Romanesco broccoli
Every bud is a cone made of smaller spiraling cones, about four levels deep. Count the spirals and you usually get Fibonacci numbers.
01 · The idea
Zoom in. It never gets simpler.
Zoom far enough into a circle and it looks like a straight line. Zoom into a fractal and you find more detail, often looking like the whole thing again.
SELF‑SIMILAR
Copies inside copies
A fern frond is made of small fronds, which are made of even smaller ones. The parts look like the whole.
ITERATION
One rule, repeated
Fractals are not drawn piece by piece. You apply one simple rule to the result, then again, and again.
DIMENSION
Between 1D and 2D
A Koch snowflake’s edge is more than a line but less than an area. Its dimension is about 1.26.
z → z² + c
That is the whole rule behind the next two pictures. Start with a number, square it, add c, and repeat. Some starting points stay small forever; others shoot off to infinity. Color shows how fast they escape.
02 · Mandelbrot & Julia
Dive into the most famous shape in math
The Mandelbrot set (left) is every c where the rule, started at 0, stays small. Each point c also has its own Julia set (right). Tap anywhere on the Mandelbrot set to see that point’s Julia set.
zoom ×1
That is as deep as this device’s numbers go
Fly to
c = …
Colors
Drag to move · pinch or scroll to zoom · double‑tap to dive in · tap to pick c. Each zoom shows detail no one had seen before computers.
03 · Step by step
Grow a snowflake, carve a triangle
Both of these start from a plain triangle. Move the slider to apply the rule once more. Watch what happens to the edge length and the area.
step 0
04 · Branching
A tree is a trunk with two smaller trees on top
Draw a branch. At its tip, draw two shorter branches, turned left and right. Do the same at each new tip. Drag sideways on the tree to bend it.
1,023 branches
Each level doubles the number of tips, so 10 levels give 512 tips. Natural adds a little randomness to every angle and length. The rule stays the same, but the tree suddenly looks alive.
05 · In the wild
Nature has been doing this for millions of years
Real fractals stop after a few levels, because atoms and cells set a limit. Within that range, branching and repeating is often the cheapest way to fill space, spread out, or move energy.
Tap to regrow
Ferns
This fern is made of four rules that shrink, tilt and copy a point, picked at random about 80,000 times. Michael Barnsley found it in 1988.
Tap where it should strike
Lightning
The spark steps through the air, splitting wherever the air breaks down a little more easily. Each fork is a smaller bolt. Its fractal dimension is around 1.5.
Coastlines
How long is a coast? It depends on your ruler. A shorter ruler follows more bays and points, so the total keeps growing. Mandelbrot asked this about Britain in 1967.
Lungs
Your windpipe splits in two about 23 times, ending in roughly 480 million tiny air sacs. Folded inside your chest is a surface of about 70 m².
06 · Glass
Break a window. Then let it freeze.
Cracks in glass and frost on a pane both branch like trees. Tap the glass to set off a crack, or switch to frost and tap to drop a seed crystal.
Tap the glass
Why cracks fork
A crack in window glass speeds up to about 1.5 km per second. Beyond that, one tip cannot release the stored energy fast enough, so it splits in two. Each half can split again.
Spokes and rings
An impact makes radial cracks that shoot outward, then rings that link them. A crack stops when it reaches another crack. Investigators use this to tell which hole was made first.
Frost ferns
Ice starts on scratches and dust. Its six‑sided crystal makes side branches at 60°. Tips that stick out catch the most water vapor, so they grow faster and branch again.