A Note on "Quasicrystal Symmetry-Angle Frequencies and Tests of GEIER’s n×6° Rule or GEIER’s nx(2xand,or3)° Rule and Φ-Related Geometry; Part 4.1 of ‘Crystallography and GEIER’s n×6° Rule’"
A Note on "Quasicrystal Symmetry-Angle Frequencies and Tests of GEIER’s n×6° Rule or GEIER’s nx(2xand,or3)° Rule and Φ-Related Geometry:
GEIER’s nx6° Rule ↔ GEIER’s nx(2 xand,or 3)° Rule;
GEIER’s nx6° Rule ↔ GEIER’s nx(2 xand,or 3)° Rule;
GEIER’s nx6° Rule (Crystals C) ↔ (Quasicrystals and C) GEIER’s nx(2 xand,or 3)°
Rule;
Part 4.1 of ‘Crystallography and GEIER’s n×6° Rule’"
From a technical, review-style perspective, the manuscript excels at turning the GEIER grid hypothesis into a clear, falsifiable arithmetic test on an openly documented benchmark (Chang LIU et al. Supplementary Table S1; n = 159; see op.cit.), with the symmetry order n mapped transparently to the fundamental increment θ=360∘/n before evaluating 6°/3°/2° divisibility.
Reference:
GEIER, Stefan et al.: “Quasicrystal Symmetry-Angle Frequencies and Tests of GEIER’s n×6° Rule or GEIER’s n×(2 xand,or 3)° Rule and Φ-Related Geometry: Part 4.1 of ‘Crystallography and GEIER’s n×6° Rule’.” February 2026. DOI: 10.13140/RG.2.2.20360.28168. Available at ResearchGate: https://www.researchgate.net/publication/400747533_Quasicrystal_Symmetry-Angle_Frequencies_and_Tests_of_GEIER's_n6_Rule_or_GEIER's_nx2xandor3_Rule_and_PH-Related_Geometry_Part_41_of_'Crystallography_and_GEIER's_n6_Rule'
The quantitative reporting is unusually disciplined for an angle-regularity claim: beyond the headline coverage rates (e.g. 156/159 on the 6° grid, 159/159 on the 3° grid), GEIER et al. also provide exact CLOPPER–PEARSON binomial confidence intervals and explicitly emphasize the curated, non-random nature of the literature list, treating the statistics as descriptive rather than over-interpreting them as population-level bounds. I also found the structure-to-interpretation pipeline well designed: per-class tables (n, θ, pass/fail) make the lone octagonal 45° exception to the 6° and 2° grids immediately legible, thereby motivating the refinement to the n×(2 ×and/or 3)° framing in a logically minimal way rather than via ad hoc “patching.”
Finally, the discussion sustains a productive balance between mathematical parsimony (e.g. the GCD perspective on canonical angle sets) and physically meaningful context (with Φ emerging via pentagonal trigonometry and Roger PENROSE-type length ratios), while consistently treating the rule as a compact descriptive mapping rather than an asserted microscopic mechanism—a restraint that substantially enhances the manuscript’s credibility and usefulness for future dataset extensions.
GEIER’s Equations as the basis of GEIER’s n×6° Rule furthermore allow a symmetry- and equilibrium-based physical near-mechanistic explanation that can be developed in greater detail in the broader series.
MGN
Reference:
GEIER, Stefan et al.: “Quasicrystal Symmetry-Angle Frequencies and Tests of GEIER’s n×6° Rule or GEIER’s n×(2 xand,or 3)° Rule and Φ-Related Geometry: Part 4.1 of ‘Crystallography and GEIER’s n×6° Rule’.” February 2026. DOI: 10.13140/RG.2.2.20360.28168. Available at ResearchGate: https://www.researchgate.net/publication/400747533_Quasicrystal_Symmetry-Angle_Frequencies_and_Tests_of_GEIER's_n6_Rule_or_GEIER's_nx2xandor3_Rule_and_PH-Related_Geometry_Part_41_of_'Crystallography_and_GEIER's_n6_Rule'
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