totalPointGroups
plain-language theorem explainer
Recognition Science fixes the total number of crystallographic point groups at 32. Researchers enumerating symmetry classes or deriving space group counts from the eight-tick structure cite this constant when summing across the seven crystal systems. The definition is a direct constant assignment with no computation or lemmas applied.
Claim. The total number of crystallographic point groups is $32$.
background
Crystal symmetry emerges from the 8-tick octave that forces three spatial dimensions. Periodic arrangements in this geometry restrict rotations to orders 1, 2, 3, 4, and 6; 5-fold and 7-fold axes cannot tile space without gaps. These rules plus combinations of rotations, reflections, and inversions produce exactly 32 point groups that cluster into seven crystal systems (triclinic through cubic).
proof idea
Direct definition that assigns the integer constant 32.
why it matters
The constant supplies the right-hand side for the summation theorem across crystal systems and the inequality showing space groups exceed point groups. It realizes the module prediction that the space-filling constraint in three dimensions yields exactly 32 crystallographic point groups.
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