Start with one octaflake. Surround it with 8 others, smaller by the factor sec(π/8)/2. (The new outline happens to be a larger octaflake). Surround that by 24 the same factor smaller, yielding a new shape. The next ring has 56 flakes. But the 2^n-8 pattern breaks down at n=7, when eight vacuoles form on the periphery, producing a ring of 8*119 vs 8*127. Corey managed to write a system of linear recurrences which defeated RSolve, but not Julian: c + a p^n + b q^n + d r^n + e s^n /. {a -> Root[131072 - 155648 #1 + 29184 #1^2 - 10432 #1^3 + 815 #1^4 &, 2], b -> Root[131072 - 155648 #1 + 29184 #1^2 - 10432 #1^3 + 815 #1^4 &, 1], c -> -(104/5), d -> Root[131072 - 155648 #1 + 29184 #1^2 - 10432 #1^3 + 815 #1^4 &, 4], e -> Root[131072 - 155648 #1 + 29184 #1^2 - 10432 #1^3 + 815 #1^4 &, 3]} /. {p -> Root[-2 - 2 #1 - 2 #1^2 + #1^4 &, 2], q -> Root[-2 - 2 #1 - 2 #1^2 + #1^4 &, 1], r -> Root[-2 - 2 #1 - 2 #1^2 + #1^4 &, 4], s -> Root[-2 - 2 #1 - 2 #1^2 + #1^4 &, 3]} For n>1 this gives 8*{7, 15, 31, 63, 119, 231, 439, 839, 1591, 3031, 5751, 10935, 20759, 39447, 74903, 142295, 270231, 513303, 974871, 1851671, 3516823, 6679703,...} This fails to tell us the limiting boundary dimension, but constrains it below log[2+sqrt(2)](4), the dimension of the octaflake boundary. Using Corey's technique, Neil RSolved the analogous problem for the Kochflake <http://gosper.org/kocharmor.png>: RSolve[{ threes[n + 1] == 2*fours[n] + 2*fives[n], fours[n + 1] == threes[n], fives[n + 1] == fours[n] + livesixes[n], livesixes[n + 1] == fives[n], deadsixes[n + 1] == deadsixes[n] + livesixes[n], threes[0] == 6, fours[0] == 6, fives[0] == 0, livesixes[0] == 0, deadsixes[0] == 0} , {threes[n], fours[n], fives[n], livesixes[n], deadsixes[n]}, n] giving a pile of nth powers of quartic surds, and the sequence 6*{7, 15, 31, 55, 109, 197, 375, 699, 1301, 2453, 4551, 8563, 15957, 29885, 55895, 104443, 195541, 365349, 683719, 1278243, 2390773, 4471469, 8361367} Neither sequence appears in OEIS. The quartic surds apparently don't denest. The closed forms for these sequence are anything but nice. (Think of the Binet Fibonacci formula gone berserk.) Probably the nicest way to define and evaluate them is with matrix powers, e.g. {{7, 15, 31, 55}, {15, 31, 55, 109}, {31, 55, 109, 197}, {55, 109, 197, 375}} . {{0, 0, 0, -2}, {1, 0, 0, 2}, {0, 1, 0, 3}, {0, 0, 1, 0}}^^16 == {365349, 683719, 1278243, 2390773}, {683719, 1278243, 2390773, 4471469}, {1278243, 2390773, 4471469, 8361367}, {2390773, 4471469, 8361367,15639467}} (four squarings). The octaflake requires 5x5 matrices. --rwg On Sun, Jul 15, 2012 at 2:48 AM, Bill Gosper <billgosper@gmail.com> wrote:
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