13.3 trillion decimal digits of pi, by someone known only as "houlouonchi" and Alexander J. Yee. We also explained what pi is actually used for in modern science and technology - for example, the binary value of pi is buried in the software controlling your mobile phone, so that it can process gigahertz signals that transmit data.

**Growing public interest in pi**

Originally conceived in 1988 as a cute joke, since 2009 in America, Pi Day has been officially sanctioned by an act of Congress.

Growing interest in Pi Day mirrors a growing interest in pi (and mathematics in general) in the popular culture. Where once only professional mathematicians and teachers cared about pi, nowadays it is not at all uncommon to see pi mentioned in the movies, on television or in social media. In 2003, the first three digits of pi (314) appeared in the Matrix Reloaded. On 9 May 2013, the North American quiz show Jeopardy! featured an entire category of questions on pi.

Pi has been featured in at least six different episodes of the North American TV show Person of Interest. In one episode, a secret entrance to an underground control room is guarded by what appears to be an abandoned vending machine. After entering the digits "3141," a door opens.

In another key episode, the show's lead character Harold Finch (played by Michael Emerson), posing as a high school teacher, explains to the class that the digits of pi are a never-ending, nonrepeating sequence, and thus all of one's personal information, such as one's telephone number or Social Security number appear somewhere in pi. (Note: This presumes that Pi is a "normal" number, which although widely assumed to be true, has never been proven and remains an area of active research.)

**New research on pi**

A seemingly random walk on 200 billion bits of Pi

Many readers might be surprised that anything new remains to be discovered about pi. After all, given that researchers have been scouring the mathematical universe for facts about pi for millennia, surely everything of any note has already been found? This sentiment was even expressed, for example, in Petr Beckmann's 1976 book A History of Pi. Yet almost before the ink was dry, two researchers (Gene Salamin and Richard Brent) independently discovered a new algorithm for computing pi, each iteration of which approximately *doubles* the number of correct digits. This simple algorithm can even be implemented on a programmable calculator. Numerous other, even more remarkable, formulas and algorithms for pi have been discovered in the years since 1976.

Similarly, Daniel Shanks who himself had computed pi to over 100,000 digits in 1961, once declared that computing pi to one billion digits would be "forever impossible." Yet this was done, by both Kanada and the Chudnovsky brothers, in 1989.

Well-known mathematical physicist Roger Penrose, in the first edition of his book The Emperor's New Mind, suggested that mankind would likely never know whether a string of ten consecutive sevens appears in the expansion of pi. But within months after the book's publication, Kanada had indeed found the string "7777777777" in a 580-million-digit computation he had done.

Even within the past 20 years, some remarkable new facts have been discovered, some of which were completely unanticipated. One example was the 1997 discovery of a formula and associated algorithm for pi that permits one to directly calculate digits of pi starting at an arbitrary position, such as the millionth or the billionth digit, without needing to compute any of the digits that came before (although it only works with binary or base-16 digits).

With advancing computer technology, even more powerful tools are being brought to bear on pi. One example is new graphical-based techniques (as shown above) that permit researchers to visually explore questions such as whether and why the digits of pi are "normal" (i.e., statistically random in a certain sense, as mentioned above). We can only expect that new and heretofore unanticipated facts will come to light, perhaps sooner than many think.

**A compendium of recent work on pi**

To celebrate Pi Day 2016, we have decided to collect 25 key technical papers that have appeared over the past 40 years with research results related to the computation and analysis of this memorable number. The collection includes papers describing the results and computations mentioned above, and many others as well. Some papers in our collection explore advanced research questions suitable for active researchers in the field, but several are highly accessible to undergraduate mathematics students. A listing and brief description of the individual papers is available here.

*Jonathan Borwein (Jon) (Ontario & Jesus 1971) read MSc and DPhil Maths and is Laureate Professor Mathematics at the University of Newcastle, Australia. He also regularly blogs:** http://experimentalmath.info/blog/. This article was co-written with David H. Bailey, Lawrence Berkeley National Lab (retired) and University of California, Davis.*

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