Thanks, Alan.
So, does this make it the same as starting with a plus sign of 5 adjacent squares, and taking 5 congruent copies snuggled together and normalized -- then iterate to the limit? (Actually the boundary of this limit.)
If you do it right, you can start with a square for Q[0] and get the "+" sign for Q[1]. See http://gosper.org/erezd.PNG . These aren't quite the same but they have the same closure and boundary dimension. A toy analogy is the approximations to the unit interval you get by varying Q[0] in the iteration Q[n+1] = (Q[n] U (1+Q[n]))/2. Allan's method, Q[0] := {0}, produces the dyadic rationals. But Q[0] := (0,1] gives the complement of the dyadic rationals. Same closure, utterly unequal measures. You can't zoom down to a self contact because of self similarity. For greater rigor, you can construct sausage links that will remain demonstrably disjoint. You can also write a function analogous to that obscure Peano Mathematica function that will exactly and continuously map the rationals in the unit interval onto one quarter of the boundary in question. There are also open and closed loop spacefills of this region, which can also be computed exactly for rationals. I like D=log_5(9) . Any takers on 3D revisitation? --rwg
If so, it's a fractal I was studying just a couple of months ago. I'll go review what I had about it.
=-Dan
<< I'm pretty sure I understand the intended construction. We'll construct a sequence of sets of complex numbers; the limit of this sequence will be (well-defined and) the intended fractal.
Let Q[0] contain only 0. Then for any nonnegative integer i, let Q[i+1] = (Q[i] + {0, 1, -1, i, -i}) / (2i + 1). Here, if A and B are sets, the set A+B is intended to mean {a+b | a in A and b in B}.
Each Q is roughly cross-shaped; RWG observes (very tersely) that dividing by 2i+1 rotates and shrinks each such cross by just enough that five crosses can snuggle together to make a meta-cross.
I think this fractal is in Mandelbrot; I cannot dig up my copy at the moment.
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