Fibonacci numbers and the Pascal Triangle

 

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Archimedes` constant PI and the Golden Section

        

PI is one of those numbers that cannot be evaluated exactly as a decimal - it is in that class of numbers called irrationals. The hunt for PI began in Egypt and in Babylon about two thousand years before Christ. The Egyptians obtained the value (4/3)4=3.160493827 and the Babylonians the value 3 1/8=3.125 for PI.About the same time, the Indians used the square root of 10 for PI.
In the 3rd century B.C. Archimedes considered inscribed and circumscribed polygons of 96 sides and deduced that 3 + 10/71 < PI < 3 + 1/7 .

Archimedes was born about 287 BC in Syracuse, Sicily. At the time Syracuse was an independent Greek city-state with a 500-year history. Probably studied in Alexandria, Egypt, under the followers of Euclid. Archimedes' father was Phidias, an astronomer . Archimedes' fields of science were : hydrostatics, static mechanics, pycnometry (the measurement of the volume or density of an object). He is called the "father of integral calculus." The order in which Archimedes wrote his works is not known for certain. The works of Archimedes which have survived are as follows: On plane equilibriums (two books), Quadrature of the parabola, On the sphere and cylinder (two books), On spirals, On conoids and spheroids, On floating bodies (two books), Measurement of a circle, and The Sandreckoner. Archimedes generally regarded as the greatest mathematician and scientist of antiquity and one of the three greatest mathematicians of all time (together with Isaac Newton (English 1643-1727) and Carl Friedrich Gauss (German 1777-1855)).

        

It is known that the area enclosed by a circle of radius 1 is:



while its circumference is:



Archimedes`constant PI has many infinite series, infinite product and continued fraction representations. As a continued fraction, PI can be written as :




This form of continued fraction was found in 1665. by William Brouncker.

Surprisingly there are several formulae that use Root-5 and Golden Section ( Phi, phi ) to compute PI.

The Golden Section occurs in Geometry and Trigonometry. So :



where continued fraction of Phi is:



Now we return to using Golden Section and Root-5 to compute PI:



The two most famous numbers in the history of mathematics, phi and pi , are exactly related to each by a several formulae, even though both are irrational numbers.For example:



Ed Oberg and Jay A. Johnson have developed a unique expression for the PI-Phi product as a function of the number 2 and an expression they call "The Biwabik Sum," a function of Phi, the set of all odd numbers and the set of all Fibonacci numbers, as follows:

p Phi = 22 {1 + [ (2/3) / (F1+F2Phi) + (1/5) / (F3+F4Phi) - (1/7) / (F5+F6Phi) ]
  -  [ (2/9) / (F7+F8Phi) + (1/11) / (F9+F10Phi) - (1/13) / (F11+F12Phi) ]
  + [ (2/15) / (F13+F14Phi) + (1/17) / (F15+F16Phi) - (1/19) / (F17+F18Phi) ]
  - }
  = 5.083203692....

Now, we return to using the golden secion number phi to compute Archimedes`constant PI. Well-known trigonometry`s two-angle tanges formula is:



Putting



where the next relation is valid,



we get



and we calculate the value of t:



In case x=1 we found the following formula styled the Oberg Formula:



The Oberg Formula is the connection between Archimedes`constant PI and the golden section.

        

        

     

     

        

  2001-2005 Radoslav Jovanovic                 updated:  January 2005.