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 2-D case and the regular heptagon and 7-pointed star in the 3-D case.

A triangular 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.

Convergent Ratios