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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.

[ 2D point groups | GA with Cinderella ]

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