@Dan, that looks very close to the right thing, but it doesn't look like it shows the sections very clearly. @RWG, I don't think I agree that the limit set depends on the initializer. I bet I can prove that any compact initializer will have the same result. I can also prove that the union of a particular constellation of 22 C-curve units has nonzero area, and I think I have found that constellation in the 14th iteration. On Thu, Mar 12, 2015 at 4:41 PM, Dan Asimov <asimov@msri.org> wrote:
On Mar 12, 2015, at 1:38 PM, Allan Wechsler <acwacw@gmail.com> wrote:
If the endpoints of the curve are 0 and 1, I think that if you zoom on (3+5i)/8, there is an open patch just to its northwest (in the -1+i direction). It starts to crystallize at 16-18 levels. If this isn't
enough
information to find the open patch, I am going to have to write some code (unless somebody already has a pan-and-zoom C-curve explorer implemented).
Don't know if this is what you have in mind, but it looks to be at least in the ball park:
< http://www.mathlab.cornell.edu/~twk6/program.html >
--Dan
The patch I've identified looks tiny (area on the order of 2^(-15)?), but it may not be the largest patch, and there may be lots of them. I'm going to guess that the area of the unit C-curve is less than 2^(-8), but I'm now convinced that, pace Neil B., it is greater than 0.
On Thu, Mar 12, 2015 at 4:06 PM, Dan Asimov <asimov@msri.org> wrote:
Appears to be an interesting article about the C curve (aka Dragon curve) by S. Bailey, T. Kim, R. S. Strichartz: "Inside the Lévy dragon", Amer. Math. Monthly, 2002.
It mentions 16 distinct known shapes that these components of open subsets can take. (Apparently some Israeli mathematician showed there is actually a total of 21 different shapes.)
--Dan
On Mar 12, 2015, at 11:58 AM, Bill Gosper <billgosper@gmail.com> wrote:
NeilB wrote a sort of pixelated IFS that convinced us that this set
contains no convex patch of positive area. The article seems to say that the open sets follow from a proven dimension of 2.
I read it the other way round: that the existence of open subsets of the curve implies the Hausdorff dimension must be 2 (since it lies in R^2).
Tentatively granting D = 2, does openness follow?
Can someone show me an open subset of the (D = 2) boundary of the Mandelbrot set? Exercise (NBickford): Must an open subset of R² have D = 2?
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