Deriving the Golden Ratio
"A straight line is said to have been cut in
mean ratio when, as the whole line is to the greater segment,
so is the greater to the lesser."
Euclid, ca. 300
Two distances are said to be at the golden
ratio if the ratio of their sum to the greater distance is equal to
the ratio of the greater to the lesser. There are countless rich examples of
how this ratio describes the proportions of things in nature and how it was
used by many artists and architects to create aesthetically pleasing works.
We can derive the actual value by considering a line segment whose parts
obey the golden ratio. By starting with the “mean and extreme ratios,” we can
solve for a, using the quadratic formula, and then solve for the ratio
of a to b.
We can reject the negative value since we’re working with measurements and
the ratio between positive numbers has to be positive. But since 0.618 is the
reciprocal of 1.618, it still plays an interesting role.
A final interesting piece is that if you start with a golden rectangle, you
can divide it into a square and a smaller golden rectangle, and then again
into a smaller square and an even smaller golden rectangle, etc. And if you
keep going, what you find is that the proportions of each new golden rectangle
follow the Fibonacci sequence.
It turns out that the ratio of two consecutive numbers in the Fibonacci
sequence approaches the golden ratio as the number in the sequence approaches
The math here is very interesting, and there’s plenty more to explore!