OOPS, eight <http://gosper.org/8ringoctaflake.svg> overlaps! (9M svg. Safari zooms and scrolls; Firefox only zooms.) Entirely Corey's & Julian's. Julian points out that, starting with the single flake, there are infinitely many ways to build the structure outward: Surround it by 2, 4, or 3^n*8. Lots more if you mix sizes. Where two surrounding flakes abut, they create a Cantor set of smaller flakes. We still don't know the limiting dimension of the 1 - 8 - 24 - 56 --- (actually 0 - 8 - 24 - 56 --- !) progression. If instead we surround the first one by 24 (of the "56" size), then we could follow with 216 of the "504" size, ... probably leading to a different boundary with a different dimension. There are apparently infinitely many progressions, and probably infinitely many possible limiting boundaries, of possibly infinitely many different dimensions. --rwg On Tue, Jun 26, 2012 at 1:42 AM, Bill Gosper <billgosper@gmail.com> wrote:
On Sun, Jun 17, 2012 at 5:20 PM, Bill Gosper <billgosper@gmail.com> wrote:
For some pictorial. mildly trick questions: http://gosper.org/foupent.pdf (1.6M) --rwg
The scheme that makes the pentagram from pentaflakes and heptagram from heptaflakes doesn't work for even polygons--your choice is untapered or curled arms (or binary trees instead of arms). I asked Julian if the enneagram worked and he said only with overlap, and made this peculiar figure <http://gosper.org/octaflake.png> showing that eight was the highest nonoverlap case, with Cantor sets of octaflakes formed where two larger octaflakes touch side-to-side. I don't recognize the implied limiting fractal, but note that the construction does not coerce ever smaller flakes. It appears you can instead start adding ever larger ones. Maybe even after reaching the fractal limit. Maybe even asymmetrically? A playworthy shape. --rwg