**Pi Day**. So, in the end of the day, I try to write a little post about this day. $\pi$ is one of the irrational numbers: we cannot write an irrational numbers like a ratio between twointeger numbers, so Pitagora's team called them

*irrational*and hidden their existence. We can define $\pi$ like the ratio between circle and his diameter, and it's impossible that they are contemporary integer. This assertion was proofed in 1761 by

**Johann Heinrich Lambert**, while in 1882

**Ferdinand von Lindemann**proofed that $\pi$ is a trascendental number, where a trascendental number is a number that we cannot calculate with usual algebric operations.

In ancient times, the first mathematician who evaluate $\pi$ was

**Archimedes**, who used the method of exhaustion, that is a method to evaluate the area of a shape by inscribing inside it a series of polygons that converge to area's shape. In this way Archimedes find $\pi = \frac{211875}{67441} = 3.14163 \cdots$

Today we know 5 triolions of digits, thanks to the work of

**Shigeru Kondo**, who modify a usual home pc in order to perform the calculations using the softwerdeveloped by

**Alexander Yee**. They design an internet site to describe methods, software and results.

Following

**David Bailey**and

**Jonathan Borwein**

^{(1)}, I find some others interesting site about the calculations of $\pi$'s digits (Inverse Symbolic Calculator, Electronic Geometry Models). Other links at Experimental mathematics. But... what is

*experimental mathematics*?

(...) namely the utilization of modern computer technology as an active tool in mathematical research.For example using the PSLQ algorithm^{(1)}

^{(2)}in 1995 was found the BBP formula for $\pi$: \[\pi = \sum_{k=0}^\infty \frac{1}{16^k} \left ( \frac{4}{8k+1} - \frac{2}{8k+4} - \frac{1}{8k+5} - \frac{1}{8k+6} \right )\]

This formula permits one to directly calculate binary or hexadecimal digits beginning at the $n$-th digit, without needing to calculate any of the first $n − 1$ digits, using a simple scheme that requires very little memory and no multiple-precision arithmetic software.You can find others BBP formulas in the^{(1)}

**David Bailey**'s compendium (pdf). (1) Jonathan M. Borwein, David H. Bailey.

*Experimental Mathematics: Examples, Methods and Implications*. Notices of the American Mathematical Society, vol.52, 5, (2005)

(2) PSLQ algorithm,

*which, compared with other integer relation algorithms in the literature, features superior performance and excellent numerical stability*,

*will produce lower bounds on the (Frobenius) norm of any possible relation for $x$*, where $x$ is an arbitrary vector defined on some number field.

Ferguson, H., Bailey, D., & Arno, S. (1999). Analysis of PSLQ, an integer relation finding algorithm Mathematics of Computation, 68 (225), 351-370 DOI: 10.1090/S0025-5718-99-00995-3

*Image source: Brainfreeze Puzzles*

What about that strange Sudoku pattern?

ReplyDeleteThe rules are in the original link:

ReplyDeleteFill in the grid so that each row, column, and jigsaw region contains 1-9 exactly once and π three times.http://brainfreezepuzzles.com/main/piday2009.html

Ah, thanks, Gianluigi!

Delete