So, e.g., the last figure of the first paper describes a "twicefilling" function, hitting every interior point at least twice by retracing most of itself. (Back in the 70s, Robert Maas called this "sausaging a tree".) But then we seem to have 12th order points!? It would be amazing to see piecewiserecursivefractal cough up a twelvetuple. On 2016-08-22 00:26, Joerg Arndt wrote:

For similar curves (that exist for orders that are odd squares) on the (3.6.3.6)-grid see http://jjj.de/3frac/p6/all-r25-b-curve-decompositions.pdf (last two pages) and http://jjj.de/3frac/p6/all-r25-b-tiles.pdf (last four pages).

Sextuple and triple points are easy to see in these.

I cannot decode the Mathca, does you curve live on the (3.6.3.6)-grid as well (morally it should)?

Best regards, jj

I don't understand your grid notation, but it should be clear from the png. If you missed the url, you should be impolitely questioning my "little white hexagons". Do my posts usually sound this delusional? --rwg

* Bill Gosper <billgosper@gmail.com> [Aug 22. 2016 08:17]:

...

using a 25-way conditional. ClearAll[piecewiserecursivefractal]; piecewiserecursivefractal[x_, f_, which_, iters_, fns_] := piecewiserecursivefractal[x, g_, which, iters, fns] = ((piecewiserecursivefractal[x, h_, which, iters, fns] := Block[{y}, y /. Solve[f[y] == h[y], y]]); Union @@ ((fns[[#]] /@piecewiserecursivefractal[iters[[#]][x], Composition[f, fns[[#]]], which, iters, fns]) & /@which[x]));

(I cannot praise this function enough.)

In[363]:= trifil25[t_] := piecewiserecursivefractal[t, Identity, Piecewise[{{{1}, #1 == 0}, {{Ceiling[25*#1]}, Inequality[0, Less, #1, LessEqual, 1]}}, {}] & , {25*#1 + 0 & , -25*#1 + 2 & , 25*#1 - 2 & , 25*#1 - 3 & , 25*#1 - 4 & , 25*#1 - 5 & , 25*#1 - 6 & , 25*#1 - 7 & , 25*#1 - 8 & , -25*#1 + 10 & , -25*#1 + 11 & , 25*#1 - 11 & , -25*#1 + 13 & , -25*#1 + 14 & , 25*#1 - 14 & , -25*#1 + 16 & , 25*#1 - 16 & , -25*#1 + 18 & , 25*#1 - 18 & , 25*#1 - 19 & , -25*#1 + 21 & , -25*#1 + 22 & , 25*#1 - 22 & , -25*#1 + 24 & , 25*#1 - 24 & }, {#1/5 & , (1/10)*(3 + I*Sqrt[3] + (-1 - I*Sqrt[3])* #1) & , (1/10)*(3 + I*Sqrt[3] + #1 - I*Sqrt[3]*#1) & , (2 + #1)/5 & , (1/10)*(6 + #1 + I*Sqrt[3]*#1) & , (1/10)*(7 + I*Sqrt[3] + I*(I + Sqrt[3])*#1) & , (1/5)*(3 + I*Sqrt[3] - #1) & , (1/10)*(4 + 2*I*Sqrt[3] + (-1 - I*Sqrt[3])*#1) & , (1/10)*(3 + I*Sqrt[3] - 2*#1) & , (-(1/10))*I*(-I + Sqrt[3])*(-2 + #1) & , (1/5)*(2 + I*Sqrt[3] - #1) & , (1/5)*(2 + I*Sqrt[3] + #1) & , (1/10)*(5 + 3*I*Sqrt[3] + #1 - I*Sqrt[3]*#1) & , (1/10)*(4 + 2*I*Sqrt[3] + #1 + I*Sqrt[3]*#1) & , (1/10)*(4 + 2*I*Sqrt[3] + I*(I + Sqrt[3])*#1) & , (-(1/10))*I*(-I + Sqrt[3])*(-4 + #1) & , (1/5)*(2 + 2*I*Sqrt[3] + #1) & , (1/10)*(5 + 3*I*Sqrt[3] + #1 + I*Sqrt[3]*#1) & , (1/10)*(5 + 3*I*Sqrt[3] + 2*#1) & , (1/10)*(7 + 3*I*Sqrt[3] + (-1 - I*Sqrt[3])*#1) & , (1/5)*(4 + I*Sqrt[3] - #1) & , (1/10)*(9 + I*Sqrt[3] + I*(I + Sqrt[3])*#1) & , (1/10)*(9 + I*Sqrt[3] - 2*#1) & , (1/10)*(8 + I*(I + Sqrt[3])*#1) & , (4 + #1)/5 & }][[1]] E.g., In[361]:= trifil25[2/25 + 2/75] // ComplexExpand

Out[361]= {1/2 + (I Sqrt[3])/10}

In[367]:= Graphics[{Hue@#, Line[ReIm@{trifil25@#, trifil25[# + 5^-5]}]} & /@ Range[0, 1 - 5^-5, 5^-5]] gosper.org/trifil25.png (Those little white hexagons are sextuple points.)

