Phi not Pi. Big difference - it explains why the windows on a CRJ-200 are so small:
Two quantities
a and
b are said to be in the
golden ratio φ if:
This equation unambiguously defines
φ.
The right equation shows that
a = bφ, which can be substituted in the left part, giving
Dividing out
b yields
Multiplying both sides by
φ and rearranging terms leads to:
The only positive solution to this
quadratic equation is
[edit] Relationship to Fibonacci sequence
For more details on this topic, see Fibonacci number.
http://en.wikipedia.org/wiki/File:FakeRealLogSpiral.pnghttp://en.wikipedia.org/wiki/File:FakeRealLogSpiral.png
Approximate and true
golden spirals. The
green spiral is made from quarter-circles tangent to the interior of each square, while the
red spiral is a Golden Spiral, a special type of
logarithmic spiral. Overlapping portions appear
yellow. The length of the side of a larger square to the next smaller square is in the golden ratio.
http://en.wikipedia.org/wiki/File:Fibonacci_spiral_34.svghttp://en.wikipedia.org/wiki/File:Fibonacci_spiral_34.svg
A
Fibonacci spiral that approximates the golden spiral, using Fibonacci sequence square sizes up to 34.
The mathematics of the golden ratio and of the
Fibonacci sequence are intimately interconnected. The Fibonacci sequence is:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, … The
closed-form expression (known as
Binet's formula, even though it was already known by
Abraham de Moivre) for the Fibonacci sequence involves the golden ratio:
The golden ratio is the
limit of the ratios of successive terms of the Fibonacci sequence (or any Fibonacci-like sequence):
Therefore, if a Fibonacci number is divided by its immediate predecessor in the sequence, the quotient approximates φ; e.g., 987/610 ≈ 1.6180327868852. These approximations are alternately lower and higher than φ, and converge on φ as the Fibonacci numbers increase, and:
Furthermore, the successive powers of φ obey the Fibonacci
recurrence:
This identity allows any polynomial in φ to be reduced to a linear expression. For example: