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. 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? On Tue, Mar 10, 2015 at 11:22 PM, Bill Gosper <billgosper@gmail.com> wrote:
The Wikipedia article < http://en.wikipedia.org/wiki/Lévy_C_curve > says that the curve contains open sets, which seems pretty believable from the construction. It also says the Hausdorff dimension *of the boundary* is roughly 1.9340, which is consistent with the curve itself containing open set.
--Dan
I suspect if Mandelbrot were alive, he could tell us that dimension exactly. But I'll bet the C curve has measure 0. --rwg I think you get different sets depending on whether you initialize the IFS to a single point or a large disk.
At MIT in the early 60s, it was always gnurd, not nerd, and the term seemed to come out of the SciFi Club, and mean someone even more reclusive, technically obsessed, and socially inept than an an ordinary Tech Tool. The adjective was "gnurdly". The name for a generic Tool was J. Random Gnurd. Rollo Silver's AI Lab login was thus JRG. In the 50s, Readers' Digest was the *only* mass medium with the courage and financial independence (and legal staff) to run antismoking articles. They were so strident and urgent and out of step with, e.g., panelists and commentators randomly smoking on live TV that they sounded crank. "Ask your doctor which brand he smokes ..." "Scientists debunk cancer scare." When my med-school brother noticed firsthand zero lung cancers among nonsmokers, he realized the magnitude of the brainwashing. The Surgeon General's Report was a political watershed. No doubt presaged by Arthur (Buy 'em by the carton) Godfrey's tergiversation.
On Mar 10, 2015, at 6:33 PM, Allan Wechsler <acwacw@gmail.com> wrote:
This is almost certainly a question for Gosper. Wikipedia claims that the C-curve contains open sets, and therefore has area, and has scaling dimension 2. I had been under the impression that it had a scaling dimension that was just a little less than 2.
Is this true? Suppose I have a C-curve in the complex plane with endpoints 0 and 1, with most of the bulk in the first quadrant. Where are some example solid areas? What is the total area of the object? How many differently-shaped "tiles" does it have?
On a couple of past occasions I have tried to solve the fairly hairy set of equations arising from the different ways little families of C-curve copies can intersect, in hopes of deriving the scaling dimension. I always failed -- there was too much opportunity for stupid arithmetic errors.
Wow. I stumbled onto it ca 1966 as a bug in a Hilbert square-fill. It never occurred to me that it might yet have positive area. It should therefore have one of those tricky recursive definitions for all rationals. A graphics trophy awaits: the maximal connected open subset. Last year Julian found some amazing technology that might apply. --rwg