
π
π ^{t} is the sweetest expression in all of mathematics
Events in the History of π
Sing a song of sixpence, a pocket full of rye. Four and twenty blackbirds, baked in a pie. When the pie was opened, the birds began to sing. Wasn't that a dainty dish to set before the king? 
The "birds" must really have been black and white birds—they just had to be magpi's! 
Printing π
Recipe for π
 176 parts  arctan series of 1/57 
 28 parts  arctan series of 1/239 
 48 parts  arctan series of 1/682 
 96 parts  arctan series of 1/12943 
 Fill the initial array of digits for each series with zeros. Place the parts value in the leading position. 
 Mix each series separately by alternating adding and subtracting terms. Long division is kneeded at least twice to obtain each term. Continue until the last term is sufficiently small. 
 Combine by adding the 1/57 and 1/239 series and subtracting the 1/682 series. Season by adding the 1/12943 series. 
 Bake thoroughly at 180°, or preferably in as hot of a computer as is available, until all series have converged. Remove from memory and format to a file. 
 May be served electronically direct from the file or arranged tastefully for the web. Small pieces may be presented spread on sheets of paper. 
 Note: Just because π = 180° does not mean you need to turn the computer backwards to perform the proper baking! 
The most satisfying real number is 8π!
Rational Approximations
Computing many digits of π is easy. It is as easy as pi!
Arctan Formulae for π
If the ratio of the circumference of an Igloo to its diameter is an Eskimo pi, then why is the ratio of the circumference of a hole to its diameter a piehole, and not a hole pi? Actually, inserting a whole pi in a piehole is wonderful!
Small pieces of π
Digits in the decimal expansions are displayed in groups of 5 with 100 digits per line and 1000 digits per block.
Values given were calculated on a notebook computer using arctan formulae programed in C.
 Pi  1,000 digits   a nibble of the filling 
 Pi  10,000 digits   a bite 
 Pi  100,000 digits   maybe a mouthful 
 Pi  1,000,000 digits   a tiny sliver 
 Pi  3,000,000 digits   a small piece 
The only thing rational about π is eating it! 
Effect of the Pie Pan
If you divide a jacko'lantern by its diameter, you get pumpkin pi! If you divide a cow by its diameter, you get .... oh nevermind! 
Historical Growth in the Known Digits of π
The record for the number of digits of π that has been computed has changed over the past few centuries as new techniques and (in the last few decades) faster computers with larger memories became available.
Currently the record is 31.4 trillion digits in 2019.
Calculations done prior to 1900 were hand calculations (527 correct digits). The mid 1940's saw the utilization of desktop calculators (808 correct digits).
From 1949 to the present, computations were performed with computers.
The largest knight at King Arthur's Round Table was Sir Cumference. He got his size and shape from too much π! 
The Mystery of π Solved

