Consider the following frieze:
How would you find the group isometries of the plane preserving this frieze?
I'm not really sure how to do this, but I assume that it would help to start by ladling the triangles and adding coordinates, as follows:
Here, to get from triangle A to triangle B, a rotation $R_{n}$ of $\pi$ rads about the point $(1,0)$ would be required, so I suppose this would be one isomorphism.
Furthermore, getting from triangle A to triangle C would require a translation $T_{n}$ of 2 units in the $x$ direction, which could be expressed as $T_{n}(x,y)=(x+2, y)$.
How would you express the rotation in the form $R_{n}(x,y)=(f(x), f(y))$?
Are there any other isometries that I've missed?