The inverse fractal, constructed more or less automatically: In[380]:= untrifil25[t_] := piecewiserecursivefractal[t, Identity, If[0 <= Re[#1] <= 1 && 0 <= Im[#1] <= Sqrt[3]/2, {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25}, {}] &, {ComplexExpand[5 #1] &, ComplexExpand[1/4 (6 - 2 I Sqrt[3] - 10 #1 + 10 I Sqrt[3] #1)] &, ComplexExpand[1/4 (-4 I Sqrt[3] + 10 #1 + 10 I Sqrt[3] #1)] &, ComplexExpand[-2 + 5 #1] &, ComplexExpand[1/4 (-6 + 6 I Sqrt[3] + 10 #1 - 10 I Sqrt[3] #1)] &, ComplexExpand[1/4 (4 + 8 I Sqrt[3] - 10 #1 - 10 I Sqrt[3] #1)] &, ComplexExpand[3 + I Sqrt[3] - 5 #1] &, ComplexExpand[1/4 (10 - 2 I Sqrt[3] - 10 #1 + 10 I Sqrt[3] #1)] &, ComplexExpand[1/2 (3 + I Sqrt[3] - 10 #1)] &, ComplexExpand[1/4 (8 - 10 #1 + 10 I Sqrt[3] #1)] &, ComplexExpand[2 + I Sqrt[3] - 5 #1] &, ComplexExpand[-2 - I Sqrt[3] + 5 #1] &, ComplexExpand[1/4 (4 - 8 I Sqrt[3] + 10 #1 + 10 I Sqrt[3] #1)] &, ComplexExpand[1/4 (-10 + 2 I Sqrt[3] + 10 #1 - 10 I Sqrt[3] #1)] &, ComplexExpand[1/4 (-2 + 6 I Sqrt[3] - 10 #1 - 10 I Sqrt[3] #1)] &, ComplexExpand[1/4 (16 - 10 #1 + 10 I Sqrt[3] #1)] &, ComplexExpand[-2 - 2 I Sqrt[3] + 5 #1] &, ComplexExpand[1/4 (-14 + 2 I Sqrt[3] + 10 #1 - 10 I Sqrt[3] #1)] &, ComplexExpand[1/2 (-3 I Sqrt[3] + 5 (-1 + 2 #1))] &, ComplexExpand[1/4 (16 - 4 I Sqrt[3] - 10 #1 + 10 I Sqrt[3] #1)] &, ComplexExpand[4 + I Sqrt[3] - 5 #1] &, ComplexExpand[1/4 (6 + 10 I Sqrt[3] - 10 #1 - 10 I Sqrt[3] #1)] &, ComplexExpand[1/2 (9 + I Sqrt[3] - 10 #1)] &, ComplexExpand[1/4 (8 + 8 I Sqrt[3] - 10 #1 - 10 I Sqrt[3] #1)] &, ComplexExpand[-4 + 5 #1] &}, {#1/25 &, (2 - #1)/25 &, (2 + #1)/ 25 &, (3 + #1)/25 &, (4 + #1)/25 &, (5 + #1)/25 &, (6 + #1)/ 25 &, (7 + #1)/25 &, (8 + #1)/25 &, (10 - #1)/25 &, (11 - #1)/ 25 &, (11 + #1)/25 &, (13 - #1)/25 &, (14 - #1)/25 &, (14 + #1)/ 25 &, (16 - #1)/25 &, (16 + #1)/25 &, (18 - #1)/25 &, (18 + #1)/ 25 &, (19 + #1)/25 &, (21 - #1)/25 &, (22 - #1)/25 &, (22 + #1)/ 25 &, (24 - #1)/25 &, (24 + #1)/25 &}]

Here it finds the preimages of the simplest sextuple point:

In[381]:= untrifil25[1/2 + I Sqrt[3]/10]

Out[381]= {8/75, 11/75, 14/75, 17/75, 4/15, 23/75} --rwg