Although this column is officially about mathematics, I will be stretching this limit from time to time. One of the most beautiful things about math is that it is essentially a study of patterns - number patterns, shape patterns, patterns found in nature, patterns found in human thought. But pattern is found everywhere, in many areas that are not commonly thought to lie within this domain. It is seen in music, literature, science, art, philosophy - in most areas of human experience. To forbid the discussion of subjects within these areas would seem somewhat artificial, so they might sneak in from time to time.

One interesting topic that skirts these boundaries rather nicely is the Fibonacci sequence. This sequence is generated by adding the two previous elements of the sequence together to produce the next number. It can be expressed like this:

where F(n) is the nth Fibonacci number. It is typically started with seed values of 0 and 1, yielding this sequence:

It is also possible to create other sequences using the same pattern with other seed values; these sequences are interesting topics in their own right. An amazing property of this sequence is that the ratio of adjacent terms approaches a set value (1.61803398...) as the terms get larger - this number is precisely the ratio resulting from the division of a line into two parts, such that the ratio of the smaller part to the larger part is the same as the ratio of the larger part to the line as a whole.

This number, an irrational number, can be found by solving a simple quadratic equation, and is called the golden mean or the golden ratio. It has a couple of very unusual and quite wonderful properties - if you subtract 1 from it, it becomes its own reciprocal, and if you add 1 to it, it becomes its own square! If you think that you can do it, try to find another number that behaves like this.

The Fibonacci sequence seems to capture the pattern seen in many areas of life, from the growth of computer databases to the behavior of the stock market. The golden ratio that can be extracted from this sequence was thought to be the most aesthetically pleasing way to divide a line, and was used liberally in art and architecture - it can be found in the Parthenon and other Greek constructions. It also shows up in areas completely outside of human control, such as the patterning of the spiral in mollusk shells and the growth of plants in the wild. It, like pi, appears in the most unexpected of places, and derives much of its charm from this fact.

Just A Thought: Do you think it is possible that the sequence of pi contains the sequence of e, or vice versa? What about both?