數(shù)學英語 14 How to Use the Golden Ratio

by Jason Marshall
Despite many people’s assumption to the contrary, math is undeniably artistic. It takes a tremendous amount of creative muscle and artistry to devise mathematical solutions. And, as made evident by their frequent battles between elegant symmetry and rampant chaos, the traditional fine arts are chock-full of math. Suffice it to say that math and art are intimately related. Today we’re going to take a look at one of these happy relationships and see how the golden ratio that we talked about last time can make you a better photographer.
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Recap of the Golden Ratio
In the last article, we discovered a surprising connection between the Fibonacci sequence developed to model the growth of rabbit populations and the golden ratio used in art and architecture. By simply dividing each element in the Fibonacci sequence by the previous one, we obtained a brand new sequence which, amazingly enough, continually converged toward the value of the golden ratio—also known as “phi.” That may not seem all that amazing, but the strange thing is that phi and the shapes derived from it—likethe golden rectangle we talked about last time—had been used artistically for thousands of years before Fibonacci ever contemplated rabbits. So the fact that this number even shows up in the rabbit problem means there must be something fundamentally important about it...and that possibility is pretty amazing.
The Fibonacci Spiral
The most beautiful rectangle—the so called “golden rectangle”—has some very special properties. As we discussed last time, when a single line is drawn inside a golden rectangle to partition off a square, the leftover interior part is yet another golden rectangle. And this partitioning off and creating smaller and smaller pairs of squares and golden rectangles can continue forever. If you create such a figure, and then draw an arc sweeping diagonally from one corner of each newly formed square to the other, you will create a beautiful Fibonacci spiral.
If you’re interested in seeing a video showing exactly how this is done, check out this week’s Math Dude “Video Extra!” episode on YouTube. Why do I call this spiral beautiful? Well, it certainly is subjective, but to me the symmetry is alluring. And it’s made even more so because this shape isn’t just some random mathematical construction—it frequently occurs in nature!
The Golden Ratio in Nature
Case 1: Compare the shape of the Fibonacci spiral to the shell of the chambered nautilus—which, if you’re curious, is an ocean-dwelling creature related to octopus. The curves of the shell and the Fibonacci spiral trace out remarkably similar patterns. Case 2: Look at the seeds in the center of a sunflower—they’re arranged in spiral patterns curling in both directions. Though that by itself isn’t so strange, the fact that the number of spirals always equals a Fibonacci number is a little curious. It doesn’t matter which way you count the spirals—right or left—either. Both directions will give Fibonacci numbers. And they’ll be sequential Fibonacci numbers too—perhaps 34 in one direction and 55 in the other. That’s kinda weird, right? Well, there are plenty more cases too. Fibonacci numbers and golden ratios show up in pinecones, pineapples, flower petals, leaves on trees, and perhaps even make an appearance in the proportioning of the human body.
We should be a little careful here, however. Some of these facts could certainly be purely coincidental—in particular those about human anatomy. But the Fibonacci numbers and Fibonacci spirals appear so frequently in nature that it seems these relationships must, at least in some cases, be a byproduct of some fundamental properties of biology—in other words, of life itself!
How to Use the Golden Ratio to Take Better Pictures
So, nature has bountifully embraced the golden ratio, artists have displayed its exquisite proportions, and now you might be wondering: Can I use it too? Absolutely. Here’s a quick and dirty tip for improving the composition of your photographs using the golden ratio! First, when taking a picture, imagine placing the Fibonacci spiral on top of the scene you’re shooting. Then, the idea is to position the most important element of your shot—perhaps a person’s eyes—not at the overall center of the image, but at the off-centered eye of the Fibonacci spiral. It’s simple, but this technique really does make for more interesting pictures—search the web for examples and see for yourself.
This trick of using the golden ratio to lay out your image is related to the well-known “rule of thirds” you may have heard of. The idea here is to divide your image into thirds both horizontally and vertically, and then to place important elements at the intersections of these lines. The rule of thirds is really just a simplified version of the golden ratio method—after all, it’s a lot easier to mentally picture dividing an image into thirds rather than a Fibonacci spiral. Either way, if you follow this rule when taking your pictures, your friends will soon be begging you to teach them your secrets—and now you know the math to do it!
Wrap Up
Okay, that’s all we have time for today. If you have a few minutes, I highly recommend watching the short video “Nature by Numbers” that’s recently been getting a lot of attention on YouTube. The video uses some stunning computer graphics to demonstrate the interplay between math and nature—including the Fibonacci spiral.
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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!