Re: [math-fun] Fractal puzzles
RWG wrote: << A private conversation raised the following: a) continuous maps from 1D onto a 2D patch visit countably many points at least three times. What is the minimum dense revisitation count for continuous volume fillers, 1D onto 3D? It was seven for that "Treano" function I discussed a few months ago.
Whst is known about the dense visitation count from 1D onto 2D ? << b) What is the dimension of the boundary of the 2D quincunx fractal, the fixed point of surrounding one by four in a square, then dividing by 2+i?
Could you please elaborate on this? I don't understand the construction. (I do know that a quincunx is the arrangement of 5 points, or dots, as the center and corners of a square.) _____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele
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. On Tue, Jun 23, 2009 at 8:50 PM, Dan Asimov <dasimov@earthlink.net> wrote:
RWG wrote:
<< A private conversation raised the following: a) continuous maps from 1D onto a 2D patch visit countably many points at least three times. What is the minimum dense revisitation count for continuous volume fillers, 1D onto 3D? It was seven for that "Treano" function I discussed a few months ago.
Whst is known about the dense visitation count from 1D onto 2D ?
<< b) What is the dimension of the boundary of the 2D quincunx fractal, the fixed point of surrounding one by four in a square, then dividing by 2+i?
Could you please elaborate on this? I don't understand the construction. (I do know that a quincunx is the arrangement of 5 points, or dots, as the center and corners of a square.)
_____________________________________________________________________ "It don't mean a thing if it ain't got that certain je ne sais quoi." --Peter Schickele
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participants (2)
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Allan Wechsler -
Dan Asimov