Ee-ther/Ai-ther: Calling the Whole Thing Off at the Science Museum

The papers (the Boston Globe, Time, and others) were abuzz yesterday about a supposed error in a math exhibit at the Boston Museum of Science. Most of the interest in the story came from the fact that the issue — described as a minus sign instead of a plus sign in a formula for the golden ratio — was pointed out by a 15 year-old. Frustratingly, none of the articles I saw included any actual math, though if you’re familiar enough with the golden ratio, you might guess even from the very brief description above that that the fuss was probably about a difference of convention rather than any kind of serious mistake.

And, right on schedule, today the Globe reports that the exhibit is correct after all. So what’s going on?

Let’s start with what we mean by “golden ratio.” I’ve posted about it before, in the context of ratios of successive Fibonacci numbers, which have the golden ratio as their limit. Let’s start with a picture:In this picture, the small rectangle (with one side length having length a and the other length b) and the big rectangle (with one side having length a+b and the other having length a) are supposed to be similar, meaning that the ratios of their sides are the same. In other words, if you write the length of the longer side on top, a/b = (a+b)/a. You could also put the length of the shorter side on top and get an equivalent equation: b/a = a/(a+b). Either way, dividing through by top and bottom, we get:

a² = b(a+b),

or

a² − ab − b² = 0.

This equation has lots of pairs of solutions (a,b). You could find them using the quadratic formula, in one of two ways. If you treat a as the variable, you can solve for it in terms of b:

a = (b ± √b² + 4b²  ) / 2 = b·(1 ± √5 ) / 2.

But the equation is pretty symmetrical, and you can also solve for b in terms of a:

b = (−a ± √a² + 4a²  ) / 2 = a·(−1 ± √5 ) / 2.

We need to pare down our solutions just a bit. Knowing that a and b are both lengths of rectangle sides, we should make sure they are both positive. 1 − √5 and −1 − √5 are not positive, so we throw them out, leaving us with

a =  b·(1 + √5 ) / 2   and   b = a·(−1 + √5 ) / 2.

Once we know this, it’s easy to talk about ratios of sides. The ratio of the longer side to the shorter side is a/b. Taking the equation a =  b·(1 + √5 ) / 2 and dividing both sides by b, we see that a/b = (1 + √5 ) / 2 = 1.61803… And the ratio of the shorter side to the longer side is b/a, which by similar logic is just (−1 + √5 ) / 2 = a/b −1 = 0.61803… (We can also deduce b/a = a/b − 1 directly from the initial equation a/b = (a+b)/a, because  (a+b)/a  is just 1 + b/a.)

Pictorially, if the square in our initial picture is 1 × 1 (a = 1), then b = (−1 + √5 ) / 2 = 0.61803… (the short side of the small rectangle), and a + b = (1 + √5 ) / 2 = 1.61803… (the long side of the big rectangle).

So what is the golden ratio? Well, which ratio do you want — long side to short side or short to long? Do you say tom-ay-to or tom-ah-to? Which of the two we call golden is unimportant; what matters is that the picture, and all the math around the ratio, are the same either way. Which should we take as the golden ratio? Ee-ther! Or maybe ai-ther!

We think of math as being about deduction and absolute right answers, but it is also full of decisions and conventions. Sometimes the decisions make a difference: we decide to make .9999… equal to 1 (by deciding on certain rules for doing math with infinite sums), and this has consequences across the subject (decimal representations are no longer unique). But sometimes the decisions are only conventions, just a way of fixing language or notation and no more, and don’t matter very much.

We do, however, need to keep track of what conventions we’re using. The 15-year old in the news stories probably learned that the golden ratio is (1 + √5 ) / 2, which is the more common formulation. Then, at the Science Museum, he saw this (photo from the latest Globe article):

It looked wrong; he was sure it should say (√5 + 1) / 2, not (√5 − 1) / 2. But read the fine print: the short side divided by the long side. That ratio is indeed (√5 − 1) / 2, as the display claims. The Science Museum just happened to frame their display in terms of the opposite ratio from the one the student learned. There’s nothing wrong with that, but we need to be aware that which version of the ratio we use is mathematical convention for us to choose, not mathematical fact set in stone.