Square and add. Forever.
One of the most photographed objects in mathematics is the visualisation of an absurdly simple rule. Below: what the rule is, what we are actually looking at, and a button straight into the Explorer for when you want to fly.
Square a number. Add another number. Take the result and square that, add the same number again. Keep going. That single recipe — square, add, repeat — is the entire Mandelbrot set. Everything else is bookkeeping.
Pick c = −0.5. Start at 0. The next values are −0.5, −0.25, −0.4375, −0.309… The sequence settles, never running away. Now pick c = 1. The sequence is 0, 1, 2, 5, 26, 677 — it leaves and never comes back. That tiny yes/no is what colours the picture.
The boundary between yes and no is the most-photographed mathematical object on Earth. Zoom anywhere on it and new shapes appear, never repeating, never simplifying — all from one line of arithmetic that a child could understand.
Every point in the picture is a complex number c. For each c we run the same little loop: start at 0, square the current value, add c, and feed the result back in. After a few dozen steps we ask a single question — has the running value flown off to infinity, or is it still hanging around near zero? Black means it stayed. The colours encode how quickly it fled.
z = 0
for step in 1..N:
z = z * z + c
if |z| > 2:
return "escaped at " + step
return "bounded"|z| > 2 is the safety net: once the running value crosses radius 2, it provably never comes back. That gives us our cheap yes/no test.
The number of steps before escape is what becomes the colour. Black means we ran out of steps without escaping.
Three c values, three completely different stories. Deep inside the set the orbit converges to a fixed point or a small cycle. Right on the boundary it dances forever, never settling, never escaping — a knife-edge between order and chaos. Just outside, the orbit spirals out and explodes within a handful of steps. Watch them race.
The dim shape behind the canvas hints at the set. Drag the bright yellow dot anywhere — inside the cardioid, into a bulb, off in the void — and watch the orbit decide. Bounded orbits trace a violet web; escaping orbits flash rose, then reset so you can try again.
Try just left of the cardioid, near c = −0.75, where the period-2 bulb attaches.
Black is the verdict: this c is in the Mandelbrot set, its orbit stays bounded forever. The colour bands outside encode escape time — how many steps the orbit took before the running value crossed the safety radius |z| = 2. Different palettes, different moods, same underlying integer. The picture is, at heart, a contour map.
We can only iterate finitely many times. Stop too early and the boundary looks crude; iterate longer and finer filaments emerge. Drag the slider to feel that trade-off live — every pixel that was 'in' at low iterations may still flee at high ones.
At low iterations, large stretches near the boundary are mis-classified as "inside" the set. Push the slider right and the filaments thin out, revealing what was always there.
The interior is calm and the exterior is calm; all of the drama lives on the boundary. Shishikura proved in 1998 that this boundary has Hausdorff dimension exactly 2 — it is so wiggly that, in a precise sense, it tries to fill a two-dimensional area despite being a curve. Zoom in by a factor of a billion and you find tiny near-copies of the whole set surrounded by filaments that never repeat. There is genuinely something new at every scale.
The big heart-shaped lobe in the middle is a cardioid — the same curve traced by a point on a rolling circle. Inside it, the iteration converges to one fixed point. Bolted to the cardioid is a perfect disc where the iteration settles into a 2-cycle, then another disc for a 3-cycle, another for a 5-cycle, and so on. The rational numbers p/q label exactly where each disc attaches. The Mandelbrot set is a directory of every cycle length, sorted geometrically.
Fix c and run the same iteration zₙ₊₁ = zₙ² + c, but now let the starting point z₀ vary across the whole plane. The points whose orbits stay bounded form the Julia set for that c. There is one Julia set per choice of c — a whole universe per point in the Mandelbrot picture. Pick c inside the set and you get a beautifully connected Julia shape; pick c outside and the Julia set shatters into Cantor dust. The Mandelbrot set is, exactly, the map of which Julia sets are connected.
The Explorer lets you zoom into the boundary down to 10¹⁰, jump to famous spots, and turn the iteration knob live. Everything you just read is one click away.
→ Open the Explorer