FCC-BCC Reciprocal Lattice Duality: A Foundational Fourier-Space Symmetry — E8 Intelligence Research

FINDING: Reciprocal lattice duality between FCC and BCC is a foundational Fourier-space symmetry, but the search results are pedagogical videos and one lattice QCD paper — no new mathematical discovery is presented. The core structural fact: the reciprocal of an FCC lattice is BCC, and vice versa, with a scale factor of \(2\pi\) and a volume ratio of \(2\). MATH: For a direct lattice with primitive vectors \(\mathbf{a}_1, \mathbf{a}_2, \mathbf{a}_3\), reciprocal vectors are \[ \mathbf{b}_1 = 2\pi \frac{\mathbf{a}_2 \times \mathbf{a}_3}{\mathbf{a}_1 \cdot (\mathbf{a}_2 \times \mathbf{a}_3)} \] (cyclic permutations). For FCC with conventional cubic side \(a\), primitive vectors: \[ \mathbf{a}_1 = \frac{a}{2}(\hat{y}+\hat{z}),\quad \mathbf{a}_2 = \frac{a}{2}(\hat{z}+\hat{x}),\quad \mathbf{a}_3 = \frac{a}{2}(\hat{x}+\hat{y}) \] Reciprocal lattice is BCC with cubic side \(4\pi/a\). Conversely, BCC reciprocal is FCC. Volume ratio: \(V_{\text{recip}} = (2\pi)^3 / V_{\text{cell}}\).

Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com