DanA>What I wrote was misunderstood. Sorry about that, Dan!
(By the way, rwg, your last two sentences are quite interesting! I wonder how small, in some appropriate sense, that uncountable set
S = {x in [0,1]^n | #(finv(x)) > 1}
can be.
Rather huge. For the 2D Peano-Hilbert squarefiller, S = KxC U CxK, where C := [0,1] and K := the dyadic rationals therein, with the triply visited being the dense set KxK. You can intuit this by noting that images of closed subintervals of C are unions of closed squares. As the spacefill continues, other closed squares will share the boundaries. If instead we're filling a fudgeflake or a france, then the shared boundaries will be fractals, which are in some sense even larger than the square edges.
prp[p_]:=Block[{q=If[p[[1]]^2>p[[2]]^2,{-p[[3]],0,p[[1]]},{0,p[[3]],-p[[2]]}]},q*Sqrt[(p.p)/(q.q)]]
Stick[p_,q_]:=Block[{m=(p+q)/2,d=prp[q-p]/16,e},e=Cross[q-p,d]/Norm[q-p]; Table[Polygon[{r,m+Cos[2*(k-1)*Pi/3]*d+Sin[2*(k-1)*Pi/3]*e, m+Cos[2*k*Pi/3]*d+Sin[2*k*Pi/3]*e}],{r,{p,q}},{k,3}]]
Graphics3D[ Table[Stick[Treano[(k + 1/3)/256], Treano[(k + 4/3)/256]], {k,0,254}]]
These revisitations can blow out the Graphics3D above with Sticks of zero length, unless we add the (gaspy?) line Stick[x_, x_]:={}
produce the graphic at http://gosper.org/treano.png .
Oops, I used the long awaited mouse tumble feature to find a much nicer pose, but the Export command captured the original one. Here btw is the 2D version of Treano: (* Exactly and continuously map the rational unit interval onto the rational unit square. Peano redefines itself twice, calls itself in four places, and has no discernable termination condition.*) Peano[t_, a1_: 1, a0_: 0] := Peano[t, b1_: 1, b0_: 0] = (Peano[t, s1_: 1, s0_: 0] = ((a0 - s0)/(s1 - a1)); Module[{t4 = 4*t, n}, n = Floor[t4]; t4 -= n; Switch[n, 0, I*(1 - Peano[1 - t4, -I*a1/2, a0 + a1*I/2])/2, 1, (I + Peano[t4, a1/2, a0 + a1*I/2])/2, 2, (1 + I + Peano[t4, a1/2, a0 + a1*(1 + I)/2])/2, 3, 1 + I*Peano[1 - t4, a1*I/2, a0 + a1]/2, 4, 1]]) Nevertheless, e.g., In[2] := Peano[7/22] 7 17 I Out[2]:= -- + ---- 66 22
So SW's intial assessment was correct: trivial. Do I need a smiley here? --rwg MENSURATIONAL NEONATURALISM