A Constructive Geometric and Algebraic Method for Higher-Dimensional Hypercube Visualization

Visualizing geometric structures in dimensions greater than three presents a fundamental cognitive challenge due to the human perceptual manifold being bound to three spatial dimensions. This paper presents a constructive, algebraic, and computational method for generating, rotating, and projecting $n$-dimensional hypercubes into real-time interactive three-dimensional visualization windows. While the theoretical and algorithmic principles scale indefinitely to an arbitrary dimension $n$, the reference computational implementation is bounded to dimensions $n \in [1, 6]$ to optimize visual clarity and screen rendering. By formalizing a recursive "sweep-and-mirror" operator alongside Hamming graph topology, multi-plane orthogonal rotation matrices in $\mathbb{R}^n$, and an iterative perspective projection with dimension-dependent scale compensation, we bridge abstract higher-dimensional topology with computational execution.