| Regular triangles (n=3) can tesselate the plane,
m of them meeting at a point, provided that 1/3 +
1/m < 1/2.
Hence m can take all integer values from 7 on. In the figure shown, move the point P until it (just about) coincides with the point a. You should see that the plane can be tiled by regular triangles, 8 meeting at a point. Continue to move the point P in towards the centre of the disc, until P and the point b coincide. This should convince you that we can have a tiling with 7 triangles meeting at a point. This m is the least we can have. Moving P further in towards the centre shows that we can never quite tile the plane with 6 triangle at a point ( the only Euclidean case) although, as the triangles become smaller, we seem to approach the Euclidean case. Moving P further out towards the edge of the disc should convince you that enough 'space' opens up for tesselations with m = 9, 10, 11, ..... and so on. |