CatSynth

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We at CatSynth once again, celebrate Pi Day on its three-digit approximation, March 14 (3-14).

We start with some interesting facts about the digits of pi. We presented statistics about the distribution in our 2007 Pi Day post. From super-computing.org, we present some interesting patterns:

01234567890 first occurs at the 53,217,681,704-th digit of pi.
09876543210 first occurs at the 42,321,758,803-th digit of pi.
777777777777 first occurs at the 368,299,898,266-th digit of pi.
666666666666 first occurs at the 1,221,587,715,177-th digit of pi.
271828182845 first occurs at the 1,016,065,419,627-th of digit pi. (that’s e for those who haven’t memorized it)
314159265358 first occurs at the 1,142,905,318,634-th digit of pi.

Last year, we showed the relationship to the Gamma function, and of course to Euler’s identity, which links pi surprisingly closely to the imaginary constant i and the number e. But it is also surprisingly easy to generate pi from simple sequences of integers. Consider the Madhava-Leibniz formula for pi:

Thus one can generate pi from odd integers and simple arithmetic. Another formula only involving perfect squares of integers comes from the Basel problem (named for the town of Basel in Switzerland):

In recognition of Pi Day, the U.S. House of Representatives passed a resolution this week:

And thus the sad history of pi in politics as exemplified by the Indiana Pi Bill of 1897 is put to rest. Now onto erasing the sad history of science and politics in general of the past eight years…

We saw this picture on meeyauw, and thought it was a good way to open our own Pi Day offering. Pi Day, or π day is celebrated on March 14 (3/14) of every year.

π does turn up in some interesting places besides circles and standard trigonometry (and LOLcat photos). There is of course Euler's famous identity:

which unites π with four of the other most famous constants in mathematics: zero, 1, i (the imaginary root of -1) and e. But it also turns up in some more surprising places. Consider the well-known factorial function, where n! or “n factorial” is the product of all the integers between 1 and n. For example:

5! = 5×4×3×2×1 = 120.

Simple enough. But of course some troublemaker is eventually going to ask for the factorial of 1/2. Not so easy. Fortunately, there is a function, called the Gamma function, that provides a solution:

Not really as simple as the original integer-only factorial. Once calculus is involved, might as well forget about it. But if you go through the trouble of plugging in 1/2 to the formula, you get the following intriguing result:

or

So the factorial of one half is one half the square root of π. Who knew?