Polyhedra in the mirrors

Drag the gold point P in the small picture on the left. It shows the triangle on the sphere that holds P, drawn flat, with its angles marked as they are on the sphere; its three sides are three mirrors. P and all its images in the mirrors are the vertices of the polyhedron on the right. Each edge from P joins it to its image in one mirror and is twice as long as P's distance from that mirror, the coral line; every other edge is a reflection of these. The seven faint dots in the small picture are the triangle's three corners, one place on each side and the middle: only at these seven does every edge come out the same length, and when you drag close to one, the point snaps to it and the lower right names the polyhedron. Switch to another kaleidoscope, with one angle of the triangle at 60°, 45° or 36° instead, and the same seven places give other solids. Wherever the point is, vertices minus edges plus faces comes to 2.