A Fractal Primer: The Math Hidden in Everything You Can Think Of.

What if I told you that Alaska's coastline is simultaneously 6,640 miles long and 33,904 miles long, and both answers are technically correct?

That's not a typo. The Congressional Research Service says 6,640. NOAA says 33,904. A 27,000 mile difference, and neither agency is wrong. If you're confused, good. You should be. Welcome to the coastline paradox, and the rabbit hole that eventually leads to one of the coolest ideas in all of mathematics: fractals.

---

The Ruler Problem

Here's the thing about coastlines: the shorter your ruler, the longer the coast gets.

Use a big ruler and you skip over bays, inlets, and all the jagged little details. Use a smaller ruler and suddenly all those nooks and crannies start adding up. In theory, if you measured down to the grain of sand, the length would just keep growing. Even Approach infinity, technically. Which sounds insane but is somehow true.

Geographer Katherine Sammler put it plainly: a giant ruler basically draws a box around your country. A smaller one catches everything it missed. And the thing is, this isn't some flaw in how we measure coastlines. It's actually a feature of the type of shape a coastline is. A shape that mathematician Benoit Mandelbrot eventually gave a name to: a fractal.

---

So What's a Fractal?

Mandelbrot coined the word in 1975, pulling it from fractus, Latin for "broken." Which is funny, because fractals are some of the most intricate, beautiful structures that exist. They share four properties that, once you understand them, kind of break your brain a little.

The first is infinite intricacy. No matter how far you zoom in, the detail never runs out. In 1861, German mathematician Karl Weierstrass built a shape so jagged it was nothing but corners at every level of magnification. Zoom in expecting smoothness and you get more jaggedness. Zoom in again, same thing. Mathematicians at the time called it "pathological." We now call it a fractal.

The second is zoom symmetry. Zoom into a fractal and you're looking at the same fractal. Think about a fern. The leaves look like tiny versions of the branches, which look like smaller versions of the whole plant. A pine tree is branches made of branches made of branches. It sounds simple but sit with it for a second. The whole thing is built from copies of itself.

Then there's complexity from simplicity, which is honestly the one that gets me the most. An infinitely complex fractal shape can be fully generated from a tiny list of simple rules. Mathematician Michael Barnsley proved this in the 1980s, and his algorithm was so efficient it was used to pack thousands of images onto a single CD in early Microsoft Encarta. The whole structure. From a short list of instructions.

And finally, fractional dimensions. A line is one dimensional. A flat surface is two dimensional. A fractal sits somewhere in between, which sounds like it should be impossible but isn't. Crumple a piece of paper. It now fills more space than a flat sheet but less than a solid ball. That crumpled shape is fractal like, hovering between dimensions. Nature, apparently, has absolutely no problem with this.

---

They're Literally Everywhere

Clouds. River deltas. Snowflakes. Broccoli. The branching of your lungs and blood vessels. The folding of your brain's cortex. Once you know what fractals look like, you cannot stop seeing them.

And it's not random. Fractal branching is an extraordinarily efficient design for maximizing surface area while minimizing material. Your lungs need to transfer oxygen across a massive surface area but still fit inside your chest. Fractal geometry solves that problem neatly. Evolution figured out fractals long before mathematicians did.

They show up in art too, sometimes without the artist even knowing it. Physicist Richard Taylor showed in 1999 that Jackson Pollock's drip paintings are fractal, as fractal as natural scenery. Ten separate research groups confirmed the finding. Pollock was tapping into visual patterns humans evolved to find beautiful, and he was doing it entirely by instinct.

Taylor's lab then measured people's stress responses to fractal patterns and found a 60 percent reduction in stress just from looking at them. Not from medication. Not from therapy. From looking at a pattern. That is a genuinely wild result.

---

Back to the Coast

So how long is Alaska's coastline? Honestly, we may never know, and that's kind of the point.

Different maps use different resolutions. Different resolutions produce different numbers. And it's not just an academic problem. International law uses coastal baselines to determine resource rights at sea, and when there's no consensus on measurement scale, those boundaries stay genuinely, frustratingly unclear.

The coast itself won't stay still long enough to settle the debate anyway. Erosion, tides, sea level rise. As Sammler pointed out, it doesn't take much sea level rise to start shifting legal boundaries in ways that really matter.

Law professor Ryan Stoa summed it up simply: "We just have to agree on what unit of measurement we're using. And the problem is that there's really no consensus."

Fractals, by their mathematical nature, get more complex the closer you look. Their measured length doesn't converge on an answer. It just keeps growing. The coastline is a fractal. And like all fractals, it keeps revealing new complexity the deeper you go.

Which is either the most frustrating thing in mathematics, or the most beautiful.

Enjoyed this post? Numbers in the Wild explores the mathematics hiding in the natural world around us. Check out the posts tab for more!

Challoner, Jack. "Fractal." BBC News Magazine, 19 Oct. 2010, www.bbc.com/news/magazine-11564766.

Ornes, Stephen. "See How Fractals Forever Changed Math and Science." Science News, 19 Aug. 2025, www.sciencenews.org/article/fractals-math-science-society-50-years.

Rose, Michael. "Explainer: What Are Fractals?" The Conversation, 7 Feb. 2017, theconversation.com/explainer-what-are-fractals-10865.

Sun, Alice. "What's the 'Coastline Paradox'?" LiveScience, 11 Nov. 2024, www.livescience.com/planet-earth/whats-the-coastline-paradox.

Taylor, Richard. "Fractal Patterns in Nature and Art Are Aesthetically Pleasing and Stress-Reducing." The Conversation, 31 Mar. 2017, theconversation.com/fractal-patterns-in-nature-and-art-are-aesthetically-pleasing-and-stress-reducing-73255.