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Boris Gourévitch
The world of Pi - V2.57
modif. 13/04/2013



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The Pi-phenomenons
Plenty of mathematical and historical frenzies !


Once human being was aware of the importance of the number Pi in mathematics' world, i.e. quite early, our favorite constant has become the subject of numerous myths and stories. From the serious anecdote to the most crazy claims, the following stories describe the exceptional "personality" of Pi and the endless interest of mathematicians in this constant. Enjoy !


Some jokes to start (sorry...)

What do you get when you divide the circumference of a pumpkin by the diameter?... Pumpkin Pi


Mathematician : Pi is the ratio of the circumference of a circle to its diameter.
Engineer : Pi is about 22/7.
Physicist : Pi is 3.14159 plus or minus 0.000005
Computer Programmer : Pi is 3.141592653589 in double precision.
Nutritionist : You one track math-minded fellows, Pie is a healthy and delicious dessert!


Top ln(e^10) reasons why e is better than pi

10) e is easier to spell than pi.
9) pi ~= 3.14 while e ~=2.718281828459045.
8) The character for e can be found on a keyboard, but pi sure can't.
7) Everybody fights for their piece of the pie.
6) ln(pi^1) is a really nasty number, but ln(e^1) = 1.
5) e is used in calculus while pi is used in baby geometry.
4) 'e' is the most commonly picked vowel in Wheel of Fortune.
3) e stands for Euler's Number, pi doesn't stand for squat.
2) You don't need to know Greek to be able to use e.
1) You can't confuse e with a food product.

Top ten reasons why e is inferior to pi

10) e is less challenging to spell than pi.
9) e ~=2.718281828459045, which can be easily memorized to its billionth place, whereas pi needs "skills" to be memorized.
8) The character for e is so cheap that it can be found on a keyboard. But is special (it's under "special symbols" in word processor programs.)
7) Pi is the bigger piece of pie.
6) e has an easy limit definition and infinite series. The limit definition of pi and the infinite series are much harder.
5) e you understand what it is even though you start learning it late when you're in pre-calculus. But pi, even after five or six years it's still hard to know what it really is.
4) People mistakenly confuse Euler's Number (e) with Euler's Constant (gamma). There is no confusion with the one and only .
3) e is named after a person, but pi stands for itself.
2) Pi is much shorter and easier to say than "Euler's Number".
1) To read pi, you don't have to know that Euler's name is really pronounced Oiler.

 

The Pi fans

** Since the late 80s, David Bailey has owned two plates celebrating Pi ! One the first one is written P314159 and on his second one is Pi in base 16. What an obtinacy ! Click on the pictires to see the whole photos.


** On 3rd october 2006, 4.30pm, Akira Haraguchi, a 60 years old japanese retired engineer, recited 100 000 digits of Pi in 16h30, beating his own previous record of 83 431 digits established in 2005. First comment of the guy: "I have felt nothing exceptional. I just emptied my memory" ! Akira Haraguchi uses a system he developed, which assigns Kana symbols to numbers, allowing for the memorization of Pi as a collection of stories. This form of mnemonics had already been known and used by ancient Greeks and Romans.


** On the web, there is a small community of people completely crazy about Pi (me for instance...). In addition to the classical clubs of people who know 100 or 1000 digits of Pi (see links), I really like the club of the friends of Pi (http://pi314.at/). As they say, "The main objective of our club is to uphold, promote and celebrate the spirit of Pi!". For example, they don't forget the international Pi day (14/3), and they do some other frenzied things. Among all, Lukas Mueller recited the digits of Pi in a Vienna's square in 1996. You should see the face of people around ! I really should become a member...

Close Encounters of the Third Pi

A mysterious "crop circle" (unexplained sizable pattern created by the flattening of a crop) has been decoded and... can you believe it ? the crop circle does represent Pi's digits !!! Two possible explanations: either some jokers are fund of geometry and mathematics or Pi is a "universal" constant celebrated (in base 10 !) by Aliens !! If we divide the circle in ten equal parts, each arc of the circle equals a Pi's digit, from the smallest circle to the biggest one.

The french paper in Direct-Matin of 6/10/2008 here, the initial paper in english there, and for people who want to know more, THE reference website about "crop circles" (in english), another in french, and some more "X-files"-like interpretations.

