Indexing the archive…
Your Universe of Digital Possibilities
The Mandelbrot set lives in the flat complex plane because multiplying complex numbers rotates and scales. There is no such multiplication in 3D — so White & Nylander built one by analogy, in spherical coordinates: raise the radius to the n, multiply both angles by n, add the starting point. The same escape test, |z| > 2, now carves a solid object with infinite surface detail. Drag to orbit it, fly in, and watch the lobes split as you turn the power.
Feed the rule its own output, starting from zero. If |z| ever passes 2 the orbit escapes to infinity; otherwise it is trapped forever. That single yes/no, asked at every point, draws everything here.
No 3D multiplication behaves like the complex plane’s, so White & Nylander defined one by analogy: write the point in spherical coordinates (r, θ, φ), multiply the two angles by n and raise the radius to the n. The escape test |z| > 2 carries over unchanged.
A solid fractal has no formula for “distance to the surface”, but the escape-time potential gives a safe lower bound. The ray leaps by that bound each step — far away it strides, near the surface it creeps — so the whole bulb resolves in a few dozen steps per pixel.
The boundary is endlessly detailed — Hausdorff dimension 2 (Shishikura) — with whole mini-Mandelbrots buried at every depth, and the same shape governs the chaos onset of any smooth family. Infinite structure, five symbols of law.
This is The Set lifted out of the plane. The same rule, the same escape test, the same lesson — an object no finite description exhausts, grown from five symbols of law — but now you can fly into it. There is a quiet honesty in how it renders: the distance estimate is a proven lower bound, so the ray can sprint across empty space and still never pierce the surface. It is the rack’s sharpest answer to the oldest question on it — if reality has a source code, it need be no longer than this.