Threefoldsymmetric

Threefoldsymmetric describes the property of possessing threefold symmetry, meaning invariance under rotation by 120 degrees about a fixed axis. If a figure can be rotated by 120 degrees and again by 120 degrees to coincide with its original position, it has a threefold rotational symmetry. The set of these rotations forms a cyclic group of order three, denoted C3 in group-theory terms. In two dimensions the axis is effectively a center of rotation; in three dimensions the axis is a line through the object.

Threefold symmetry can be accompanied by additional symmetries. An object with a threefold rotation axis and mirror planes is said to have C3v symmetry; with more complex reflection arrangements you may encounter D3h or higher symmetry groups. If only rotations are present, the symmetry group is C3; if reflections are included, the group can be dihedral or larger, depending on the exact combination of symmetries.

Examples include an equilateral triangle, which is threefoldsymmetric about its center. A regular tetrahedron has four C3 axes through each vertex and the center of the opposite face. A triangular prism has a C3 axis along its length, around which a 120-degree rotation maps the prism to itself. In crystallography and molecular chemistry, 3-fold axes appear in hexagonal and trigonal systems, and screw axes such as 3_1 and 3_2 combine rotation with translation.

Threefold symmetry is a common motif across geometry, crystallography, chemistry, and design, reflecting a balance of structure around a central axis.