### Reflections in the symmetry group 2H_4 of a square

The group 2H_4 has eight distinct (oriented) reflections x -> x' = rxr, rr=1. Each reflection is characterized by the unit direction vector r of the line of reflection. These reflections are oriented, because each line of reflection has two possible opposite directions, characterized by the sign of the unit direction vector r.
We have:
r = a, aba ... blue
r= -a, -aba ... light blue
r= b, bab ... green
r= -b, -bab ... orange.

Each reflection can be visualized by dragging the red unit direction vector of the line of reflection to a particular value of r. It is easily verified that other choices of r, not in the above list, will not produce a reflection symmetry of the square. The combination of two such reflections at r and r' will yield an (oriented) rotation y->R~yR=(r'r)y(rr') from the dicyclic group 2C_4.

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[ 2D point groups | GA with Cinderella ]

Soli Deo Gloria. Created with Cinderella by Eckhard Hitzer (Fukui).