Controls: Move the mouse up and down to move the viewpoint up and down. Click to stop and start the spinning. Use the slider to change the curvature. Press the button to toggle between a fixed scale and a scale that keeps the edge length fixed.

This is a demonstration of the connection between hyperbolic plane tiling and the polyhedra. By varying the curvature we can smoothly change between the hyperboloid model (projecting onto the Poincaré disk) and the sphere (projecting onto the stereographic plane).

The edge points A and A' are projected onto the curved surface at B and B' using stereographic projection from point S.

Points B and B', together with the center of the curved surface at O, define a plane which intersects the curved surface to give the geodesic - the shortest path across the curved surface between B and B' is shown in green. For a sphere this is part of a great circle. For the hyperboloid it is part of a hyperbola.

Reflecting B and B' in point O gives the antipodal points D and D' that lie on the same plane. When the curvature is negative they lie on the upper sheet of the hyperboloid.

We project D and D' onto the plane at M and M'. We now have four points A, A', M, M' that define a circle. Arcs in this circle are straight lines in the stereographic plane.

We can use circle inversion to mirror things across the edges. We reflect our central tile across each of its edges in this way.

To see the construction more clearly, there's a 1D version here. To understand why negative curvature leads us to a hyperboloid, there's an explanation here. For exploring the many tilings of the sphere and hyperbolic plane, including hyperbolic space tessellations, there's a three-slider demo here: here.

For more demonstrations, see the hyperplay page: https://github.com/timhutton/hyperplay.