The structure of crystal cells is essential to describe material properties of many elements, e.g. aluminium, copper or iron. One can define the structure of a crystal cell in 3D by stating three axes and their angles or by considering its symmetries. The latter form a group in mathematical sense and are called space groups (or crystallographic groups).
The symmetry operations of a crystal cell can be described with basic linear algebra and simple matrix multiplications or vector additions. However, with the theory of conformal geometric algebra one can also generate all symmetry operations (including translations) with a number of reflections at specific hyperplanes. The Space Group Visualizer software utilizes this theory to perform symmetry operations in a homogenous way.
For more on geometric algebra and its application in space groups visit the website of GA-Net.