The Golden Fibbo,

Nuck Nuck

by ARV, Uncle in Cairo (inside joke)

Proportions

The classical Greeks thought of a golden proportion in terms of a point on a line segment.

The point P defined for them a Golden cut (ratio) or Golden Section in the following sense:

suppose a and b are real numbers such that a b = a + b a

A simple proof of this fact can be found by clicking root5proof.htm.

The Golden Ratio has some interesting expansions in terms of continued fractions and continued roots which highlight the self-similarity aspect.

The self similarity aspects can best be seen geometrically by plotting an logarithmic spiral within a configuration of Golden rectangles [1,2,3].

Sequences

These continued processes suggest a relationship with sequences which will now be investigated.

This sequence is the prototype for all sequences of the following type:

Definition:

It turns out that the sequence S above is THE MOTHER OF ALL FIBOS in the sense that any fibo is a term by term linear combination of a shifted and doubly shifted Fibonacci sequence. To see the precise statement and proof of this claim, click hereThm2.htm.

It is of course no surprise that the first two terms of any fibo serve to determine the entire sequence. It is also immediately clear that:

  1. If the first two terms of any fibo are integers, then all the terms are integers
  2. any fibo can have at most one zero term.

Finally, the relationship of every fibo to the Golden section may be a bit more of a surprise. One might say that the ratio of successive terms in a fibo always turns out golden. The Theorem can be found hereLimThm3.htm.

Application

Here is a simple application of the Theorem to the sequence

Hence it is a fibo. Next we compute

follows from the previous theorem.

We can "normalize" (or rather "Fibo-nuck-nuck-chize") this sequence [4, p. 452] by observing that any linear transformation of a fibo is still a fibo, and putting

Matrix Expression

By adding a null term at the beginning of the sequence, there is a suggestive matrix expression for the Fibonacci sequence:

It seems pleasing that there should be such a natural relationship between Fibonacci sequences and Golden Ratios.

References

[1] H.S.M. Coxeter, Introduction to Geometry, Wiley, 1969

[2] Matila Ghyka, The Geometry of Art and Life, Sheed & Ward, 1946, republished by Dover, 1977

[3] H. E. Huntley, The Devine Proportion: a Study in Mathematical Beauty, Dover, 1970

[4] Konrad Knopp, Theory and Applications of Infinite Series, Second English edition, Blaclie, 1964