Cube Packing Inside a Sphere: A Layer-Based Approach

We present a discrete, axis-aligned cube-packing algorithm that fills a 3-D sphere (and its 2-D analogue, a circle) with high volumetric efficiency. Key points * Layer-by-layer (“slice”) formulation converts the 3-D problem into repeated 2-D square packings. * A strict corner-inclusion test ensures every cube stays completely inside the spherical boundary. * A conditional “> 0.5” centering rule adds offset layers when the fractional radius permits, boosting density. * Exact counting formulas are given in closed form; brute-force simulations confirm densities of 94 % + for r=5, n=0.1. * As n ⁣→ ⁣0 the packing density tends to 1, outperforming classical sphere-packing limits for analogous problems.