🌌 Mandelbulb 3D Fractal

Raymarching advances a ray step by step until it hits a surface, but without knowing how far away that surface is, it would need thousands of tiny, wasteful steps. The distance estimator uses the escape-time formula's radius r and its accumulated derivative dr to compute a mathematically safe upper bound on how far the ray can travel without overshooting the fractal — this simulator's shader marches forward by exactly that distance each of its up to 96 steps, which is what makes real-time frame rates possible on ordinary GPUs.

The power n controls how many times the angular coordinates θ and φ are multiplied (θ→nθ, φ→nφ) and how the radius is exponentiated (r→rⁿ) on every iteration of z→zⁿ+c. Small changes in n shift where points escape versus stay bounded, which reorganises the whole self-similar structure — n=8 gives the famous bulbous, cathedral-like form this simulation defaults to, while other values of n between 2 and 12 produce noticeably different lobe counts and surface textures.

No — the 2D Mandelbrot set relies on complex-number multiplication, which has a clean geometric meaning (multiply magnitudes, add angles) that does not extend uniquely to three dimensions. The "triplex" spherical-coordinate formula this simulator implements — r→rⁿ, θ→nθ, φ→nφ — is one popular choice among several competing 3D generalisations, valued for producing rich, organic detail rather than for being a unique mathematical extension of the original.

Bailout sets the escape radius: once a point's distance from the origin exceeds this value (checked as r² > bailout² in the shader), it is classed as having escaped and iteration stops early. Max iterations caps how many times z→zⁿ+c is applied before giving up on convergence. Points that escape quickly sit far outside the fractal's surface; points that survive the full iteration budget are deep inside it — the boundary between those two behaviours is the fractal surface the raymarcher renders.

Every raymarching step calls the distance-estimator function, and that function itself loops up to the current Max iterations value computing z→zⁿ+c and updating the derivative dr. More iterations let the escape-time test distinguish finer and finer filigree near the fractal's boundary, but since this inner loop runs for every one of the roughly 96 marching steps per pixel, per frame, doubling the iteration budget roughly doubles the GPU work and lowers the FPS counter shown in the Stats panel.