Golden Ratio

date Jul 8, 2007
authors Mario Livio
reading time 2 mins
  • Book Title: The Golden Ratio - The story of Phi, the World’s Most Astonishing Number
  • Author: Mario Livio
  • Year written/published: 2002
  • Summary: This book talks about the various occurrence of the number phi and also its history of discovery
  • Contents Page:
  1. Prelude to a Number
  2. The Pitch and the Pentagram
  3. Under a Star-y-pointing Pyramid?
  4. The second Treasure
  5. Son of a good number
  6. The divine propertion
  7. painters and poets have equal license
  8. From tiles to the heavens
  9. Is God a mathematician?

Buffon’s needle…

… … Suppose you have a large sheet of paper on the floor, ruled with parallel straight lines spaced by a fixed distance. A needle of length equal precisely to the spacing between the lines is thrown completely at random onto the paper. What is the probability that the needle will land in such a way that it will intersect one of the lines? Surprisingly, the answer turns out to be the number 2/pi.

The number 666…

With a scientific pocket calculator calculate…

The number you will obtain is a good approximation of the negative phi. Incidentally, President Ronald Reagan and Nancy Reagan changed their address in California from 666 St. Cloud Road to 668 to avoid the number 666…

Some maths with Golden Ratio…

Golden Fibbonaci…

Examine the Fibonacci sequence; 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987…

1/1=1 2/1=2 3/2=1.5 5/3=1.66666666666… 8/5=1.6 13/8=1.625 21/13=1.61538446154… 34/21=1.619047619… 55/34=1.6176470588… 89/55=1.6181818182… 144/89=1.6179775281… 233/144=1.6180555556… 377/233=1.6180555556… 610/377=1.6180371353… 987/610=1.6180327869… 1597/987=1.6180344478…

As we go further and further down, the Fibonacci sequence, the ratio of 2 successive Fibonacci numbers oscillates about, but comes closer and closer to the Golden Ratio.

logarithmic Spiral with the Golden Rectangle and the Golden Triangle

a few intriguing examples of the great treasury of primes…

  • the number 1234567891 which cycles through all the digits is a prime
  • the 230th pargest prime, which has 6400 digits is composed of 6399 9s and only one 8
  • the number composed of 317 iterations of the digit 1 is a prime.