3D Quadratic Map (Discrete Iteration)

Overview

While continuous differential equations generate smooth, continuous ribbons and tubes, discrete maps operate through instantaneous algebraic transformations. At every iteration step $n \to n+1$, the point jumps across space according to a polynomial transformation rule.

In three dimensions, the general quadratic map folds 3D space like kneading dough: stretching coordinates along expanding eigenvectors, and folding them back along contracting directions. Over millions of iterations, points settle onto intricate fractal manifolds that exhibit self-similarity at all magnifications.

The Mathematical Framework

The recurrence relation is expressed vectorially as:

$\vec{v}_{n+1} = \mathbf{C} \cdot \vec{T}(\vec{v}_n)$

where:

$\vec{T}(\vec{v}) = \begin{bmatrix} 1 & x & y & z & x^2 & xy & xz & y^2 & yz & z^2 \end{bmatrix}^T$

Distinctive Visual Character

• Translucent Membranes: Because discrete iterations jump across space without drawing solid connecting curves, renderings look like diaphanous veils, planetary nebulae, or frozen smoke rings.
• Cantor Dust & Fractal Cross-Sections: Slicing through any section of a discrete 3D attractor reveals a Cantor-like fractal dust of infinite complexity.
• Extreme Parameter Space: The 30 coefficients define a vast mathematical universe containing billions of unique geometric species.