On 2014-05-30 18:23, Dan Asimov wrote:
Is there a standard name for the fractal G that's the *boundary* of Bill Gosper's Flowsnake?
I've been calling it "the map-of-France". Not sure where that originated,
Straight from Mandelbrot's (previous?) book.
but it's kind of apt, since the actual map of France is hexagonalish and has a fractalish boundary.
Omitting technical details:
One way to define it is to start with the 2D regular hexagon H_0 in the complex plane C.
Inductively, define H_(n+1) as
a) First taking the union of 6 translated copies of the previous stage H_n, along with H_n itself, so that the center of one of them lies at 2+tau, where tau := exp(2pi*i/6) so they all fit snugly together. To complete this step, and then
b) shrink this union of 7 copies of H_n by the effect of the function z |-> z/(2+tau).
At this and each successive stage, we have the option of complex-conjugating the figure before recursing, providing a continuum of variations. .
The result is H_(n+1).
Then we define
H_oo := lim H_n n->oo
It's not too hard to show this actually converges in the Hausdorff metric on compact subsets of the plane.
Finally, define the fractal G := bd(H_oo), the boundary of H_oo.
It has the lovely property that a rosette of 7 translated copies of it fitting snugly together forms a magnified and rotate copy of H_oo, as the regular hexagon doesn't quite do.
It's easy to see the Hausdorff measure d of G is the solution to sqrt(7)^d = 3, so
d = log_7(9) = 1.1291500681....
Reminder: http://gosper.org/wikifrac.gif
QUESTION: Does G have an official name?
--Dan I'm satisfied (≠ thrilled) with Franceflake. --rwg