New Algebraic Modeling of Autonomous Fractal Trees in D-Dimensional Spaces and Their Optical Rendering via Cascaded Perspective Projection.

Abstract

This paper introduces a unified algebraic framework for the generation and visualization of complex binary, ternary, and quaternary fractal branching structures embedded in arbitrary D-dimensional Euclidean spaces (R^D). While traditional fractal tree algorithms rely on discrete recursive coordinates or explicit trigonometric summations, we formulate the structural growth of vertices (nodes) and continuous parametric line segments (edges) through chained products of homogeneous transformation matrices of size (D+1)×(D+1). Furthermore, to bridge the gap between higher-dimensional fractal structures and physical 2D display interfaces, we develop a non-linear cascaded projection operator. This operator models multi-stage directional rotations using Givens matrices intertwined with focal perspective distortions (d − x)^(−1) to progressively collapse the spatial geometry from R^D → R^2. Computational validation in Python confirms that our mathematical formulation avoids branch occlusion, guarantees structural self-similarity, and visually renders hyperspace properties. This framework offers significant applications in multidimensional data visualization, procedural natural modeling, and computer animation.