This work introduces Hyperspheric Chemistry, a new theoretical framework in which atoms, orbitals, bonding, chirality, and chemical reactivity are understood as 3D cross-sections of higher-dimensional hyperspheres (S³ and S⁴). In this model, electron shells correspond to nested hyperspheric resonance layers, and chemical bonds arise from overlapping hyperspheric intersections that form coherent phase regions. Orbital shapes, hybridization angles, and molecular geometries emerge directly from deformation modes on the hypersphere. Chirality is a natural consequence of S³ intrinsic handedness. Reaction pathways, transition states, and catalysis follow from higher-dimensional curvature changes and ∇T² minimization. Hyperspheric Chemistry unifies atomic structure, molecular bonding, hybridization, chirality, and biological complexity under a single multidimensional resonance framework.