4mmm

4mmm is a symbol that appears in crystallography and solid-state chemistry and is used to denote symmetry related to tetragonal systems. In standard notation, the symbol is part of a family that includes both C4v and D4h point groups, which are distinguished by the presence or absence of certain symmetry elements.

The C4v interpretation is often written as 4mm in Hermann–Mauguin notation and corresponds to the point group with a fourfold rotation axis and four vertical mirror planes. The D4h interpretation is written as 4/mmm in Hermann–Mauguin notation and corresponds to a higher-symmetry group that includes, in addition to the C4v elements, a horizontal mirror plane and an inversion-centered operation. Because some texts omit punctuation or use 4mmm informally, the symbol can be ambiguous and its meaning may vary by source.

In practical terms, 4mm (C4v) applies to objects with a single fourfold axis and vertical mirror symmetry, such as certain molecular geometries and some crystal surfaces. 4/mmm (D4h) applies to higher-symmetry objects in which horizontal symmetry and additional operations are present, a common situation in many tetragonal crystals.

The distinction matters for applications such as vibrational spectroscopy and group-theoretical analysis, where the symmetry governs selection rules and the degeneracy of representations. Each interpretation yields a different character table and different predictions for infrared or Raman activity.

See also: point group, crystallography, Schoenflies notation, Hermann–Mauguin notation, symmetry operations.