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Dicyclic (point) group of order 4: 2C_2

- oriented rotations of a dumbbell

The group 2C_2 has 2 rotations x->x'= R~xR (aa=1, bb=1) with "positive" sense represented by the rotors R:

1=(ab)^0, ab

and two corresponding rotations with "negative" sense represented by the rotors R:

ba=-(ab), (ba)^2=(ab)^2=-1.

Here the rotations are realized by double reflections x->x' -- similar to the applet
Rotations by reflection.
The results after the first reflection (at the blue line through a) are not shown explicitly.
The final results x' after the second reflection (at the red line of reflection) are shown as rings of the
corresponding color marked by numbers with a prime.
*One can "see" all the above 4 rotations by interactively changing the
position of the bright red point* (this changes the second red line of reflection).
We get symmetries only when the red vector coincides with one of the [positive] vectors
a or b,
or
one of the two orange [negative] vectors.
These symmetry operations are in 1:1 correspondence with the rotors listed above.
The single reflections, which pairwise yield
the rotations x->x' are members of the symmetry group of a dumbbell.

[ 2D point groups | GA with Cinderella ]

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