golden proportion of recursion

The golden proportion of recursion, also known as the golden ratio or golden mean, is a number often encountered when taking the ratios of distances in simple geometric figures such as a pentagram, with the tips of the pentagram behaving as golden triangles. It is denoted

For our solar system such geocentric observed conjunction can be discretely designated mathematically by the general recurrence relation.

S(n) = aS(n-1) + bS(n-2) |
(1) |

(2) |

If ,
then

(3) |

setting and

(3) |

Solving the quadratic equation, we can obtain the numerical solution obtained by Pythagoras with his calculations for the legs of a golden triangle

(4) | |||

The powers of the golden proportion of recursion also satisfy

(5) |

where is a
Fibonacci number (Wells 1986, p. 39).

It was noted by Kepler (1571 - 1630) where
is the *n*th Fibonacci number.

The Fibonacci numbers were noted by Leonardo of Pisa in about 1200 in trying to predict the number of pairs of rabbits in successive generations. Fibonacci numbers are often found in nature. They describe the arrangement of leaves in plants (Phyllotaxis) and a sunflower head contains spirals of florets with, typically, S(9) = 34 anti-clockwise spirals and S(10) = 55 clockwise spirals. Some smaller sunflower heads have S(8) = 21 and S(9) = 34, (or 13 and 21) spirals and a giant sunflower was recorded with S(11) = 89 and S(12) = 144 spirals. In addition to their natural occurrences the Fibonacci numbers have many applications in computer science.

The sine of certain complex numbers involving gives particularly simplex answers,

(6) | |||

(7) |