The crystallographic restriction theorem characterizes the possible orders of rotational symmetry a lattice can have, showing that in two or three dimensions rotational symmetry is restricted to two fold, three fold, four fold and six fold. The theorem's name comes from geological crystals, whose rotational symmetries are generally limited to these same values, though quasicrystals with other symmetries, such as five fold, were not discovered until 1982 by Dan Shechtman.
Facts
StatementThe crystallographic restriction theorem states that a discrete two or three dimensional lattice can only have rotational symmetry of order 1, 2, 3, 4 or 6, ruling out an exact 5-fold or 7-fold or higher rotational symmetry for a periodic lattice, though such orders can appear in non-periodic quasicrystals. 1 Classification
Statement FormCharacterization Theorem 1 Connections
Has Statement Form
Entity-backed identity for the statement-form enum value this theorem already carries, resolved to a mathematics concept by an explicit value-to-entity map (phase 3 bucket conversion, docs\design_entity_backed_browse_buckets_20260928.md). The statement-form fact itself stays on the theorem unchanged.
Sources
1. Crystallographic Restriction Theorem (Wikipedia)
Wikimedia FoundationLedeQuote, Lede
The crystallographic restriction theorem characterizes the possible orders of rotational symmetry in a lattice. In 2 or 3 dimensions, the rotational symmetries are restricted to 2-fold, 3-fold, 4-fold, and 6-fold.
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