Reflecting a Regular Polygon across Its Sides in the Hyperbolic Plane

polygon sides n

3

4

5

6

7

8

9

number of reflections k

1

2

3

4

radius r

0.3

tile

In the hyperbolic plane, given

m

and

n

, there is a unique positive real number

s

such that the regular

n

-gon

P

of side length

s

tiles the whole plane, with

m

copies of

P

touching at each vertex. This Demonstration shows how the tiling fails when the side length differs from

s

.