Today, March 14, is one of the days that is sometimes celebrated as “Pi Day”, in honor of the best-known irrational and transcendental number, the ratio of the circumference of a circle to its diameter, usually written as the Greek letter π (pi). The date, 3/14, is chosen because the approximate value of π is 3.14159265… Another date that is sometimes observed as Pi Day is July 22, from the rational approximation to π, 22/7. Legend has it that the value was named π because pi is the first letter of the Greek word “περίμετρος”, meaning perimeter.
Mathematicians thought for many centuries that π was irrrational, although this was first proved in the 18th century by Lambert. Ferdinand von Lindemann proved that π was transcendental (that is, it is not the root of any polynomial with rational coefficients) in 1882, a proof that also implied the impossibility of “squaring the circle”, constructing with a compass and straightedge a square with the same area as a given circle. (This did not stop cranks from trying, just as the laws of thermodynamics did not put paid to the search for a perpetuum mobile. )
Despite all this, the value of π crops up here and there in surprising ways. It is related to the integers by the infinite series:
It also turns up as the average value of the sinuousity of rivers: that is, the total length of a river, including its meanderings, divided by the straight-line distance from the river’s source to its mouth. If you have a floor in your house made of parallel wooden boards, you can estimate the value of π with a toothpick. If we call the width of each board L, and the length of the toothpick k (we require that k < L), then the probability that the toothpick will cross a line between boards is 2 k / L π. This is known as Buffon’s needle experiment, after George-Louis Leclerc, Comte de Buffon, who first proposed it in the 18th century. Although the math behind the estimate is correct, this is an extremely inefficient way to estimate the true value of π.
Pi also was involved in the case of one of the candidates for all-time legislative lunacy. In 1897, a rural Indiana physician and amateur mathematician, Dr. Edwin Goodwin, decided that he had figured out how to square the circle. He also decided, in the spirit of civic-mindedness, to share his discovery with his fellow Hoosiers, and prevailed upon his representative in the state’s General Assembly, Mr Taylor Record, to introduce a bill to accomplish this. The preamble of the bill reads:
A Bill for an act introducing a new mathematical truth and offered as a contribution to education to be used only by the State of Indiana free of cost by paying any royalties whatever on the same, provided it is accepted and adopted by the official action of the Legislature of 1897.
The bill contains several incorrect, and in some cases mutually exclusive, mathematical claims, including one that seems to say that the value of π would be set by law at 3.2. It is otherwise notable chiefly for being almost incomprehensible.
The bill passed the lower house of the legislature unanimously. Fortunately, at the point that the state Senate was going to take up the bill, the head of the Purdue University math department was present on other business. He was told about it, and offered the opportunity to meet the learned doctor. I especially liked his response:
He declined the courtesy with thanks, remarking that he was acquainted with as many crazy people as he cared to know.
He spent some time “coaching” members of the state Senate, and the bill was ultimately postponed indefinitely.
Finally, today is also, by coincidence, the 131st anniversary of the birth in 1879 of Albert Einstein. His reaction to events in Indiana was, alas, not recorded as far as I know.