Hyperbolic tilings

Set q to 3, then add triangles at each corner one at a time. With three, four or five they close up around a sphere, as the tetrahedron, octahedron and icosahedron (seen here in stereographic projection); six lie exactly flat; from seven on, each angle has to be smaller than 60°, which only triangles in the hyperbolic plane can manage. In the Poincaré disc they are all the same size, only looking smaller towards the rim: walk along the axis and a small triangle from the rim comes to the middle at full size. Change p to use squares or pentagons instead.