Re: [math-fun] fractal optimizer
I love the fractal I call the map-of-France (but which RWG can probably correct my terminology as he may have done in the past): Start with a hexagon (stage 0) centered at 0 in C, and with vertical sides 1 unit apart. Now surround that with six copies of itself and normalize this rosette by dividing the whole thing by sqrt(7) to get stage 1. Repeat ad infinitum. The thing would converge to a fractal shape with 6-fold symmetry except that the stages keep rotating a bit. So the normalizations should divide not just by sqrt(7) but by a complex number of length sqrt(7) that counteracts this tendency to rotate. (Does it work if that number is always 2 + exp(2pi*i/6), or does the rotation angle need to vary with the stages?). The result is a fractal-boundary hexagon, a rosette of 7 copies of which have the identical shape. So, the boundary B_n of stage n converges to a fractal curve B of Hausdorff dimension d(B) = log_7(9) = 1.129.... It seems as if no matter how this is done, B will not have mirror symmetry, just rotational. BUT there is a lot of choice in how 6 copies of each rosette are placed around itself, so this method can give a plethora of different curves in the limit. (Probably continuum many, given a countable number of discrete choices.) QUESTION: Among all these choices, is there some optimization problem that will result in a much smaller collection of maps-of-France, or ideally only essentially one? Perhaps the Hausdorff dimension log_7(9) *measure* of the limiting shape can be minimized? I'm fuzzy about just how to compute Hausdorff *measure* from its usual definition. --Dan Veit asks: << Are fractal sets ever the solution of an optimization problem? Sometimes the brain has a mind of its own.
From http://www.mathcurve.com/fractals/gosper/gosper.shtml the angle is arctan(sqrt(3)/5).
Hausdorff mesaure is described in the last few pages of the Anthony Barcellos article at http://mathdl.maa.org/images/upload_library/22/Hasse/07468342.di020711.02p00... Benoit B. Mandelbrot wrote about this fractal in The Fractal Geometry of Nature, (1982) on page 46, illustrated by plates 46 and 47. Based his bibliography ("Gardner 1976") it was in Scientific American December 1976, pages 124-133. The French source (below) says "Fractal studied by Gosper in 1973." Plate 46 looks like: http://mathworld.wolfram.com/images/eps-gif/flowsnake_700.gif Plate 47 is figure 7 in: http://mathdl.maa.org/images/upload_library/22/Hasse/07468342.di020711.02p00... It looks kind of like this: http://en.wikipedia.org/wiki/File:Gosper_Island_Tesselation_2.svg but with each of the 7 pieces filled in like this: http://upload.wikimedia.org/wikipedia/commons/thumb/6/6e/Gosper_curve_3.svg/... The filling is a different fractal (the Peano space-filling curve called the Gosper curve, which Gosper calls "flowsnake".) Here is what Mandelbrot wrote: "This variant of the Koch island is due to W. Gosper (Gardner 1976): the initator is a regular hexagon, and the generator is: [ Snowflake generator figure with 3 line segments; N=3; 1/r=sqrt(7); D=log(3)/log(sqrt(7))~1.1291 ] PLATE 46. In this plate, several stages of construction of the "Gosper Island" are drawn as a bold line "wrapping." [...] PLATE 47. This is an advanced construction stage of the "wrapping". [...] Observe that, contrary to Koch's original, the present generator is symmetric with respect to its center point. It combines peninsulas and bays in such a way that the island's area remains constant throughout the construction. [...] TILING. The plane can be covered using Gosper islands. This property is called *tiling*. PERTILING. Moreover, the present island is self-similar, as is made obvious by using variable-widths hatching. That is, each island divides into seven "provinces" deducible from the whole by a similarity of ratio 1/sqrt(7). I denote this property by the neologism *pertiling*, coined with the Latin prefix *per-*, as used for example in "to perfume" = "to penetrate thoroughly with fumes." Most tiles *cannot* be subdivided into equal tiles similar to the whole. For example, it is a widespread source of irritation that hexagons put together do not quite make up a bigger hexagon. The Gosper flake fudges the hexagon just enough to allow exact subdivision into 7. Other fractal tiles allow subdivision into different numbers of parts. FRANCE. A geographical outline of unusual regularity often described as *the* Hexagon, namely the outline of France, resembles a hexagon less than it resembles Plate 47 (although Brittany is undernourished here.) See also: http://mathworld.wolfram.com/GosperIsland.html http://en.wikipedia.org/wiki/Gosper_curve#Properties On Sun, Jul 31, 2011 at 16:43, Dan Asimov <dasimov@earthlink.net> wrote:
I love the fractal I call the map-of-France (but which RWG can probably correct my terminology as he may have done in the past): [...]
BUT there is a lot of choice in how 6 copies of each rosette are placed around itself, so this method can give a plethora of different curves in the limit. (Probably continuum many, given a countable number of discrete choices.)
QUESTION: Among all these choices, is there some optimization problem that will result in a much smaller collection of maps-of-France, or ideally only essentially one?
Perhaps the Hausdorff dimension log_7(9) *measure* of the limiting shape can be minimized? I'm fuzzy about just how to compute Hausdorff *measure* from its usual definition.
--Dan
Veit asks: << Are fractal sets ever the solution of an optimization problem?
Sometimes the brain has a mind of its own.
_______________________________________________ math-fun mailing list math-fun@mailman.xmission.com http://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- Robert Munafo -- mrob.com Follow me at: gplus.to/mrob - fb.com/mrob27 - twitter.com/mrob_27 - mrob27.wordpress.com - youtube.com/user/mrob143 - rilybot.blogspot.com
participants (2)
-
Dan Asimov -
Robert Munafo