Some math is functional. Some math is fun. And some math is simply stunning. If that last description sounds improbable to you, then today just might change your mind. Because now that we’ve covered enough ground, we’re going to take a look at some of the surprising, elegant, and downright mysterious ways that the Fibonacci sequence shows up in the world around you.

Recap of the Fibonacci Sequence

In the last article, we talked about how a seemingly innocent question about the growth of rabbit populations led Fibonacci to the sequence of numbers that now bears his name—the Fibonacci sequence:

1, 1, 2, 3, 5, 8, 13, 21, …

Each successive number in this sequence is obtained by adding the two previous numbers together. And, save a few complicating details like the fact that rabbits eventually grow old and die, this sequence does an admirable job at modeling how populations grow. But the numbers in Fibonacci’s sequence have a life far beyond rabbits, and show up in the most unexpected places.

What is the Golden Ratio?

One such place is particularly fascinating: the golden ratio. So, what is this golden ratio? Well, it’s a number that’s equal to approximately 1.618. This number is now often known as “phi” and is expressed in writing using the symbol for the letter phi from the Greek alphabet. Phi isn’t equal to precisely 1.618 since, like its famous cousin pi, phi is an irrational number—which means that its decimal digits carry on forever without repeating a pattern. If you’re interested in seeing how the actual value of phi is obtained, check out this week’s Math Dude “Video Extra!” episode on YouTube. But how did this number come to be of such importance? Oddly, it started as a question of aesthetics.

The Golden Rectangle

What’s the most beautiful rectangle? More specifically: What’s the ratio of this “most beautiful” rectangle’s height to its width? This question seems strange, but it isn’t crazy. We won’t go into the details right now, but there is evidence that people tend to perceive one particular shape of rectangle as being most pleasing to the eye. Of course, the Greeks knew this long before modern psychologists tested it, which is why they used golden rectangles, as well as other golden shapes and proportions adhering to the golden ratio, in their architecture and art.

For example, almost 2500 years ago, a Greek sculptor and architect named Phidias is thought to have used the golden ratio to design the statues he sculpted for the Parthenon (note the word “phi” in Phidias’ name—that isn’t a coincidence and actually inspired the naming of the number in the 20th century). And since Phidias’ time, numerous painters and musicians have incorporated the golden ratio into their work too—Leonardo da Vinci, Salvador Dalí, and Claude Debussy, among many others.

But back to the problem of figuring out the shape of the most pleasing rectangle. If you simply draw what you believe to be the most beautiful rectangle, then measure the lengths of each side, and finally divide the longest length by the shortest, you’ll probably find that the ratio is somewhere around 1.6—which is the golden ratio, phi, rounded to the nearest tenth. It won’t be exactly 1.6, but it should be pretty close. Besides being “beautiful,” the resulting shape has an intriguing characteristic: If you draw a golden rectangle, and then draw a line inside it to divide that rectangle into a square and another smaller rectangle, that smaller rectangle will amazingly be another golden rectangle! You can do this again with this new golden rectangle, and you’ll once again get a square and yet another golden rectangle.

Connection Between the Golden Ratio and the Fibonacci Sequence

Okay, but what about the Fibonacci sequence? How does that figure into this? I know it might seem totally unrelated, but check this out. Let’s create a new sequence of numbers by dividing each number in the Fibonacci sequence by the previous number in the sequence. Remember, the sequence is

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, …

So, dividing each number by the previous number gives: 1 / 1 = 1, 2 / 1 = 2, 3 / 2 = 1.5, and so on up to 144 / 89 = 1.6179…. The resulting sequence is:

1, 2, 1.5, 1.666…, 1.6, 1.625, 1.615…, 1.619…, 1.6176…, 1.6181…, 1.6179…

But do you notice anything about those numbers? Perhaps the fact that they keep oscillating around and getting tantalizingly closer and closer to 1.618?—the value of phi: the golden ratio! Indeed, completely unbeknownst to Fibonacci, his solution to the rabbit population growth problem has a deep underlying connection to the golden ratio that artists and architects have used for thousands of years!

Applications of the Golden Ratio

But the golden ratio isn’t just for mathematicians, Greek sculptors, and Renaissance painters—you can use it in your life too. In fact, in the next article we’ll talk about how you can use the golden ratio to help you take better pictures. And there’s even more. Not only do these pleasing shapes show up in human art, they also show up in the “art” of the natural world—in everything from shells to sunflowers! We’ll talk about all that next time too.

Wrap Up

That’s all the math we have time for today. Thanks again to our sponsor this week, >Go To Meeting. Visit click this link and sign up for a free 45 day trial of their online conferencing service.

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Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!