Fibonacci Type Relationships as Fractals

The mystery of the Fibonacci ratio (phi = 1.618...) is often expressed geometrically in terms of a Nautilus Shell which is a good example of a self-repeating spiral pattern (found frequently in nature).

Therefore from this perspective the very structure of the number phi is inherently dynamic and of a fractal nature (that endlessly repeats a simple pattern).

Now the Fibonacci ratio itself can be obtained as the positive solution to the simple polynomial equation,

x^2 - x - 1 = 0.

We can use a fascinating way to approximate this solution - and indeed any polynomial equation with an algebraic solution - through an easy iterative procedure.

In general terms for the equation x^2 + bx + c = 0, we start with the two numbers 0, 1 and add 1 * (- b) + 0 * (- a). So for the Fibonacci equation, this gives (1 * 1) + (0 * 1) = 1.

So we now have in the sequence 0, 1, 1 .

Continuing on in the same fashion the next term = (1 * 1) + (1 * 1) = 2.

So we now have 0, 1, 1, 2,

In this manner the well known Fibonacci sequence can be derived

0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597,...

Now, phi can be approximated as the ratio of a term and its preceding term (which approximation continually improves with higher terms).

So if we take the ratio of the last two terms we get 1587/977 = 1.61803444782..

This is already an extremely good approximation to the true value of phi = 1.6180339887.. .

However phi can equally be approximated in terms of a simple formula relating to the terms in the sequence that involve powers of 2.

So phi = 1/t1 + 1/t2 - 1/t4 - 1/t8 - 1/t16 - ......

Thus calculating up to t16 we get

phi = 1 + 1 - 1/3 - 1/21 - 1/987 = 1.61803444782..

So the value of this series up to t16 gives the same result for phi as t17/t16!


Just as the Golden Ratio (phi) can be represented in dynamic terms as a number fractal, in principle every (real) algebraic irrational number can be expressed in like manner as a fractal. This follows from the fact such an algebraic irrational must correspond to some polynomial equation with (real) integer coefficients. And as all such equations give rise - like the Fibonacci - to unique number sequences with recursive features, we can use these numbers to approximate (to any required degree of accuracy) the irrational numbers involved.


For example for the equation x^2 - 2x - 1 = 0, we derive the following sequence

0, 1, 2, 5, 12, 29, 70, 169, 408, 985, ...

The solution to this equation 2.414213562373... is approximated again by ratio of successive terms.

Thus using the last two terms we obtain 985/408 = 2.41421568... (which already is a pretty good approximation).

In exactly the same manner as with the Fibonacci, we can use this sequence of numbers to drive a simple expression to derive the square root of 2.

So square root of 2 = 1/t1 + 1/t2 - 1/t4 - 1/t8 - ...

= 1 + 1/2 - 1/12 - 1/408 -....

= 1.41421568... (i.e. 2.41421568... - 1).