Pi in history

** In the paragraph 1.Rois 7.23 of the Bible, we can read the following statement:

"And he [Hiram on behalf of King Solomon] made a molten sea, ten cubits from the one brim to the other: it was round all about, and his height was five cubits: and a line of thirty cubits did compass it round about." A similar account is given in the parallel passage in 2 Chronicles 4:2.

From the definition of Pi , we obtain Pi==3. To explain this amazing result, several hypothesis were proposed by some people anxious about the putative religious controversy. Indeed, if God does not know maths, that's a shame !
For instance, some say that Bible's authors were not interested in giving all the details of the construction, and they used an integer approximation. Some others claimed that the perimeter of the external border is not the same than that of the internal border. See this link for instance, where the truth is of course. However, M.D. Stern proposed something completely different, based on an ancient text and some special letters in the original text in Hebrew. It's really far-fetched, but you know when it comes to religion....

Before adressing the problem, we have to note two points:

1- In the old Hebraic language, some words can be spelled and read differently. In this verse, for instance, there is such a word: the word line is spelled qwh but you read it qw.

2- Since Greeks and Jews of the ancient world used letters to represent numbers, each word had a numerical value as the sum of its letters.

If we suppose everything makes sense in the Bible, the difference between qw and qwh is obvious. But why is there this additional h ? A few computations will turn the light on...

In Hebrew, letters q, w and h are 100, 6 and 5 respectively. Then the word line has a value of 111 when you write it (qwh) but is only 106 when you read it (qw). If we use the quuotient of these two numbers as a way to correct the given value for Pi , we then obtain : = 3,141509... , which is true for the first 4 digits !
But why use the quotient after all, this remains unclear... I would really like to say to these people: please, don't care about these details. Don't be ridiculous. Even a value with 4 digits is really wrong. Exact values of irrational numbers were not the priority of people at this time and that's it. It does not weaken the Bible. Please don't miss the real message in the Bible.
And that was not a mathematical message...


** In 1836, LaComm claimed he had just discovered that tre true value of Pi is 3+1/8. Although the value of pi was known to more than 100 digits at this time, LaComm was awarded several medals of numerousfrench scientific societies for his discovery !


** Through history, the value of Pi has been a constant issue. With the proof of the transcendance of Pi by Lindemann in 1882, one may think that proposals for eccentric values would faint. Not at all ! For instance, a guy named Edward Johnston Goodwin (1828-1902) proposed in 1897 some formulas for curves lengths and inscribed surfaces in a project for a law in the State of Indiana (House Bill #246). From this work, we could get all this Pi=4, Pi=3.1604, Pi=3.2, and finally Pi=3.232 ! Of course, this ingenious author had also resolved the squaring of the circle, the trisection of an angle, and the duplication of the cube, which had been proved impossible a few years before.
Fortunately, the senators concluded that scientific truth shouldn't be decided in laws and rejected the text at the very last moment. And that's a good thing because from such statements, we could get 1=0 which is very annoying in economy !
In "Le fascinant nombre Pi", J.P. Delahaye adds that Goodwin had even offered the free use of his discoveries, and had succeeded in publishing two eccentric papers in American Mathematical Monthly, new at that time. This famous journal checks the submissions more carefully nowadays...



** In the 60's in some arabic schools, the arabic letter "Ta" was still used for Pi !
The notation for Pi has appeared late actually. Oughtred (1574-1660) and Barrow (1630-1677) used to refer to the perimeter of a circle of radius R, being the first letter of the word "perimeter" in greek. However, William Jones has introduced the notation in a modern sense in 1706 in "A New Introduction to Mathematics". Euler used "c" in 1736 as well as Jean Bernoulli, then "p" in 1747 and finally in his "Introduction to Infinitesimal Analysis" in 1748, which helped to standardize the notation.

Why being simple ?

** Our german friends found a not so simple way - but very elegant - to remember the first digits of Pi. They remember the sentence : "Drei Komma Hus verbrannt und Brennabor bringen die Zahl Pi hervor", which means : "Three points Hus burns and Brandenburg, conquered, will give Pi". Indeed, we know that during the Council of Constance, in 1415, Johannes Hus was condemned to be burnt. moreover, Brandenburg was conquered in 927.

