Fibonacci

Some numerical methods for calculating the zeros of a polynomial. (part 3)

The approximants of the golden ratio

are : F₂/F₁, F₃/F₂, F₄/F₃, F₅/F₄, ... .
The approximants of the golden ratio converge slowest of all continued fractions.
{{This property is often the cause of the appearance of Fibonaccinumbers in nature (sunflowers, leaves of plants, etc.).}}
Now we draw a picture. We represent the appoximant F_n+1/F_n by the point (F_n,F_n+1).
In the graphics we also draw a line through (0,0) and the point (1,x). (the golden ratio x = (1 + 5)/2 ).

We now regard the graphics as a view from above. The gridpoints are bars and the drawn line through (0,0) are in reality 2 coinciding ribbons.
We now pull one of the ribbons to the left so that the ribbon touches bar (1,2). The other ribbon is pulled to the right against bar (1,1).
The two ribbons will only touch the red colored bars.
We may interpret this as follows: Of all fractions, the approximants of an irrational number will approximate that number best.
Notice that the approximants are alternately left and right from the straight line.
This property of approximants is used in the following famous theorem:
Let x be an irrational number. Then there are infinite many fractions a/b so that |x - a/b| <1/(5 b²).
This theorem can often be used to show whether a number is a fraction or not.
It is possible to proof that, from every three consecutive approximants there is at least one which fulfils that inequalty.
In the theorem the number 5 is mentioned. That number cannot be replaced by a bigger number. In that case, the theorem is incorrect for x = (1 + 5)/2 (the golden ratio).
When we remove the golden ratio and its friends from the set of numbers, then:
There are infinately many fraction a/b so that |x - a/b| <1/(8 b²).
Now again it is not allowed to replace 8 by a larger number. The wrong-doers are 2 and its friends.
We can go on. The roots which appear in the denominator have the following form (9 - 4/n²). Now n cannot be any positive integer; n has to be a number from the following (infinite) figure.

(Note: the numbers (9 - 4/n²) are all smaller than 3. After that there are infinite many numbers larger than 3).
The numbers around a three-forked point are the x,y,z solutions in positive integers of the equation x² + y² + z² = 3 xyz.
Notice that the numbers around the area with number 1 are the Fibonacci numbers with odd index.
Another property: The sum of the numbers at the end of a line segment equals 3 times the product of the numbers on both sides of that line segment. E.g. 2+13 = 3*1*5 and 5+37666 = 3*29*433.