“π-Day” (“Pi-Day”) Guest Post Written by Bulent Atalay

Ink drawing by the author, inspired by a Yousuf Karsh photo hanging in the Physics Department at Princeton University

Last year for “π-Day” (“Pi-Day”) I wrote a guest blog for Documama about Albert Einstein, Time Magazine’s choice for the “Individual of the 20^th century.” The physicist, whose name has become synonymous with genius, was born in Ulm, Germany on March 14 (3.14) one hundred and thirty-four years ago. Frequently physics students have celebrated the day in homage to the birthday of the venerable scientist, and these days Pi-Day has become a bit more mainstream.

In his “miracle year,” 1905, Einstein had written four papers, three of which could have won the Nobel Prize. It was his paper with the most obscure title of all, “On the Electrodynamics of Moving Bodies,” that he changed the paradigm for physics that had prevailed since Isaac Newton published his masterpiece, the Principia, almost 230 years earlier. Better known as the “Special Theory of Relativity,” Einstein’s theory rejects the three fundamental undefinables of length, mass and time as being invariant, and in their place posits the speed of light as the unique invariant. Length, mass and time could increase or decrease, when the body travels at different velocities. Then ten years later he published his masterpiece, the “General Theory of Relativity,” which offered a refinement to Newton’s theory of gravitation. The Big Bang Theory, stars collapsing into black holes, quasars, pulsars… are all manifestations of the General Theory. Einstein’s legacy is as seminal, and as staggeringly consequential to the physicist’s understanding of physical reality as his theories are inscrutable to the non-physicist.

TEACHING YOUR CHILD π (Pi): 3.14…

π is the symbol for the number representing the ratio of the circumference of a circle to its diameter. It is a universal constant, the same for all circles and indeed everywhere in the universe. In the language of mathematics, it is also an irrational number, and as such cannot be expressed exactly by the ratio of two numbers. Finally, it is also a transcendental number, that is, not algebraic — not a solution a non-zero polynomial equation with rational coefficients. A ramification of this last statement is that geometrically speaking “a circle cannot be squared,” a circle cannot be constructed with exactly the same area as a specified square using only a compass and a straight edge, and accomplished in a finite number of steps. The proof of this conjecture is so complicated that it was not achieved rigorously until 1882.

What Children are taught in elementary school:

Trick One:

A good approximation and an easy way to remember the number still comes from the mnemonic, “How I need a drink, alcoholic of course, after the heavy lectures involving quantum mechanics,” 3.141 592 653 589 79… Good to 15 places, it comes from counting the letters in each successive word. (For children, substitute “pepsicola” for “alcoholic”.)

Trick Two:

Again, π is the symbol for the number representing the ratio of the circumference of a circle to its diameter. At first pale, it is roughly equal to 3. Expressed to

photo from March 1926 National Geographic Magazine

two decimal places, it is 3.14. To seven places after the decimal, the correct value of π is 3.141 592 7 As an irrational number, however, π cannot be expressed exactly by the ratio of two numbers; however, elementary school students are often taught 22/7, as a crude approximation. The ratio yields 3.142 857, correct to just two places after the decimal.

The Ancient Egyptians building the Great Pyramid about 4600 years ago had the value of π to two decimal places, 3.14. After laying out a circle (points equidistant from a center), they measured its radius. Then they physically “squared the circle,” presumably by having four groups of workers pulling in four directions, with four equal sides and two equal diagonals. (This is not “squaring the circle” in the mathematical sense discussed in the last paragraph. The perimeters of the two figures are equal, but the areas encompassed by the two are not.) After the square base of the pyramid was laid out, then the radius of the original circle served was adopted as the height of the pyramid, 455 ft (139 m). The Great Pyramid, essentially a man-made mountain serving as a mausoleum for Pharaoh Khufu, rises at 52° relative to the plain of the base.

Trick Three:

Take the six integers 1 2 3 4 5 6, and subtract from them 0 1 0 1 0 1. Thus

1 2 3 4 5 6

 —0 1 0 1 0 1

1 1 3 3 5 5                                                                                                                                                             

Dividing the last three digits by the first three, 355/113, the ratio is obtained as 3.141 592 9. This is good to six decimal places.

EPILOGUE

About 1940 the π was computed to ten thousand significant figures.

In 1960, a computer was used to apply an algorithm to calculate π to one million decimal places, where it was found to terminate with 5.

In 2011, a most determined Japanese gentleman, Shigeru Kondo, collaborating with the Northwestern University graduate student, Alexander Yee, computed π to ten trillion places, where its value was found to be 5 again. This, however, is nothing more than a happy coincidence!

Bulent Atalay

Bulent Atalay is my brilliant father-in-law and a retired physics professor. He is also the author of two books, Math and The Mona Lisa, and Leonardo’s Universe. You can find out more about the amazing man my kids call Buyukbaba (Turkish for grandfather ) at his website  and on his blog for National geographic.