Convergent Ratios

-- the golden mean in context

John Hendry

Synopsis

The golden mean is the prime example of a class of mathematical objects we propose to call convergent ratios. They are the limits of certain numerical sequences as the number of terms increases indefinitely. For the golden mean, this sequence is known as the Fibonacci sequence.

The Fibonacci sequence can be generated by a triangular matrix acting on a 2-D base vector. Analogous sequences exist for base vectors with more than two dimensions.

Convergent Ratios have common features with geometric figures, such as the regular pentagon and 5-pointed star in the case of 2-D vectors and the regular heptagon and 7-pointed star in the 3-D case.

A triangular (cumulator) matrix together with its inverse, zero and identity matrices generates a field of matrices which maintain the convergent ratios for the 2-D, 3-D, 5-D, 6-D, 8-D and 9-D cases. In these cases we can obtain equations to relate the dimensions of the ratio vector.

Not all n-D cases behave similarly. Only if 2n + 1 is prime is there such a field.

The Golden Mean

If the sides of a rectangle are in this proportion then adding or subtracting a square leaves a rectangle in the same proportion.

This diagram might show subtracting the square from the big rectangle to leave the smaller one: or it might show adding the square to the smaller rectangle to get the bigger one. The sides of both rectangles are in the golden ratio.

The golden mean is also closely related to the pentagon. If we form a 5-pointed star by joining alternate vertices of a regular pentagon then the side of the pentagon to the side of the star is in the golden mean.

Imagine unfolding the star across BC so that E is reflected to the other side. Then triangles BCE and ADE would be similar and show the sides to be in the golden ratio.

Another interesting geometrical feature of the golden mean is demonstrated by the spiral in figure 3.

The Fibonacci Sequence

In the Fibonacci sequence successive terms are the sum of the two preceding terms: 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233 etc. As the number of terms increases, the ratio of the last two terms approaches the golden ratio (or mean). Since we are concerned with the ratio of two numbers, we might express it as a 2-dimensional vector

The inverse of the golden mean is formed by adding 1 for s/l or subtracting 1 for l/s:

r**-1 = r + 1

1 = r + r**2 (multiplying by r)

(r + 1/2)**2 = 5/4 (adding 1/4 to both sides)

r + 1/2 = +- 5**1/2 / 2

r = 5**1/2 - 1 / 2, (-5**1/2 - 1 / 2)

So the golden mean is an irrational number, thus bewildering the ancient Greeks, who much admired it.

The Silver and Bronze Means

The 4-D Case

The 5-D Case

The 6-D Case

The 8-D Case

The 9-D Case

Review