due 4apr
square:  c=16
              ratio of c:d  :  4:1
             verification: c=(pi)d
                                16=(4)4

triangle: 12:4=3
hexagon:ratio:3

To expand on this idea, are there non-euclidean spheres?

The reason there are only 3 shapes that tesselate is they are the only shapes whose interior angle measures divide evenly into 360.