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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 .
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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)).
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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) ] |
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- [ (2/9) / (F7+F8Phi) + (1/11) / (F9+F10Phi) - (1/13) / (F11+F12Phi) ] |
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+ [ (2/15) / (F13+F14Phi) + (1/17) / (F15+F16Phi) - (1/19) / (F17+F18Phi) ] |
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- … } |
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= 5.083203692....
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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.
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