fourfoldsymmetric

Fourfold symmetry, or fourfold rotational symmetry, is a property of an object or pattern that is invariant under a rotation of 90 degrees about a central axis. Consequently, the object looks the same in four orientations: 0°, 90°, 180°, and 270°. The rotation group generated by 90-degree rotations is a cyclic group of order four, denoted C4. If reflections are also symmetry operations, the full symmetry group may be larger, such as the dihedral group D4 for square-shaped objects.

In geometry, any square and any circle-like pattern composed of four identical quadrants can exhibit fourfold symmetry. The Greek cross and plus sign are common simple examples. In crystallography, fourfold rotational symmetry is characteristic of the tetragonal crystal system and of certain Bravais lattices, where the lattice is invariant under a 90-degree rotation around a principal axis.

Fourfold symmetry is distinct from bilateral symmetry or from higher-order rotational symmetry; however, objects with fourfold symmetry may also be symmetric under additional rotations (e.g., 180°) or reflections.

Applications of fourfold symmetry appear in pattern design, tiling, crystallography analysis, and molecular chemistry to describe stereochemistry. Limitation: real materials may exhibit approximate rather than perfect fourfold symmetry due to imperfections.