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A tiling of a 1D line is shown under varying curvature. Move the mouse to change the curvature between spherical geometry (positive curvature) and hyperbolic geometry (negative curvature).

For positive curvatures the tiling lives on a circle. The mapping to the line is provided by stereographic projection from point S.

For negative curvatures the tiling lives on one sheet of a hyperbola. Stereographic projection is still used to map onto the line, giving a 1D version of the Poincaré disk model.

While this 1D case is pretty boring, the details of construction are exactly the same when we move into 2 and 3 dimensions. The details are given below.

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So we want to make a tiling. We're going to use Wythoff construction, where we put down a single tile and then repeatedly reflect it in its edges.

The red line between C and A is our first tile. The first edge, at C, is easy to make a mirror for: the scene is already symmetric around this point so we simply reflect across it. The second edge of the tile, at A, is more tricky.

We project edge A onto our curved surface, to B. The light green line shows the geodesic of B - the plane that passes through the edge at B and the origin at O. All points on this geodesic will be unaffected by the mirroring. In our 1D example the only other such point is D but in 2D there will be a whole circle of such points, and likewise a whole sphere in 3D.

Point D projects onto the stereographic plane at M. Points A and M define a circle of inversion in the stereographic plane that we can use to reflect the tile. It has its center at point K that in this 1D case is halfway between A and M.

We can then reflect the red tile C-A using the circle of inversion A-K-M. This leaves A where it is and puts C onto point G, making our new tile A-G, drawn in blue.

Reflecting the original tile C-A in C gives the dark green tile. Reflecting the dark green tile over A gives the pink tile on the other side of G, and so on.

In 2D and 3D the process is the same - the tile is reflected in the circle (or sphere) of inversion defined by each of its facets. For certain values of curvature the tiles all line up and we get a regular tiling of the hyperbolic plane or of the sphere. In the spherical case we call this a polyhedron or a polychoron.

So what is negative curvature? And why a hyperbola? These questions are addressed in another demo here. There's a 2D version of this here.

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