Tiling H2 with regular figures

    A  regular tessellation (n, m) denotes a possible tiling of the plane by  regular  n-gons (n-sided figures with equal sides and angles) where m of them meet at a point. In H2 the following remarkable result holds.
     
    There is a regular (n, m) tessellation of H2  for all integers n and m such that  1/n+1/m < 1/2   (*)
     

    Regular Triangles (n=3)

    In the Euclidean plane there is only one regular triangle, the equilateral triangle, and we can fit 6 of them about a point. This is the only regular triangle tesselation possible. Regular triangles in H2 are different in that the three equal angles can take on all values from 0 up to (but not equal to) 60 degrees. From (*) we see that a regular tesselation (3, m) exists for all integer m from 7 onwards.
     
    The animated gif is taken from a Cabri figure constructed by drawing a regular triangle (using the command 'non E regular triangle') with vertex at the origin of the disc and then reflecting this triangle (using 'edge reflect') about an edge and continuing the process with  the resulting image triangles, all of which are congruent to the first. The centre point P of the first triangle can be moved in towards the centre. As this happens the point a1 approaches a2 and b1 approaches b2. When they meet we have a regular tessellation (3, 8), eight regular triangles meeting at a point. Moving P further in towards the centre cause some triangles in the construction to overlap until we reach the point of the (3, 7) tessellation. 
    This is as far as we can go, (3, 6) being a  tessellation of the Euclidean plane by regular triangles (the only one), but not of the hyperbolic plane since 1/3 +1/6 = 1/2.

    Download

     You can download a Cabri file for this construction.  tri.tess.fig

    Proof

    While the construction is a good visual confirmation of the result (*) the proof  that (*) holds in H2 is fairly straightforward.