FCC-BCC Reciprocal Lattice Duality: A Foundational Fourier-Space Symmetry — E8 Intelligence Research
- 1Independent Researcher, United Kingdom
Abstract
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 a1, a2, a3, reciprocal vectors are b1 = 2π (a2 × a3)/(a1 · (a2 × a3)) (cyclic permutations). For FCC with conventional cubic side a, primitive vectors: a1 = a/2(ŷ+ẑ), a2 = a/2(ẑ+x̂), a3 = 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
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BibTeX
@misc{caldin2026fccbcc,
title = {{FCC-BCC Reciprocal Lattice Duality: A Foundational Fourier-Space Symmetry — E8 Intelligence Research}},
author = {Andrew Stewart Caldin},
howpublished = {Zenodo},
year = {2026},
month = oct,
doi = {10.5281/zenodo.23610273},
url = {https://zenodo.org/records/23610273},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - FCC-BCC Reciprocal Lattice Duality: A Foundational Fourier-Space Symmetry — E8 Intelligence Research
AU - Andrew Stewart Caldin
PY - 2026
DA - 2026/10/24
PB - Zenodo
DO - 10.5281/zenodo.23610273
UR - https://zenodo.org/records/23610273
AB - 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: V_recip = (2π)^3 / V_cell. Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com
ER -