Reflecting a Regular Polygon across Its Sides in the Hyperbolic Plane

In the hyperbolic plane, given and , there is a unique positive real number such that the regular -gon of side length tiles the whole plane, with copies of touching at each vertex. This Demonstration shows how the tiling fails when the side length differs from .

Select the number of sides of and choose its side length via the Euclidean radius of its circumscribed circle. Also select the number of stages you want to see; by default, you see only the central tile and its reflections over its sides. By increasing the number of stages , you can also see the reflections of reflections (and so on) of the central tile over its sides.

The button "tile" calculates such that the graphic is indeed the tiling. Other values cause the reflections to have gaps or to overlap.