The 3D analog of the Mandelbrot set: instead of squaring complex numbers, we use "spherical 3D squaring" with a power n (typically 8). The escape-time fractal in 3D is the Mandelbulb. Daniel White and Paul Nylander discovered it in 2007 after decades of searching for a satisfying 3D fractal. It looks like a psychedelic interpretation of cathedral architecture — and it has zero relationship to anything in nature, yet looks intensely organic.
how to multiply 3D numbers (when you can't)
Frobenius proved (1877) that the only finite-dimensional associative division algebras over ℝ are ℝ, ℂ, and the quaternions ℍ. There's no "3D number system" with multiplication — so there's no direct analog of z² + c in 3D.
The Mandelbulb cheats. Map (x, y, z) to spherical (r, θ, φ). Define "squaring" as:
(r, θ, φ)² = (r², 2θ, 2φ)
i.e., square the radius, double both angles. Generalize to power n:
(r, θ, φ)ⁿ = (rⁿ, n·θ, n·φ)
Apply v ← vⁿ + c iteratively. Points where the magnitude stays bounded form the Mandelbulb set.
This is not a holomorphic dynamical system — the construction doesn't respect a complex/quaternionic structure. The "fractal" properties are heuristic. But the visual result is extraordinary: cathedral-like spires, lily-flower curls, bulbous protrusions, infinite zoom complexity. Each n gives a different shape:
• n = 2: smooth, rounded blob — not very fractal
• n = 8: the canonical Mandelbulb with 7-fold spire symmetry
• n = 10+: increasingly "pointy" with sharp tessellations
The Mandelbulb is one of the most-rendered 3D mathematical objects of the 21st century. Despite having no relation to physics or biology, it's been a steady source of computer graphics, generative art, and mathematical curiosity since 2009.