What are Fractals?
A Beginner's Guide
An accessible introduction to fractal geometry - from simple self-similarity to the infinite complexity of the Mandelbrot set. Designed to spark curiosity without requiring any mathematical background.
What are Fractals?
Infinite complexity. Simple rules.
You've Seen These Before
You've seen fractal images - on posters, screensavers, album covers. The Mandelbrot set, with its seahorses and spirals, has become one of the most recognisable images in mathematics.
But here's the thing: most people who recognise a fractal couldn't tell you what actually makes it a fractal. What's special about these shapes? Why can you zoom in forever? And how does such complexity come from such simple rules?
That's what we're going to explore.
Patterns Within Patterns
The defining feature of fractals is self-similarity - zoom in, and you find smaller copies of the same shapes you just saw. Zoom in on those copies, and you find even smaller copies. This continues forever.
The remarkable thing is that this infinite complexity usually emerges from very simple rules. Let's see how.
The Koch Snowflake
The Koch snowflake is one of the simplest fractals to understand. Start with an equilateral triangle. Take each straight edge and replace it with a kinked version: remove the middle third and add two sides of a smaller triangle pointing outward.
Now repeat. Every straight edge gets the same treatment. After just a few iterations, the simple triangle transforms into an intricate snowflake. Continue forever, and you get a shape with infinite perimeter but finite area.
Each iteration replaces every straight edge with four smaller edges. The perimeter grows by 4/3 each time, approaching infinity, while the area remains finite.
The Koch snowflake has infinite perimeter but finite area. Its fractal dimension of ~1.26 means it fills more space than a line but less than a plane.
How Long is Britain's Coastline?
In 1967, mathematician Benoit Mandelbrot asked a deceptively simple question: how long is the coast of Britain?
The surprising answer: it depends on your ruler.
Measure with a 100km ruler, skipping over bays and peninsulas, and you get one number. Measure with a 10km ruler, following more detail, and the coastline is longer. Use a 1km ruler, and longer still. Use a metre stick, tracing around every rock...
There is no "true" length. The closer you look, the more detail you find - just like the Koch snowflake edges. Coastlines are fractal.
The same coastline measured at different scales yields different lengths.
More Than a Line, Less Than a Plane
A straight line is one-dimensional. A filled square is two-dimensional. But what about a coastline? Or the Koch snowflake?
These shapes are too wiggly and complex to be one-dimensional - they're trying to fill more space than a simple line. But they don't fill a whole plane either.
Mathematicians measure this in-between quality with something called fractal dimension. The Koch snowflake has a dimension of about 1.26 - more than a line, less than a plane. It's a measure of how thoroughly a shape fills the space around it.
Britain's coastline? About 1.25. The more crinkled and space-filling, the higher the dimension.
Nature's Favourite Pattern
Fractals aren't just mathematical curiosities - they're everywhere in the natural world.
Look at a fern frond: each branch looks like a smaller copy of the whole. River deltas branch and rebranch in patterns that echo at every scale. Lightning bolts, blood vessels, tree branches, broccoli florets - all fractal.
Why? Because fractal branching is efficient. It's how nature solves the problem of reaching lots of places from one source, or fitting lots of surface area into limited space. Your lungs contain about 300 million air sacs, reached through a fractal tree of branching airways - all packed into your chest.
The Icon
The Mandelbrot set is the most famous image in mathematics. It's what was zooming in the video you saw at the top of this page.
It's generated by an absurdly simple formula: take a number, square it, add a constant, repeat. Colour each point based on how quickly (or whether) it escapes to infinity.
The magic is at the boundary - that's where all the complexity lives. Zoom in on the edge, and you find spirals, seahorses, and intricate filigree. Zoom deeper, and the detail never ends. And scattered throughout, you'll find tiny copies of the whole set - self-similarity again.
Try it yourself.
Higher values reveal more detail at deep zoom
Click and drag to pan. Scroll to zoom. The black region is the Mandelbrot set itself. The colorful boundary contains infinite detail.
The Mandelbrot set is defined by a simple formula: z = z² + c. Points that stay bounded are in the set (black). The boundary contains infinite complexity.
Now You See It
Look at a fractal image now, and you see something different than you did ten minutes ago.
You see self-similarity - the pattern containing copies of itself at every scale. You understand why zooming works: the detail never runs out because the same structures repeat infinitely.
You know that this infinite complexity comes from simple rules - a few lines of mathematics generating boundless intricacy. And you know that nature discovered fractals long before we did.
That's what a fractal is. Not just a pretty picture, but a window into how complexity emerges from simplicity.
Going Deeper
For the curious - you've got the main idea, this is extra.
Julia Sets
The Mandelbrot set has a secret twin - actually, infinitely many twins.
Every point on the Mandelbrot set corresponds to a different Julia set. Choose a point inside the Mandelbrot set, and the Julia set is connected (one continuous shape). Choose a point outside, and the Julia set shatters into disconnected dust.
The Mandelbrot set is, in a sense, a map of all possible Julia sets.
Each Julia set is defined by a constant c. Points inside the Mandelbrot set produce connected Julia sets. Points outside produce fragmented "dust."
Julia sets are generated using the same formula as the Mandelbrot set, but with a fixed constant c. Each point on the Mandelbrot set corresponds to a different Julia set.
Beyond Two Dimensions
Fractals extend into three dimensions and beyond. The Mandelbulb (discovered 2009) is a 3D analog of the Mandelbrot set - infinitely detailed surfaces you could explore forever.
More Than Pretty Pictures
Fractals aren't just beautiful - they're useful.
Antennas
Fractal designs pack more electrical length into small spaces - that's why your phone can receive multiple frequencies.
Computer graphics
Games and films use fractal algorithms to generate realistic terrain, clouds, and vegetation.
Medicine
The fractal patterns of blood vessels and airways help doctors spot abnormalities.
Finance
Mandelbrot's later work revealed fractal patterns in market price movements - something traditional models miss.
Further Exploration
Recommended Reading
- The Fractal Geometry of Nature - Benoit MandelbrotThe foundational text
Watch
- 3Blue1Brown: Fractals are typically not self-similarA deeper dive into what fractals really are