That was more or less the point. If you zoom in on a boundary point of the Mset you see pretty fractal images forever. If you zoom in on an interior or exterior point, you eventually see a full or empty region. I was interested if there was a fractal where you would see fractal images no matter where in the plane you zoomed. Obviously, there are sets like Q^2 such that both the set and its complement are dense on R^2, but a image of these would be homogeneous. I was looking for something with fractal-like density variations across the entire plane.
-----Original Message----- From: math-fun [mailto:math-fun-bounces@mailman.xmission.com] On Behalf Of Kerry Mitchell Sent: Saturday, February 06, 2016 1:17 PM To: math-fun Subject: Re: [math-fun] plane-filling fractal
With the Mandelbrot set, the fractal is the boundary. For every point on the boundary, there's a neighborhood containing points outside of the set, so it seems to me that if the fractal is dense, its complement must be dense, too.
On Sat, Feb 6, 2016 at 11:03 AM, David Wilson <davidwwilson@comcast.net> wrote:
Is there a fractal set M a la the Mandelbrot set where both M and its complement are dense on the plane?
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