Three mirrors

Choose each of the three angles, every one of them 180° divided by a whole number: two mirrors meet at each corner, and any other angle would leave a seam there. When the angles add up to exactly 180°, the reflected triangles fill the plane, and only three triangles do: 60° · 60° · 60°, 90° · 45° · 45° and 90° · 60° · 30°. When they add up to more, no such triangle can be drawn on the plane, so the three mirrors are set up through the centre of a sphere, and the triangles they reflect close up on it; the count in the corner, made by generating the reflections one by one, is exactly 720° divided by the excess. When they add up to less, the triangle fits only on the hyperbolic plane, drawn here as a disc, where its straight lines, the mirrors among them, appear as arcs; the triangles look smaller and smaller towards the rim, yet in that geometry every one of them is the same size.