Tim Lexen has a created a family of 'triangles' with curved sides that he calls tricurves. What makes tricurves visually and mathematically interesting is their abundant variety and the fact that all sides have the same curvature thus providing an opportunity for various tessellations. There are magnificent mosaics, or tilings, if you prefer, that continue to amaze us.

Other types of tricurves, such as those derived from the Reuleaux triangle and deltoid, will also tessellate, albeit in groups of four tiles. Tessellation with fewer than four different tiles has not yet been demonstrated and is open for discovery.

Our Figure 1 shows a 30-60-90 Tricurve tiling .

The key part is the segment, i.e., the area bounded by a chord and its associated arc. This segment area is the difference between the sector area and the triangular area bounded by the chord and the two radial lines. This is shown below for a 90 degree angle/arc.