Boundary of the Mandelbrot Set
Robert P. Munafo, 1999 Feb 3.
The boundary of the Mandelbrot Set contains all of the chaotic behavior in the iteration algorithm: all points that iterate indefinitely without a period are in the boundary. All "interesting" views of the Mandelbrot Set contain points in the boundary.
The boundary can be mapped one-to-one onto a circle (see external angle), but at the same time it is infinitely convoluted, having a Hausdorff dimension of 2.0.
The boundary of the Mandelbrot Set is a fractal by Mandelbrot's definition, but not by the simple "dimension" definition since its dimension is 2.0. The Mandelbrot Set itself (boundary plus interior) is not a fractal by any definition.
From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo. Mu-ency index
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