Julian's curvaceous Fourierization of the usual 4-for-1 triangle fill gosper.org/trifil125%25.png looks improbable when polygonally sampled instead gosper.org/polygonal trifil0.png gosper.org/polygonal trifil.png (just standard connect-the-dots plots.) --rwg It appears that the actual spacefill has no triple points-- only singles, doubles, and quadruples! {1/2 + i^(2/3), 1 + i^(2/3)/2, 1/2 + i^(2/3)/2, 3/4 + i^(2/3)} In[257]:= lifirt /@ (% /. i -> I) {{3/8, 11/24, 9/16, 17/24}, {5/16, 5/12, 13/16, 11/12}, {1/12, 3/16, 7/24, 7/16}, {11/32, 35/96, 23/32, 71/96}} The centroid is not even double: In[258]:= lifirt[1 + I/√3] Out[258]= {2/5}