I can't find such a group of symmetries in 2 dimensions. I only know of so many groups of symmetry that are of infinite order (circle, line, infinite dihedral group), but I can't figure out how I can combine a reflection (order 2) and a rotation (an order 4 rotation) to produce the entire group. It always seems to result in only 8 symmetries, not infinite.
Maybe there is something in 3 dimensions that works?
Best Answer
Start with the right isoceles triangle $$T = \{(x,y) \in \mathbb R^2 \mid x \ge 0, y \ge 0, y \le 1-x\} $$ Let $P = (0,0)$ which is the location of the right angle.
Now take the group $G$ generated by the rotation of angle $\pi/2$ centered on $P$, with order $4$; and the reflection across the hypotenuse, with order $2$. This group $G$ is infinite, and it is a symmetry group of an infinite tiling of the plane by right isosceles triangles.
This group $G$ is not the full symmetry group of the tiling depicted in that link, because there is an additional dihedral symmetry of the triangle $T$ itself which is not in $G$.
There's two things you could do about that.
One is that you could enlarge $G$ by adding that dihedral symmetry; you would still get an infinite symmetry group containing an order 4 rotation and an order 2 reflection, although those two elements would no longer generate $G$.
As an alternative, you can break the dihedral symmetry by adding a little non-dihedral decoration inside $T$, such as a little spiral, and then transport that decoration over the entire tiling by the action of $G$. The group $G$ is now the full symmetry group of the decorated tiling.