Magnetic behavior across Fe₂O₃ polymorphs varies widely despite identical chemistry, highlighting crystal architecture as a key determinant of exchange topology, magnetic anisotropy, and ultimately magnetic order. Here, using neutron and synchrotron X-ray diffraction, we establish the magnetic ground state of the poorly understood bixbyite β-Fe₂O₃ polymorph and uncover the structural origin of its strong frustration. Below the Néel temperature, a noncollinear antiferromagnetic state emerges through activation of the mH₁⁺ irrep at the H-point [k=(1,1,1)] and the antitranslation (1'|1/2,1/2,1/2), breaking the body centering and yielding two interpenetrating primitive cubic magnetic subcells with inverted moments and nonpolar type-IV symmetry. Under exclusively antiferromagnetic Fe³⁺-O-Fe³⁺ interactions, β-Fe₂O₃ exhibits a large frustration index (f ≈ 7.56). This behavior originates from the intrinsic geometry of the bixbyite lattice: two magnetic sublattices with distinct point symmetries and anisotropy constraints are embedded in a three-dimensional exchange network containing interconnected triangular and hexagonal motifs. In {111} planes, Fe2 ions form hexagonal rings interconnected by frustrated triangular units, while locally Ising-like Fe1 ions occupy the ring centers. Our results thus identify the bixbyite architecture as a promising general platform for frustrated noncollinear magnetism. Extending this structural framework to other magnetic transition-metal or 4f ions opens a materials space for engineering competing exchange interactions and anisotropies, potentially stabilizing new noncollinear and field-tunable magnetic states.