to use his tools to invert the semihexagons spacefiller. Unfortunately, Mac problems lost him access to Mathematica for a while. But now he's back with a vengeance. And I don't feel so stupid. The two semihexagons, closed loop & open curve, are defined in terms of each other in a simultaneous recursion. For which Julian's fractal inverter makes no provision. He somehow worked around this with an incredible contortion I will probably never understand. Here it is finding multiple points by inverting the semihexes at successive Wilf-Calkin fractions. (In[182]:= NestList[1/(#/(# - 1) + 2 Floor[1/(1 - #)]) &, 0, 11] Out[182]= {0, 1/2, 1/3, 2/3, 1/4, 3/5, 2/5, 3/4, 1/5, 4/7, 3/8, 5/7} ) foo = 0; Do[ If[Length[invsemi[semihexes[foo = nunu@foo][[1]]]] > 1, Print@{semihexes[foo][[1]], invsemi[Identity[semihexes@foo][[1]]]}], {999}] (Legend: {complex value, preimages}) {1/2+(I Sqrt[3])/2,{1/10,1/4,1/2,13/20}} {1/2+(I Sqrt[3])/2,{1/10,1/4,1/2,13/20}} {-1,{3/4,9/10}} {1+I Sqrt[3],{1/5,3/10}} {5/4+(I Sqrt[3])/4,{27/80,3/8,7/16,19/40}} {-(1/4)+(I Sqrt[3])/4,{47/80,5/8,11/16,29/40}} {-(1/2)+(I Sqrt[3])/2,{1/20,3/40,27/40,7/10,37/40,19/20}} [Sextuple!] . . . Here <http://gosper.org/hex&quad.png> is a triple point on a side of the hexagon, and a quadruple point interior. The interior corner of the little gold trapezoid is either triple(??), quad or hex. Three colors at a vertex guarantees at least a triple point, because spacefilling functions map closed intervals onto closed areas. Julian has shown that the preimages of all the sextuple points have denominators of the form 5⨉2ⁿ. Above, we saw that some quadruple point denominators are just 2ⁿ. But some double point denominators are more exotic: {2/3+(2 I)/Sqrt[3],{14/65,37/130}} {13/10+(I Sqrt[3])/5,{513/1360,37/85}} {23/14+(I Sqrt[3])/7,{41/105,1313/3360}} {17/10+(I Sqrt[3])/5,{33/85,1069/2720}} Uncountably many double points are transcendental, but I don't know how to exhibit any. It would probably involve implementing the semihexes as a finite-state machine, then finding a Thue-Morse-like preimage. . —rwg