Symmetry orbits of the triangle

These six images

illustrate the symmetry orbit of the three-color triangle. Note that:

• Any two triangles in different rows are single-reflection images of each other;
• Any two triangles in the same row are double-reflection images of each other;
• Any two triangles in the same row are rotated images of each other.

As a consequence, the symmetry orbit of the triangle is generated by single and double reflections, or, equivalently, by reflections and rotations. 

The separation angle between adjacent reflection axes in the triangle is 2Pi/6 and is referred to as its dihedral angle. The resulting rotations are in angles that are multiple of twice the dihedral angle.

The rotations in one row are in the opposite direction relative to the rotations in the other row. This may hint us to connecting symmetry orbits with space orientation: For example, the arms of  a transparent clock hanging in a transparent wall rotate in opposite directions when viewed from opposing sides of its wall. Conversely, your position relative to the wall may be determined or label by the direction of rotating arms.
More to come on this.

Any one of the six triangles generates all the remaining ones after the reflections and iterated reflections are applied to that initial triangle. Therefore, the initial choice, or the generating element, is arbitrary. This property of arbitrariness is present in any symmetry orbit and will be explored later on in future postings.

Last revised of 05/14/2011
These  postings are based on "Symmetry Studies  An  Introduction to the Analysis of Structured Data in Applications"  Cambridge Press (2008)