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 \(2π\) and a volume ratio of \(2\). MATH: For a direct lattice with primitive vectors \(a_1, a_2, a_3\), reciprocal vectors are \[ b_1 = 2π a_2 × a_3/a_1 · (a_2 × a_3) \] (cyclic permutations). For FCC with conventional cubic side \(a\), primitive vectors: \[ a_1 = a/2(ŷ+ẑ), a_2 = a/2(ẑ+x̂), a_3 = a/2(x̂+ŷ) \] Reciprocal lattice is BCC with cubic side \(4π/a\). Conversely, BCC reciprocal is FCC. Volume ratio: \(Vrecip = (2π)^3 / Vcell\). Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com