### Reflections in the symmetry group 2H_3 of a triangle

The group 2H_3 has six 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=-bab ... blue
r= -a ... orange
r= b, bab=-aba ... light green
r= -b orange.

Each reflection can be visualized by dragging the red vector in the direction 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 triangle. 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_3.

Please enable Java for an interactive construction (with Cinderella).
[ 2D point groups | GA with Cinderella ]

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