Theorem

Crystallographic Restriction Theorem

Geometry

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

Statement

The 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 Form

Characterization Theorem 1

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Characterization

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Geometry

Source Crystallographic Restriction Theorem (Wikipedia)

Sources

1. Crystallographic Restriction Theorem (Wikipedia)

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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.
In Branch: Geometry, Lead section
restriction theorem characterizes the possible orders of rotational symmetry in a lattice. In 2 or 3 dimensions, the rotational symmetries

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Crystallographic Restriction Theorem

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