Eye-Catching Math: Fractals

Eye-Catching Math is a series about beautiful mathematical images. Read more about it here.

A fractal is a pattern that contains smaller and smaller versions of itself. In a fractal, those smaller versions contain even smaller versions of the whole pattern. Theoretically, these smaller versions continue on forever, although when we draw them, we have to stop at some point.

Here is an example of a fractal, called a Koch curve. To make this, draw a particular pointy line, _/_. Then, replace each edge of _/_ with a mini version of _/_. Do that again with each of the new, smaller edges. And so on.

When we zoom in on this fractal, we see that it is made up of smaller and smaller versions of itself.

Fractals are a wonderful source of artistic inspiration.


Fractals are the key idea behind self-similar pictures like this one.

Fractal patterns are found in nature surprisingly often.

In trees and clouds:


Peacock tails:

Even broccoli:

Questions about fractals? Feel free to ask in the comments.

More Eye-Catching Math posts coming soon! They will be interspersed with regular blog content.


Eye-Catching Math

Imagine a world where kids are never exposed to the sound of music. Instead, they learn musical notation: how to read it, memorize it, and reproduce it. To these kids, notes, chords and songs are all just arrangements of dots on a page.

You would expect these kids to associate music with boredom and resentment. How else could they feel? They missed out on everything that makes music beautiful. They learned music as a written language, when really, music is a language in and of itself.

Fascinating? Not unless you can hear the music.

To a lesser extent, this happens to some people who feel that they don’t “get” math. In school, we learn mathematical notation, as well as how to get solutions to certain kinds of problems. But in this environment, many of us miss out on what makes math interesting.

Math is full of rhythms and harmonies that can only be perceived in the language of math itself. When someone says that e^(iπ)+1=0 is “beautiful,” they aren’t referring to the ink on the page or the pixels on their computer screen: they are referring to the beautiful pattern that those symbols represent. “e^(iπ)+1=0″ isn’t math. It is what you get when you translate math into written symbols.

Just as an expert composer can look at musical notation and hear the song in his head, a mathematician can look at written symbols and “hear the math” in her thoughts. The difference is that, while music can be performed on a piano or a bass guitar, there is no instrument on which math can truly be played. (The real medium for math is the human mind.)

Mathematical notation is a serious barrier for people who would otherwise enjoy math ideas. So in the Eye-Catching Math series, I will post and discuss beautiful mathematical images. This is a way to try to “play some math” without technical symbols. Images never fully capture how grand or astounding math can be, but they do a little.

The first post of this type will come out later this week. (The peacock and broccoli are actually examples of the first topic we’ll be discussing.) Enjoy!

The music analogy was inspired in part by Keith Devlin’s The Language of Mathematics.

Update, Nov. 30 2014: This post is accidentally similar to the introduction to Paul Lockhart’s A Mathematician’s Lament. No plagiarism intended! I hope to write about Lockhart’s lament in more detail in a future post.

Featured Links: Vi Hart

Welcome to Featured Links! This is a blog category about interesting math-y stuff offsite.

Vi Hart is a self-proclaimed “recreational mathemusician.” To put it simply, she doodles about math for fun (and makes music, too).

What’s great about her videos is that they never seem to discuss anything particularly educational or useful. They’re just pretty spirals, animals, stick figures, and weird triangles… until suddenly you realize that you’ve been looking at math all along.

Vi Hart’s YouTube channel is a great place to (1) waste time on the Internet and (2) have your mind absolutely blown. Warning: she talks reallyreallyreallyfast. Don’t worry if you don’t catch everything she says – I know I never do. Here are links to a few of her videos.

Doodling in Math Class: DRAGONS

Doodling in Math: Spirals, Fibonacci, and Being a Plant

 Open Letter to Nickelodeon, Re: SpongeBob’s Pineapple under the Sea. This one makes more sense if you’ve already watched the video above.

 Sound Braid. This one’s got music!

Is Math Fiction? (part 2)

The last post asked, “Is mathematics fiction or nonfiction?”  Here’s a recap of the discussion so far:
  • One definition for math is “the science of patterns.”
  • Science and technology depend on math. Disciplines like chemistry, physics, and computer science are made possible by the quantitative study of real-world patterns. This makes it seem like math is nonfiction.
  • Some mathematical scenarios, like nine-dimensional space, do not correspond to the real world in any obvious way. Since it is possible to study patterns that do not exist, it would seem that math is fiction.

Who needs Pythagoras' theorem?

If you went to a typical American high school, you almost certainly learned Euclidean geometry. This geometry was formalized* by the ancient Greek mathematician Euclid. The principles of Euclid’s system seem straightforward enough. Any two points can be connected by a straight line. All right angles are equal. Given a line and a point not on that line, there is exactly one line that can be drawn through the point that is parallel to the original line.

*This was important for a lot of reasons and will hopefully be discussed in a future post.

So innocuous-looking, yet so controversial.

Screenshot from here.

Our real-world experience tells us that making things bigger won’t change their shape. A triangle with sides of 3 feet, 4 feet and 5 feet should be the same shape (and have the same angles) as a triangle with sides 30, 40 and 50 feet. Fortunately, Euclidean geometry meshes with our idea of reality. It also does a fantastic job describing things like tables, beach balls, and skyscrapers. There is just one problem: apparently, the universe doesn’t work that way.

Spacetime curvature

Space is curved. You may have heard this before—the idea became famous when Einstein developed his theory of relativity. Einstein did not use Euclidean geometry in his model. He used a curved Riemannian geometry. Riemannian geometry is just as mathematical as Euclidean geometry is, but it clashes with concrete intuition. Small changes in its basic structure shove curved geometry outside the realm of experience. But curved geometry, not Euclidean geometry, fits relativity. In other words, physics says that curved geometry describes our universe, and Euclidean geometry does not. Now, which of these ideas is a fiction? Is Euclidean geometry “nonfiction” because it is true to human experience? Is Riemannian geometry “nonfiction” because it is used in physics? Are both nonfiction, or maybe neither?

Candle & Book

In the end, it depends what you mean by “fiction” and “nonfiction.” Math is not information we gather about the natural world. You can’t pick up a piece of math or look at it under a microscope. No scientific experiment can prove a theorem. No historical record contains math that “happened.” Every bit of math in the history of civilization occurred in someone’s mind. If “nonfiction” is defined as “a description of something physical,” then math is fiction. However, math gives us insight into the natural world. We use math to organize the information we gather with our senses. Math can be used to explain and predict physical patterns. It makes the difference between drinking “coffee, then coffee, then coffee” and drinking “three cups of coffee.” It’s what takes you from “This coin sometimes comes up heads” to “What are the odds it will happen?” If nonfiction includes “that which humans use to understand reality,”then math is nonfiction.

A Story Book comes to life

So, that is an answer. It depends on how you define nonfiction. But the debate is far from over. For one thing, these posts have not discussed the controversial idea of “mathematical truth.” The tremendous enjoyment of seeing the order and harmony in math is often accompanied by a sense that math is “real” in some way: that it is something humans perceive, not imagine. This adds another facet to the fiction vs. nonfiction debate. Maybe it will get its own post later on.

Thank you for reading!

There were many, many people whose ideas went into the formulation of this series. I tried to present the points smoothly, without pointing out whose idea was whose. But to everyone who discussed this with me: thanks for your input!