How does one line, z² + c, grow endless detail?

Take a complex number $c$, start from $z_0=0$ and keep computing

The sequence $0,\ c,\ c^2+c,\ \dots$ is the orbit of $c$. If it never runs off to infinity, $c$ belongs to the Mandelbrot set.

There is a shortcut for deciding: once any $|z_n|>2$, the orbit is sure to escape. So a picture only needs to iterate until $|z|$ passes 2, or until a step limit is reached.

The first point in the story, $c\approx-0.1226+0.7449i$, is special: after three steps it is back at 0, so its orbit is a triangle repeated forever. The second, $c=-0.75+0.1i$, sits just outside the set and escapes only at step 33.

Points in the set are black. Every point outside escapes eventually, just at different speeds: the closer to the set, the slower. The colour shows when a point escaped.

To get smooth colour instead of rings of steps, the picture uses the smooth escape value

where $n$ is the step at which the orbit passes radius 16. The closer to the edge, the larger $\mu$ and the brighter the colour.

In the third part of the story the step limit climbs from 1 to 220. With a small limit, many points that will escape later have not escaped yet and are wrongly counted as inside, so the outline starts as a large blob and tightens ring by ring.

The inside is solid; the complexity is all on the edge. Zoom into any part of it and new spirals, tendrils and patterns appear, at every magnification.

• Douady and Hubbard proved in 1982 that the set is connected: the islands that seem scattered are all joined by thin filaments.
• Shishikura proved in 1998 that its boundary has Hausdorff dimension 2, so a "curve" that is as thick, in a sense, as an area.
• Its area is about 1.5066, but the exact value is still unknown.

The copy at the end of the story lies deep in seahorse valley at $c\approx-0.7436442+0.1318254i$. It is only about a millionth wide, so you need a zoom of over 300,000 to see it. It has the same heart-shaped body, the same discs, the same edge, but it is not an exact copy: the spirals and filaments around it differ from the original. A much larger copy sits on the antenna sticking out to the left, near $c\approx-1.7549$; "Places" flies you there.

There are infinitely many such copies scattered along the edge. Their centres are called nuclei: starting from 0, the orbit returns exactly to 0 after some number of steps. The nucleus of the copy in seahorse valley returns every 78 steps; the three-step point at the start of the story is the nucleus of a disc.

Graphics cards are fast but work in 32-bit floats, about 7 significant digits. Around a 10,000× zoom, neighbouring pixels can no longer tell their values of $c$ apart and the picture breaks into blocks.

This page uses perturbation: one reference orbit $Z_n$ is computed at the centre of the view in 64-bit precision, and each pixel only tracks its difference $\delta_n$ from it:

$\delta$ is small, so 32 bits hold it accurately. Whenever $|Z_n+\delta_n|<|\delta_n|$, or the reference runs out, the current value becomes a new start and is lined up with the beginning of the reference again, so errors do not pile up.

That takes the page to a zoom of about 16 trillion, where 64-bit precision itself runs out.