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· Pi et le Nombre d'Or : apparitions des décimales non aléatoire
· Pi and Golden Section: the decimals occurrences are not random
· Suites universelles de Fibonacci : Phi n’est qu’une des infinies variantes de la constante P
· Universal Fibonacci sequences: Phi is one of the infinite variations of the constant P ©
· La Nouvelle Pétanque

 

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Pi and Golden Section: the decimals occurrences are not random

Pi and Golden Section: the decimals occurrences are not random

Publié le 10/11/2008 à 12:00 par jyboulay
Pi and Golden Section: the decimals occurrences are not random
Version française en deuxième partie de blog
Introduction.
In Pi and the Golden Section (Phi), two fundamental constants of mathematics, first occurrence order of the 10 figures of digits in the decimal system is not random but part of a logic arithmetic. This arithmetic logic is the same for Pi, for 1/Pi and for the Golden Section. The same phenomenon occurs arithmetic in many other numbers whose square roots of numbers 2, 3 and 5, the first three prime numbers.

Four occurrence areas.
In these constants, figures appear into 4 areas always identical occurrence of 1, 2, 3 and 4 figures. Total values of figures (intermingled in numbers) of each of these 4 areas are always a multiple of a divider 45. The number 45 is the sum of ten digits (intermingled in numbers) of the decimal system. These areas are: 1 figure area : rank 4 (of occurrence); 2 figures area : rank 2 - 3 ; 3 figures area: rank 1 - 5 - 6; 4 figures area: rank 7 - 8 - 9 - 10. The divider is under constant: 3, 5 or 9, the three possible divisors of 45.

Pi and Phi.
Order of digits in Pi, 1/Pi and Phi (a = constants b = ranking appear c = ranking first figures appear d = arithmetic configurations):

Thus, among rank 4 (area 1 of occurrence), is 9 for Pi and 0 for Phi: these two numbers are multiples of 9. Among ranks 2 and 3 (occurrence area 2) are 4 and 5 for Pi and 1 and 8 for Phi: the respective sum of these numbers are multiples of 9. This is true for areas 3 and 4 (respective occurrence ranks: 1 - 5 - 6 and 7 - 8 - 9 -10): sums of those occurrence areas are multiple of 9.

Square roots of numbers 2, 3 and 5.
The order of digits in the square roots of 2, 3 and 5:


As for Pi and Phi, in the square roots of numbers 2, 3 and 5, the values of the same groups described above (4 areas of emergence of figures) are always multiples of the same divisor: 3 for square root of 2, 5 for square root of 3 and 9 for square root of 5. These three different values are the three possible divisors of 45, the sum of the ten-digit decimal system.

Probability of 1/18 and of 1/350.
The probability of occurrence of such configurations which are multiples of numbers 3, 5 or 9 (the three 45’ dividers) is 1/18. Therefore only 5.55% of all possible combinations (of figures occurrences) have these properties. It is strange that this phenomenon occurs precisely for Pi, Phi (and reverse) and square root of the first three prime numbers. For Pi, 1/Pi and Phi (Golden Section), the probability of occurrence organized in four areas which are multiples of number 9 is to 1/350.

Square root of 4.5.
This phenomenon also occurs for the square root of the number 4.5, which is precisely the average of 10 digit decimal system.

The order of numbers in the square root of number 4.5:


Also, a singular phenomenon appears to this number: from first to tenth place, the figures are perfectly symmetrical to form groups of two numbers whose value is always equal to 9. The probability for this phenomenon is 1/945.

Singularity into 1/Pi and 1/Phi.
The order of numbers in 1/Pi and 1/Phi:


In constant 1/Pi and 1/Phi, the same figures appear in the 4 occurrence areas defined above. This singular phenomenon has a probability to occur only to 1/12600.

Constant incorporating Pi, Phi, e and i.
In this constant integrating Pi, Phi, e and i:

the first six and last four digits are identical to the constant 1/Pi and 1/Phi. Also, this formula incorporating Pi, Phi, e but also the imaginary number i, four basic mathematical constant, produces a number whose first appearance decimal numbers organized in the same four areas arithmetic multiple divider 45. This is the variant of a continued fraction of Rogers-Ramanujan:


Appearances decimals not random.
In an article most complete (http://pagesperso-orange.fr/jean-yves.boulay/pi/index.htm) the author describes more phenomena where many other numbers derived from Pi, Phi but also e (Euler's constant) have the same arithmetical properties described here. This article suggests that the order of decimal places of raised constants can be random because the first decimals are not. Also, it is suggested in this article to consider a new number family with these arithmetical properties.
En soumettant ce formulaire, j'autorise le site centerblog.net à diffuser mon commentaire sur ce blog sans limitation de durée.

:: Les commentaires des internautes ::

lumene le 02/10/2011
Phi² = 2.618... = Gematria value for "Chay YHWH" in Hebrew (YHWH the Alive)
ALLAH gematria value is 1 + 12 + 12 + 1 + 8, with 12+12=24 & 2+4=6
So, ALLAH is 1.618.

Is GOD based on The GO(L)D Number ?

lumene
In Frenc : http://www.basic-sombre.com


john tiffanyy le 19/06/2012
go to http://www.artmusicdance.com/vaspi/proof.htm


PANAGIOTIS STEFANIDES le 06/02/2013


http://www.stefanides.gr/pdf/2012_Oct/PHOTO_12.pdf
http://www.stefanides.gr /pdf/2012_Oct/PHOTO_09_PCST_GEOMETRY.pdf

http://www.stefanides.gr/pdf/BOOK%20 _GRSOGF.pdf


http://http://www.stefanides.gr.centerblog.net


Anonyme le 20/09/2019
http://jyboulaypublications.e-monsite.com/