Imagine zooming further and further into a photo of a snowflake. For an ordinary shape, things would look simpler and simpler the more you zoomed in — but for certain shapes, an equally complex shape just keeps appearing, no matter how far you zoom. Pull out one small piece and zoom in, and that piece turns out to be strikingly similar to the whole shape. This property is called self-similarity, and a shape with this property is called a fractal.
One of the most famous fractals is the Koch snowflake. The way it's made is very simple. Start with an equilateral triangle, and replace the middle third of every edge with a pointed triangular bump. Then, for every edge of the newly formed shape, repeat the exact same process again. Every time you do this, the number of edges multiplies by 4. It starts with just 3 edges, but after only a few repetitions, you get an intricate snowflake shape with thousands, even tens of thousands, of edges.
Here's where something truly strange happens. With every repetition, the perimeter keeps growing by a factor of 4/3, so repeated infinitely, the perimeter becomes infinite! But what about the area the snowflake takes up? Since the area can never grow larger than the circle enclosing the original triangle, it never grows past a certain value, no matter what. An infinitely long line ends up entirely contained within a finite area — it feels strange to our intuition, but it's a fact that can be proven precisely, mathematically.
Another famous fractal is the Sierpiński triangle. Draw an equilateral triangle, connect the midpoints of its three sides to form a small triangle in the middle, and then punch out and remove that middle triangle. Then, for each of the three remaining small triangles, repeat the exact same process again. Keep doing this, and the number of remaining triangles multiplies by 3 each time, while the total area keeps shrinking to three-quarters of what it was, getting closer and closer to 0. And yet, a strange shape emerges that never fully disappears.
Fractals aren't something mathematicians only dreamed up in their imagination. The way tree branches spread out, the shape of a lightning bolt, mountain ridgelines, the branching of blood vessels or the airways inside lungs, the jagged shape of a coastline — all of these come remarkably close to being fractals. In fact, it was the mathematician Benoit Mandelbrot, who once asked the question "how long is a coastline, really?", who first named and popularized the concept of fractals — and the idea came to him from the fact that a coastline's measured length keeps changing depending on how small a ruler you use to measure it.
On our activity page, you can increase the depth one step at a time with a slider, and see for yourself exactly how many edges the Koch snowflake grows to, and how finely the Sierpiński triangle keeps subdividing. Watch with your own eyes just how complex a shape can get from repeating the same rule alone.