The Jewish value of pi
Guest posted by Brian Zimmerman
It's only my second blog for People of the Book, and already I feel like I'm coming full circle. That's because this post is about pi, the mysterious mathematical concept that has been fascinating scholars (even Biblical scholars) for centuries. In fact, pi is such an impressive number that British author and mathematician Alex Bellos devotes a whole chapter to it in his cleverly titled new book on mathematics, Here's Looking at Euclid. In his book, Bellos asks a simple but perplexing question: "What's so special about pi? Well, with the help of an ancient Egyptian farming technique, a Biblical swimming pool, and a supercomputer in a New York studio apartment, that's what we're going to find out.
First up to explain the pi phenomenon: Rabbi Yitzchak Ginsburgh, an American-born rabbi now living in Israel. In his article "The Story of Pi," he suggests that pi "represents the most basic intermediate spanning the gap between the human mind and nature's curve." What Rabbi Ginsburgh means is that circles and curves appear nearly everywhere in nature, yet as humans our minds tend to understand the world in straight lines. Because we're a scientific culture, we love to measure, quantify, and analyze the natural phenomenon down to its smallest atom. But pi - and the circle's they represent - often defy our angular expectations.
In case it's been a while since your last geometry class, here's a quick refresher. Pi is the ratio between a circle's circumference (the measurement around the outside of the circle) and its diameter (the measurement across from one side to another). This ratio is always the same, no matter the size of the circle, and its represented by the Greek letter ∏. In numerical terms, it's approximately 3.14.
As Bellos describes in this book, the peculiarities of this pi have been noted since ancient times. In Egypt, farmers used a similar ratio to measure the size of their circular crop fields. That ratio was 3 1/8. Later on, the Babylonians would use a similar ratio to measure their wagon wheels and whetting stones.
Pi is also appears in Jewish history and is even referenced in Jewish scripture. In Sepher Melachim, the Book of Kings, chapter 7 verse 23 describes a circular pit of water that King Solomon builds near the edge of the sea. The description is as follows:
"And he made the molten sea, ten cubits from brim to brim; it (was) round all about, and the height thereof (was) five cubits; and a line of thirty cubits did compass it round about."
Did you spot pi? Look closely. The circular pit had a diameter of ten units across, or "brim to brim." That's the pit's diameter. If we were to calculate the circumference using pi, we'd end up with 3.14 units. And what did the Book of Kings say? 30 units! For Biblical times, that's a pretty close estimate.
But the Vilna Gaon, an 18th- century Talmudist, wanted to do better. Using gematria, the Jewish study of word-number relations, he provided key insight into the King Solmon's pit problem. What he noticed was that the Hebrew text for the word "perimeter" was written as kanah - a distinctly two-syllable word. But when reading this word from the Book of Kings, it has become custom to pronounce it as kahn - a one syllable word. The Vilna Gaon thought this was incredible. He noted, quite accurately, that for a circle with a diameter of ten cubits, the circumference would not be exactly 30 cubits. There'd be a remainder left over. That's where the difference in syllables comes in. The Vilna Gaon says that the added syllable that appears in the written form but not in the spoken form indicates the presence of pi's remainder - the extra 1.14! It was there all along, but hiding in plain sight.
As it turns out, pi's elusive remainder has been hiding from us for a long time. We still haven't figured it out after 2000 years of mathematical history. That's because pi is a transcendental number, meaning its digits literally go on forever, and with no discernable pattern. Still, pi's never-ending quality hasn't deterred us from trying to write it all out.
The Chudnovsky brothers, both math professors at Columbia University, once programmed a supercomputer in their tiny New York apartment to calculate pi to the 2 trillionth digit. Today's computers can calculate pi much quicker, and our current enumeration of pi is somewhere around 8 quadrillion digits. (In case you were wondering, the Guiness record holder for Most Digits of Pi recited by memory is 24-year-old Chinese graduate student Lu Chao, who remembered pi up to 67,890 digits. It took him 24 hours.)
Maimonides, in his Perush Ha-Mishna, chimes in on the transcendence of pi. Commenting on humankind's inability to understand the number in its entirety, he says the following:
"This is not due to a lack in our knowledge…but it is in its nature that it is unknown, and there is no way to know it, but it is known approximately."
This is what makes pi such a beautiful metaphor for Judaism's embrace of the unknown. In Kabbalah, God is often referred to in terms of transcendence, in which a divine presence is seen to be outside of the material world, and immanence, in which a divine presence is seen to be within the material world. For centuries, Chasidic and Kabbalistic sages argued over whether God's true being was transcendent or immanent. In the end, however, they came to the realization that a true Divine Presence could exist in both realms - finite and infinite, yet always beyond human understanding. This, they said, was an attribute of perfection.
Such is the case with pi. In a world where nearly everything can be charted on a graph, map, or screen, we should find great joy in the things that will forever remain out of our reach - and be happy that they'll remain there. Let's call it the ultimate peace of pi.