# Mandelbulb 3D Fractal Explorer

Interactive 3D Mandelbulb fractal explorer with ray-marched WebGL — rotate the true three-dimensional generalization of the Mandelbrot set

> Canonical page: https://elysiatools.com/en/visualizations/mandelbulb-3d-fractal-explorer

- **Category:** Math

## Overview

Interactive 3D Mandelbulb fractal explorer rendered with a real-time WebGL distance-estimated ray-marcher — the only true THREE-DIMENSIONAL fractal in the collection (every existing fractal case — Mandelbrot set, Julia set, Burning Ship, Multibrot, Newton fractal, IFS, fractal-dimension, fractal-chaos — is strictly 2D). The Mandelbulb, proposed by Daniel White and Paul Nylander (2009), generalises the Mandelbrot iteration z→z²+c by raising z to an integer power n in SPHERICAL coordinates: with r=|z|, polar angle θ=acos(z/r), azimuth φ=atan2(y,x), the power step gives zⁿ magnitude rⁿ, polar angle n·θ, azimuth n·φ, so the orbit z_{k+1}=z_k^n+c (z_0=0) either escapes or stays bounded. The 3D view is produced by a fullscreen-quad fragment shader that ray-marches the SAME distance estimator as the CPU model: alongside z we accumulate the running-derivative magnitude dr (recurrence dr=r^{n−1}·n·dr+1), and for escaped points DE(c)=½·ln(r)·r/dr is a lower bound on the distance to the fractal surface, letting rays step by DE(c) without overshooting; surface normals come from tetrahedron finite-differences of DE, with diffuse + ambient + ambient-occlusion (step-count) shading and four color palettes (fire/ice/aurora/mono). Controls: orbit the camera by dragging (mouse + touch), zoom with the wheel, power-n slider (2–10), max-iterations slider (4–20), light azimuth, ambient-occlusion toggle, auto-rotate, reset view, and 6 presets (classic n=8, power 2, power 3, spiky n=9, deep zoom, side slice). A second 2D-SLICE view evaluates the model CPU-side on a fixed plane (x/y/z = const, adjustable position), colouring the interior black (bounded orbit) and escape-time bands outside — showing how the 3D volume's cross-section changes with power n and plane position, with live inside-fraction and mean-escape-iteration stats. Verified by 72 model tests (power-step identity/magnitude/axis behaviour, origin-inside, distance-estimator monotonic approach, outward normal near the +x lobe, slice-axis mapping, slice stats, ray-march hit/miss, presets, power-2-vs-8 geometry difference). Educational content covers the spherical power-n formula, the distance estimator and ray marching, why this is a genuine 3D fractal (non-integer Hausdorff dimension, no single 'correct' 3D vector product), and how to explore (zoom reveals self-similar filigree). Multi-language support (zh, en, es, fr, de, ru, pt).

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