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?