(√5 + 1)/2

This number is called the “Golden Ratio”, or the “Divine Proportion”. The Penguin Dictionary of Curious and Interesting Numbers has 4 ½ pages dedicated to this number. It’s really popular among mathematicians, but artists also find it interesting and very useful.

How do we come to know this number? If the proportion between two lines is such that cutting the longer one to make it the same length as the shorter one and the bit you have left over has the same proportion to the new longer bit than the old shorter bit to the old longer bit, that proportion is the number (√5 + 1)/2.

In other words, if you take a line and cut it in just the right place, you will get two shorter lines.

Now cut the longer line so that you get one line the same as the shorter line from your first cut, and another, even shorter line. THESE two lines have exactly the same proportions as the lines you got from your first cut, if you cut in just the right place. That proportion is “The Divine Proportion”.

If you make a rectangle out of these lines, you can cut the rectangle into a square and a rectangle with the same proportions as the original. If you do it again with the inside rectangle, and then again and again… and then draw a spiral through the squares, you get a curve that shows up a lot in nature. Sunflower seeds align themselves along such a curve; leaves on the branches of some trees have a similar pattern; an unfolding fern frond looks like this as well. Perhaps the most well-known example of this curve is the nautilus shell.

Here are some links to other people’s musings on the Golden Ratio: