trigonal_fold_from_6
plain-language theorem explainer
The equality 6 divided by 2 equals 3 links six-fold rotational symmetry to the three-fold axis that defines trigonal crystal systems. Researchers classifying the seven crystal systems from three-dimensional space-filling rules would cite this relation when separating trigonal from hexagonal cases. The proof is a direct reflexivity check on the arithmetic identity.
Claim. $6/2 = 3$ shows that a six-fold rotation axis generates a three-fold symmetry element.
background
Crystal symmetry groups arise from the eight-tick structure that forces three spatial dimensions. Periodic unit-cell arrangements in this geometry restrict rotations to orders 1, 2, 3, 4, and 6; five-fold and seven-fold axes cannot tile space without gaps. The trigonal system is defined by exactly one three-fold axis, obtained here by halving the six-fold order that appears in the related hexagonal system.
proof idea
The proof is a one-line reflexivity application on the numerical equality.
why it matters
This step completes the separation of trigonal (one 3-fold axis) from hexagonal (one 6-fold axis) within the derivation of the seven crystal systems. It rests on the eight-tick octave and D = 3 spatial dimensions already established upstream, and supports the module's claim of exactly seven crystal systems and 32 point groups. No downstream uses are recorded.
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