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,

While planeary conjunctions as observed from the Earth involve a many body problem - a complicated calculus - for which there is no exact solution defined in continuous spatial and time co-ordinates. Yet using discrete mathematics we can define these conjunctions not in external parameters but in terms of the history of the phenomena itself, using linear second-order recurrence relations. A case where a future state is simply related to the present and past state or occurences, as  ancient civilisations observed with planetary conjunctions.

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) 
If the initial conditions are knwn ie. S(0) and S(1) are given then S(2), S(3),… are uniquely determined. In the case of  transits of  venus a, b = 1, the recurrence, satisfies a simple specific form of this equation. S(n) = S(n-1) + S(n-2)  Substituting with , the equation can be written,


If , then

setting and as initial conditions

Solving the quadratic equation, we can obtain the numerical solution obtained by Pythagoras with his calculations for the legs of a golden triangle
where the plus sign root has been taken to give a solution  > 1.

The powers of the golden proportion of recursion also satisfy

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

It was noted by Kepler (1571 - 1630) where is the nth 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,


Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, pp. 36-49, 1986.