Take a number c. Start at zero. Square, add c. Square, add c. Forever. For some numbers this stays small forever; for others it flies off to infinity. Color the map of numbers by which fate they meet, and you get: this. The most complicated object in mathematics, drawn by the shortest rule in it. Drag to sail, scroll to dive. Every bay hides smaller bays. No two are the same. It does not end.
For the curious kid.
In 1980, Benoît Mandelbrot asked a computer to draw this map, and nobody believed the printout: they thought the specks around the island were dust in the machine. Zoom toward any speck and it turns out to be a perfect miniature of the whole island, moored offshore, wearing its own coat of smaller specks. Mandelbrot had spent years asking an odd question: how long is the coastline of Britain? The odd answer: it depends on your ruler. Measure with a mile-long ruler and you skip the coves; with a yardstick you must trace them; with a matchstick, every pebble. The shorter the ruler, the longer the coast, without limit. He coined a word for shapes like that: fractal. This island is the purest one ever found. Its inside holds a finite amount of ink. Its coastline is infinitely long. Both of those are true at once, and you can sail it forever.
Deeper.
The rule is z → z² + c, iterated from z = 0; the set M is every complex c whose orbit stays bounded. The glow you see is not decoration: each exterior point is lit by its mathematically guaranteed distance to the set (|z|·ln|z|/|z′|, from the derivative carried through the iteration, right to within a factor of four, always), so the filaments you see faintly webbing the darkness are real structure, not rendering artifacts. And those filaments are load-bearing: Douady and Hubbard proved M is connected: every offshore islet is secretly moored to the mainland by a thread of boundary too thin for any screen. The boundary is so wrinkled that its fractal dimension is exactly 2 (Shishikura, 1998): a curve as thick as a surface, the mathematical limit of coastline-ness. Meanwhile the area inside is finite, roughly one and a half square units, and, delightfully, its exact value is still unknown to mathematics. Perimeter: infinite, proved. Area: finite, unmeasured. It hangs in this museum with its dimensions honestly labeled "we don't know."
Museum policy note: the arithmetic here is validated headless: known members and escapees, the interior shortcuts checked against 35,000 grid points with zero disagreement, the smooth coloring confirmed continuous across iteration bands, the distance estimate checked at the antenna, at the tip of i, and at the cusp (where it politely under-glows, never over). One honest confession: past magnification ×10¹⁰ this page's floating-point numbers run out of decimals, so the museum stops the boat there. The coastline continues without us. It always will.
The island's paperwork lists a finite area no one has measured and a shoreline no one could. Both entries are correct.