Some time ago we discussed the Fibonacci sequence which is the following ordering of numbers:
0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, …
where the dots at the end mean the numbers go on forever. The numbers in it are called Fibonacci numbers. Here each number, beginning with the third, is the sum of the two previous ones. Let’s divide each number of this sequence by the one preceding it. It’s a total no-no to divide by 0, so we should start with the two 1’s. However, to save a little space let’s begin with the 13 and 8.
21/13 is about 1.61538
34/21 is about 1.61905
55/34 is about 1.61765
89/55 is about 1.61818
144/89 is about 1.61798
233/144 is about 1.61806
377/233 is about 1.61803
610/377 is about 1.61804
987/610 is about 1.61803.
It looks as if these ratios are getting closer and closer. That is in fact true, and the number they are approaching is 1.61803398875… where the ending dots say I’ve given only part of the number. This number is known as the golden ratio.
That’s quite a fancy name for such an ugly number. Is there really anything special about it? When we discussed the Fibonacci sequence, we saw the entries in it often played a role in nature such as the number of seeds in a spiral of a sunflower which is often a Fibonacci number. Hence the ratio of number of seeds in one spiral to the number in the next smaller spiral is close to the golden ratio.
Suppose I take a line segment and divide it into two parts of longer length a and shorter length b so the total length of the segment is a+b. Now, if I demand that a/b be equal to the golden ratio, then it is also true that (a+b)/a is equal to the golden ratio. What, not astounded? Sigh.
Well, what about this? The idea can be extended to rectangles. Suppose we have a rectangle with long side a and short side b such that a/b is the golden ratio. Such a rectangle is called a golden rectangle.
So, who cares? Oh, that question hurts. Mathematicians care because paragraphs like the preceding few thrill them. But I suppose there’s the slightest chance others might not be as enchanted. But maybe there’s reason to be if you like art.
A game of those who study art is to place the outlines of golden rectangles on portions of paintings and architectural structures such as Da Vinci’s Last Supper, the Parthenon, and a host of other masterpieces such that each rectangle contains a significant more or less self-contained portion of the work. The amazing thing is that you can actually do that. It seems to indicate that the artist employed knowledge of the golden rectangle in his planning.
There’s debate, though, on whether that is true or not, although Dali is said to have done exactly that. But it doesn’t matter whether the artist was aware of golden rectangles. There is no denying that the rectangles can frame important portions of masterpieces. The conclusion seems undeniable that planning art via golden rectangles, either purposely or intuitively, produces aesthetically pleasing results. So the artist’s genius recognized this even if he’d never heard of the mathematical concept.
Here’s an experiment for you. Take a few measurements of your body: height (indicated by H), distance from top of head to fingertips (F), distance from top of head to navel (N), distance from top of head to pectoral muscles (P), and distance from top of head to base of skull (S). It has been pointed out that the ratio of any successive two of these measurements is approximately the golden ratio.
When I read this I just had to try it, so I made admittedly imprecise measurements of me with the help of an accommodating wife. My values in roughly determined inches are H=70, F=42, N=29, P=17, and S=9. The ratios are
Not bad, considering the undoubted inaccuracy arising from approximating to full inches, or even reading the tape measure correctly.
How close are your numbers to the golden ratio?
There is a wealth of readable nonmathematical material about the golden ratio on the internet. A simple Google search will find it. Let me warn you, though, it might be interesting, even fun!
Note: I’m going to take a break for the holidays, returning January 15. I hope you’ll rejoin me then. Meanwhile, have a wonderful holiday season.