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Dicyclic (point) group of order 2: 2C_1

- oriented rotations of a non-symmetric figure

The group 2C_1 has 2 rotations x->x'= R~xR (aa=1, bb=1). One with "positive" sense represented by the rotor

R = 1 = (ab)^0

and one corresponding rotation with "negative" sense represented by the rotor

R = ba = -ab = -1.

Here the rotations are realized by double reflections x->x' -- similar to the applet
Rotations by reflection.
The final result x' after the second reflection (at the red line of reflection) is shown as another blue ring
marked by "1'".
*One can "see" the above 2 rotations by interactively changing the
position of the bright red point* (this changes the second red line of reflection).
We only get a symmetry when the red vector coincides with the [positive] blue vector a,
or with the [negative] green vector b.
These 2 symmetry operations are in 1:1 correspondence with the 2 rotors listed above.
Combining the 2 single reflections, we get
the (identity) rotations x->x'=R~xR (=x) of the
symmetry group of a non-symmetric figure (R=+1, R=-1).

**Remark:** What "puzzles" me is that in the above figure a reflection about the a-line leaves the figure invariant.
It seems therefore that the object is not completely void of symmetry. If you have any idea about this, please tell me:
Eckhard Hitzer

[ 2D point groups | GA with Cinderella ]

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