3D Fractal Worlds

Mandel-Bulb is considered the 'Holy Grail' of 3D fractals. Since complex numbers are only two-dimensional, this fractal uses spherical coordinates and high powers (usually v^8 + c) to project the Mandelbrot set into 3D space. Benefit: Creates an organic, extremely detailed structure reminiscent of plants, coral reefs, or alien landscapes. Wikipedia

Mandelbox is based not on smooth curves, but on geometric 'folding' and scaling (Box-Folding & Sphere-Folding). Space is repeatedly folded and mirrored like origami. Benefit: Produces strictly geometric, mechanical-looking structures that resemble endless futuristic cities, the Borg cube, or complex sci-fi architecture. Wikipedia

Julia-Bulb is the 3D counterpart to the 2D Julia set. While the Mandelbulb is a 'map' of all fractals, here we fix the parameter 'c' and vary the starting point and it changes over time. Benefit: Unlike the solid Mandelbulb, Julia Bulbs are often hollow, forming complex tunnel systems or bubble-like structures perfect for flying through with the camera. Wikipedia

Original work

• Daniel White, Paul Nylander (2009): Mandelbulb
• Tom Lowe (2010): Mandelbox

How it works

There is no geometry. A fragment shader sends one ray per pixel from the camera into the scene and walks it with sphere tracing (raymarching): a distance estimator says how far the surface is at least, and the ray jumps by that amount (times 0.8 as a safety factor), at most 128 steps. Before that, a sphere test (radius 6) limits the ray to the relevant range.

• Mandelbulb: 8 iterations in spherical coordinates with power 8 (z → z⁸ + c). The distance is 0.5 · ln(r) · r / dr, where dr is the running derivative.
• Mandelbox: 15 iterations of box fold (mirror at ±1), sphere fold (scale by radius) and scale 2.8.
• Julia bulb: like Mandelbulb, but with a fixed constant c that moves slightly over time, so the shape moves.
• Normals come from four extra distance queries. Lighting: light from the camera direction plus 40 % ambient, no shadows, no ambient occlusion. Color: orbit trap (smallest radius during iteration) mapped to a cosine palette.
• Cost per frame: pixels × ray steps × iterations. At full HD with the Mandelbox that is about 4 billion iteration steps in the worst case. It runs as a fragment shader on the GPU only; there is no CPU path.

Source code:fractal3d.shader.ts

Note This visualization uses a technique called 'Raymarching' (Sphere Tracing). This means:

• No Polygons: There are no triangles or meshes. The shape is calculated mathematically for every pixel in real-time.
• Distance Estimation: The algorithm 'marches' light rays forward by calculating the safe distance to the nearest object without hitting it immediately.
• Infinite Detail: Since the surface is mathematically defined, it never pixelates when zooming in – new structures always emerge.
• Light: depth comes from the surface normal: faces that point at the camera are brighter, plus a fixed ambient term. Shadows and ambient occlusion are not computed.