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
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,
| S(n) = aS(n-1) + bS(n-2)
and as initial
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
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,