### Reflections in the symmetry group 2H_5 of a pentagon

The group 2H_5 has ten distinct (oriented) reflections x -> x' = rxr, rr=1. Each reflection is characterized by a 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, a(ba)(ba)=-b(ab)(ab) ... blue
r= -a, -aba ... light blue
r= b, bab, b(ab)(ab)=-a(ba)(ba) ... light green
r= -b, -bab ... green.

Each reflection can be visualized by dragging the red vector perpendicular to the line of reflection to a particular value of r. It is easily verified that any other choice of r, not in the above list, will not produce a reflection symmetry of the pentagon. 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_5.

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Remark: Instead of r we could also use the dark red vector s = r I2 (where I2 is the [positively] oriented pseudoscalar of the plane) to describe the reflections instead by x-> x'=-sxs = rxr. s is perpendicular to the line of reflection. This description seems indeed to have been chosen by D. Hestenes in his paper "Point Groups and Space Groups in Geometric Algebra".

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