# 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 $aandbare\; real\; numbers\; such\; that\frac{a}{b}=\frac{a+b}{a}$

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:

- If the first two terms of any fibo are integers, then
all the terms are integers
- 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