Musica, maestro !

    ** Jean-Philippe Fontanille, composer and guitar teacher in Paris, composed a piece called Harmonisation de pi. You can listen to the digits of Pi through this piece. At first, he indeed used the following principle: the scale has seven notes, he started to write Pi in base 7. He then could convert each digit in a note with the following code : 0 for C, 1 for D , 2 for E, 3 for F, 4 for G, 5 for A and 6 for B.

    To improve the result which was somewhat limited, he used each digit to code a whole chord whose the note associated to the digit will be its root. The following code was chosen : 0 for C major (C-E-G-C), 1 for D minor (D-F-A-D), 2 for E minor (E-G-B-E), 3 for F major (F-A-C-F), 4 for G major (G-B-D-G), 5 for A minor (A-C-E-A) and 6 for B minor diminished fifth (B-D-F-B). Then only remains the work about harmony. It is quite beautiful, amazingly ! (see music page)

Pi me a river

** Skolum (1996) checked that the ratio of the real length of a river and the length as crow flies (distance between the source and the mouth of the river) is roughly equal to Pi in average. This value would be more often found in Brazil or in the siberian toundra, but it still has to be checked... Personnaly, in France, I rather found that the ratio is closer to 2 (but is it a coïncidence by the way) ?

faster, higher, stronger !

** In a conference, Simon Plouffe reminds us of the following claims :

-- Daniel Shanks in 1962 : (the guy who computed Pi to 100 265 digits himself)

"We will NEVER reach the billion'th digit of Pi"

-- Borwein et Borwein (1988) :
"We will NEVER reach the 101000 digit of Pi"


-- Spock, Star Trek (1967) :

"Computer !, compute Pi to the LAST digit."
And the computer crashed !


Among the digits

** From James Davis. In the alphabet written in circle, the letters with a vertical symmetry axis are colored in red. Then, the black letters from the J are in blocks of 3,1,4,1,6 letters !




** Monte Zerger noticed that, at the positions 7, 22, 113, 335 in the digits of Pi, we find the same number 2. Of course, these positions refer to the famous fractions 22/7 and 335/113 which approximate Pi.


    ** A very staggering fact was discovered by T.E. Lobeck de Minneapolis. From the magic square 5x5 below (the sum of colums and rows is the same, here 65), we replace each number n of the square by the n-th figure in the digits of Pi. For instance, for the box containing a 25, we replace it with a 3 since it is the 25-th digit of Pi. Then, in the resulting square, the sum of one row is also the sum of one column !

  17 24 1 8 15  

2

4

3

6

9

(24)

  23 5 7 14 16  

6

5

2

7

3

(23)

  4 6 13 20 22  

1

9

9

4

2

(25)

  10 12 19 21 3  

3

8

8

6

4

(29)

  11 18 25 2 9  

5

3

3

1

5

(17)

             

(17)

(29)

(25)

(24)

(23)

 



** The "Palais de la Découverte" (palace of discovery) in Paris hosts a very special room. Indeed, the 600 first digits of Pi are displayed on the walls of the top of the room. Originally, the digits found by William Shanks were used. However, D.F. Ferguson discovered Shanks' digits were wrong after the 528th place in 1945. The correct digits were installed in the room a few months later.



** In 1997, some scientists claimed that we will technically never be able to compute and store more than 1077 digits of Pi since it is about the number of atoms that the universe contains !



Is the knowledge of so many digits of Pi really useful ? Well, yes ! The computation skills of some computers would have been tested through the calculation of the digits of Pi ! One rumor in computation history says that two serious errors were detected in the computers IBM 590 and R8000.


G. Stanley Smith found outthat = 1.778846... is an interesting approximation for the square root of Pi (1.77245...). Yet, if we take a look at the numbers used in the fraction, we can notice some figures already known in a famous approximation of Pi. Indeed, , that is written from right to left.


In the same vein than the auto-reference of Pi in the previous example, we know that a possible approximation of Pi is 3,141593.... Using the figures of this number, D. Castellanos found out that is a good approximation of !



With the help of the master thesis of Eric Doddridge
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