quality:corruption
Symmetry

The square has Dih symmetry, order 8. There are 2 dihedral subgroups: Dih, Dih, and 3 cyclic subgroups: Z, Z, and Z.

A square is a special case of many lower symmetry quadrilaterals:

These 6 symmetries express 8 distinct symmetries on a square. John Conway labels these by a letter and group order.

Each subgroup symmetry allows one or more degrees of freedom for irregular quadrilaterals. r8 is full symmetry of the square, and a1 is no symmetry. d4 is the symmetry of a rectangle, and p4 is the symmetry of a rhombus. These two forms are duals of each other, and have half the symmetry order of the square. d2 is the symmetry of an isosceles trapezoid, and p2 is the symmetry of a kite. g2 defines the geometry of a parallelogram.

Only the g4 subgroup has no degrees of freedom, but can be seen as a square with directed edges.