Dynamic Mandelbulb fractals: A mathematical framework for time-evolving 3D fractals using distance estimation

This paper presents a novel approach to generating a time-dependent Mandelbulb fractal by introducing dynamic spatial transformations within its iterative construction process. By embedding smoothly varying sinusoidal translations-scaled by both global time and iteration index and synchronizing a phase-driven rotation in the XY-plane with Greenwich Mean Time, the fractal’s internal geometry undergoes continuous, organic deformation while preserving its core self-similar patterns. Leveraging a distance-estimator based raymarching pipeline in GLSL, our implementation renders these evolving structures in real time, allowing detailed exploration of their richly evolving forms.

The dynamic behavior introduced by in-loop time modulation enhances the fractal’s complexity and offers new opportunities for applications. Specifically, the time-dependent Mandelbulb provides an intuitive tool for scientific visualization, mirroring the transient structures of turbulent flows, plasma instabilities, and other chaotic systems. It also lends itself to chaos-based cryptography: by encoding secret keys into time-varying translation amplitudes, rotation phases, or exponent values, one can generate non-repeating, sensitive pseudo-random sequences for secure communication. This work not only advances real-time fractal visualization but also lays the groundwork for future exploration in dynamic simulations and fractal-driven security systems.