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Valentin Albillo · 03-14-2021, 07:00 PM

Welcome to my SRC #009 - \(\pi\) Day 2021 Special, a small affair to commemorate that most universal and ubiquitous constant \(\pi\), The Mother of All Constants.

Why, \(\pi\) is such an über-ubiquitous constant that it appears everywhere in everything from pure math to applied sciences to stochastic processes and beyond, embedded in the very fabric of the Universe. You'll find \(\pi\) as the result of an uncountable infinity of non-trivial mathematical expressions, including finite and infinite series and integrals and as a root of equations (not polynomial, mind you, \(\pi\) 's a transcendent constant), you'll find it in the innards of fractal sets such as the Mandelbrot set, you'll find it by throwing needles on a grid, you'll find it by rounding numbers, you'll find it when dealing with quantum mechanics or financial instruments, ...

Thus, though it would seem difficult to find new, interesting appearances of \(\pi\), actually that's not the case at all and I can tap that uncountable infinite set of appearances I mentioned to get new ones for this SRC, so get your HP calculator (physical or virtual as you see fit) and use its built-in manual and/or programming capabilities to deal with what follows, you don't need anything else ... Smile

Note: this SRC isn't intended as a challenge or anything of the sort. If you're like me, you'll probably welcome the chance to check for yourself a surprising evaluation or to calculate an intriguing root or getting to know various math-related trivia. This is what the following is all about.
Also, please do NOT include CODE panels in your replies to this thread, as it makes it difficult for me to generate the online PDF document from it. I expect you'll kindly comply with this requirement but otherwise you'll risk your carefully crafted code appearing truncated or not at all in the final PDF and thus being irretrievably lost from the online document and making your posting it moot.

New assorted appearances of \(\pi\) and other trivia:

a. Find a real root x in [3, 4] of the following equation:

where for aesthetic considerations x! is considered to also apply to real x, not just integers, and the units are in radians.
b. Find a real root x in [3, 4] of the following equation:

where Ω = LambertW(1), the real root of y e^y = 1, log is the natural logarithm and the units are in radians.
c. Another most famous transcendental constant also appearing everywhere is e = 2.718+. We know that \(\pi\) and e are related by e ^{i \(\pi\)} + 1 = 0, but we may ask ourselves: is there any other simpler way to get \(\pi\) from e which does not involve complex numbers ? Yes, there is, simply evaluate:

4 * ( Arctan e - Arctan \(\frac{e - 1}{e + 1}\) )
d. Conversely, the volume enclosed by the n-dimensional sphere of radius R is given by:

Go on and evaluate the \(\pi\)-th root of the summation for even dimensions from 0 to \(\infty\) of the volumes enclosed by the respective n-dimensional unit spheres (R = 1).
e. \(\pi\) also features in a song by Kate Bush (included in her 2005's album "Aerial") about a man who's utterly obsessed with the calculation of \(\pi\) (that could describe some of us here at the MoHPC). She sings more than a hundred digits of \(\pi\) and comments the following about the experience:
- "I really like the challenge of singing numbers, as opposed to words because numbers are so unemotional as a lyric to sing and it was really fascinating singing that. Trying to sort of, put an emotional element into singing about ... a 7 ..., you know, and you really care about that 9.
  
  I find numbers fascinating, the idea that nearly everything can be broken down into numbers, it is a fascinating thing; and I think also that we are completely surrounded by numbers now, in a way that we weren't, you know, even 20, 30 years ago, we're all walking around with mobile phones and numbers on our foreheads almost; and it's like, you know, computers...
  
  I suppose, um, I find it fascinating that there are people who actually spend their lives trying to formulate \(\pi\); so the idea of this number, that, in a way is possibly something that will go on to infinity and yet people are trying to pin it down and put their mark on and make it theirs in a way I guess also I think, you know, you get a bit a lot of connection with mathematism and music because of patterns and shapes..." (Ken Bruce show, BBC Radio 2, 31 Oct 2005)
f.  Finally, a little serving of trivia. About two years ago a researcher at Google set out a new world record by computing some 31 trillion digits of \(\pi\), namely 31415926535897 digits, to be exact, and was surprised to discover that the very first digits in the output were 31415926535897... What a coincidence !!

Also, here you are, a bilingual joke I concocted for this SRC that probably only those of you who understand both English and Spanish will get:

       Q:  Fear of number 13 is called "Triskaidekaphobia". How would you call  Fear of number \(\pi\) ?

       A:  "Trescatorcephobia"   (select to see)

Sorry for that. Last, for a really good laugh have a look at just a sample of modern papers on \(\pi\) published in what they say are reputable, peer-reviewed journals:
- Paper A, B and C in the IOSR Journal of Mathematics
- Paper D in the International Journal of Engineering Inventions

All your comments are welcome and appreciated. My own results and comments in a few days.

And remember: please do NOT include CODE panels in your replies to this thread, as it makes it difficult for me to generate the online PDF document from it. I expect you'll kindly comply with this requirement but otherwise you'll risk your carefully crafted code appearing truncated or not at all in the final PDF and thus being irretrievably lost from the online document and making your posting it moot.

Researching, testing and formatting these SRC takes considerable time and effort. Hence, if you do enjoy them and would like to see more posted in the future, consider participating or at least commenting on them so that I get feedback of your appreciation. Saying "Hey, I never post a thing but I do read and enjoy them very much !" doesn't quite cut it with me, as then I have no way to tell apart sheer laziness from blatant disinterest. Your move.

V.