What follows is a proof of the symmetry operator for a threefold rotation in a Cartesian coordinate system for those who like mathematics: this page is very much supplementary course material.

Since z coordinate is unaffected by the rotation, one needs to consider only the effect on the (x, y) coordinates in the XY plane as shown in the figure below:

It is simplest to work in polar coordinates, so the initial coordinates (x, y) coordinates can be written as:

After rotation by 120°, the new coordinates can be written simply using polar coordinates as:

Expanding the double-angle sine and cosine terms gives:

Substituting for rcosθ (= x) and rsinθ (= y) gives:

Finally, substituting values for sin120° (= √3/2) and cos120° (= −1/2), and reordering gives:

quod erat demonstrandum

In addition, the symmetry operator for a threefold rotation may be represented in matrix form, both for Cartesian:

and for crystallographic coordinate systems:

An extended form of the latter (to include translational components) is used by virtually all crystallographic computer software.

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