The Clifford-Mandelbrot Set: A Higher-Dimensional Generalization of the Mandelbrot Set via Geometric Algebra

Based on the vector invariance result established in previous work for Clifford Julia iterations {Zaka2026arXiv}, we introduce and study the Clifford-Mandelbrot set Mp,n⃗ in an n-dimensional real inner-product space V with the geometric product. For V=R² and n⃗=e⃗₁, the set reproduces exactly the classical Mandelbrot set. For higher dimensions, we derive explicit polynomial formulas for the iteration map, prove basic properties including rotational symmetry and an escape radius, and establish the relationship with the corresponding Clifford Julia set. Explicit computations are carried out for R³ and R⁴, revealing a clean pattern: the squared term becomes\[(x⃗⃗_1)^2⃗_1= (x_1^2 - \|x⃗_⊥\|^2)e⃗_1 + 2x_1x⃗_⊥,\]where x⃗_⊥ is the component orthogonal to e⃗₁. This provides a unified algebraic framework for generating 3D and 4D Mandelbrot-type fractals without artificial projections. Computational aspects and escape-time algorithms are discussed, and open problems concerning the topology and fractal dimension of Mp,n⃗ for n≥3 are outlined.