Reflecting a Regular Polygon across Its Sides in the Hyperbolic Plane

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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
.
Contributed by: Gasper Zadnik (April 2013)
Suggested by: George Beck
Open content licensed under CC BY-NC-SA
Snapshots
Details
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.
Permanent Citation