Phi is an irrational number that seems to be life's archetypal architecture. Reflecting nature's balance between symmetry and asymmetry and the ancient Pythagorean belief that reality is a numerical reality, except that numbers were not units as we define them today, but were expressions of ratios.
The Fibonacci series: 0 1 1 2 3 5 8 13 21 34 55 89 144 233...
Starting with 0 and 1, each new number is simply the sum of the two before it.
The ratio of each successive pair of numbers in the series approximates Phi 1.618*, as 5/3 = 1.666*, 8/5 = 1.6.
The successive numbers quickly converge on Phi. After the 40th number, the ratio is accurate to 15 decimal places: 1.618033988749895...
Greek philosophers considered the Golden Mean to be the middle between two extremes, one of excess and the other of deficiency.
According to legend, the Greek Philosopher Pythagoras discovered the concept of harmony when he began his studies of proportion while listening to the different sounds given off when the blacksmith’s hammers hit their anvils. The weights of the hammers and of the anvils all gave off different sounds. From here he moved to the study of stringed instruments and the different sounds they produced. He started with a single string and produced a monochord in the ratio of 1:1 called the Unison. By varying the string, he produced other chords: a ratio of 2:1 produced notes an octave apart. (Modern music theory calls a 5:4 ratio a "major third" and an 8:5 ratio a "major sixth".) In further studies of nature, he observed certain patterns and numbers reoccurring. Pythagoras believed that beauty was associated with the ratio of small integers. With this discovery, the Pythagoreans saw the essence of the cosmos as numbers and numbers took on special meaning and significance. The symbol of the Pythagorean brotherhood was the pentagram, in itself embodying several Golden Means.
"Nature is not mute! Man has become deaf!"
The Golden Ratio in the World
(More divine mathematics in nature!)
Worlds Within Worlds
Fractals: Colors of Infinity docu with Arthur C. Clarke
Fractals Google Video
Amazing Fractal animation