. . . . "A regular point assemblage (lattice) can possess only axes of binary, ternary, quaternary or senary symmetry." . "theoretical" . . "Formalization: one. The crystallographic restriction. A Lean target: 2 cos(2 pi/n) in Z implies n in {1,2,3,4,6}." . "Related claim (premise-of): Thompson's argument that the regular dodecahedral and icosahedral radiolarian shells are not crystal-like space partitions takes the crystallographic restriction as its premise: five-fold axes cannot occur in a lattice, so a regular pentagonal dodecahedron in an organism must arise another way. The organism is the contrast case, not a counterexample to the theorem." . "A lattice can only have two-, three-, four- or six-fold symmetry axes" . "n \\in \\{1,2,3,4,6\\}" . . "Bravais A, “Mémoire sur les systèmes formés par des points distribués régulièrement sur un plan ou dans l'espace”, Journal de l'École Polytechnique 19 (cahier 33), 1850; read in Études cristallographiques, Gauthier-Villars, Paris, 1866 (1850); Mémoire sur les systèmes de points, théorème XLVI, p. 73. Scope: All three-dimensional lattices." . "Bravais A. Mémoire sur les systèmes formés par des points distribués régulièrement sur un plan ou dans l'espace. Journal de l'École Polytechnique 19 (cahier 33), 1850; read in Études cristallographiques, Gauthier-Villars, Paris, 1866 (1850), 1-128" . "Auguste Bravais" . "1850"^^ . "The posthumous 1866 collection reprints the 1850 memoir on lattices and the 1851 memoir on polyhedra of crystalline form. French." . "Public domain (1850; collection 1866)." . "Mémoire sur les systèmes formés par des points distribués régulièrement sur un plan ou dans l'espace" . . . . "Reading copy of Mémoire sur les systèmes formés par des points distribués régulièrement sur un plan ou dans l'espace" . . . "delphisci crystals-and-form claim extractor" . . . . "Claude Fable 5.1" . . . . . . . . . . . . "Page 73 of the reading copy; paragraph 1443 of the normalised text 1850-bravais.txt (sha256 d7a7910b838d96a1c4b37d979c0ba877c254fd83a256922580fa367c168d3cbd), produced by build/fetch.py and checked by build/check-quotes.mjs. The text itself is not committed." . . . "Un Assemblage ne peut posséder que des adcs de symétrie binaire, ternaire, quaternaire ou sénaire." . . "d7a7910b838d96a1c4b37d979c0ba877c254fd83a256922580fa367c168d3cbd" . "Checksum of the normalised one-paragraph-per-line text derived from the reading copy, not of the reading copy itself." . "crystals-and-form" . "2026-11-07T06:06:25.330Z"^^ . . . . . . "Auguste Bravais 1850 claim: A lattice can only have two-, three-, four- or six-fold symmetry axes" . . "RSA" . "MIIBIjANBgkqhkiG9w0BAQEFAAOCAQ8AMIIBCgKCAQEAyFHtPB6p0mCjDozxlOY9CmO3nBm7Te3jFbXyLmM+G1tGe47Da0jYIbhkl9Oo1ZjxmCa3LCwlEqK8gj6hEIcL4ahW30JXh9HycfKVk1bugkzMEV15tJdTzAU3sGMFnbNsn43xY2LFLy45tApkRm5byEiM7oRRrooIdwoQKnP4Wzq/bfAoJ9qvW+MGkelyOTd1jbSt+KfmT7xKmDfWzAHddUlWwMX2QJOLxDaffOteTop6ZRgPiNmPJKA/S62Gac9XtSYT3oDkZ6/nr5cxxNLpOyxTf5FPISiCe11xYO/unYjY+mwnOfA8yhJIyx1l0O9I0vpOpn+E5TvyHJV+oSTrvwIDAQAB" . "qxhsMGEpCQ+0BxI7n9p4t6gAIjDxGCgzTVUnej2F0GISbtuwiIkuWP/36hh+HaUq9ked9tV+vrDI4MRGIm9oAEoWDDNBFy72o5mnqd7Pbio6RQdgQsWWgy0PcMgQE6Zaji49zVDJmISrfLQB6kefvS2/kVbIpB67vuccmIlkifEB/UuEC7kAfpcE3v8c6c4nQNth/6etxStADe27t4h/ypR0DGARjPMsxZk87rCO7nUfrO7keKWZSVMwi0EGlTadRu4fDgzj7KVW+CVq+wgOYs5caiR6MdgZlYQ5E1tG8oCiZzclFp66sfYaj1Hv6c+SC6bm0YQayoHBI8Yp3fK8og==" . . .