# 19 What is the dimension of the Mandelbrot Set?

## Description

This article is from the sci.fractals
FAQ, by Michael C. Taylor and Jean-Pierre Louvet with numerous
contributions by others.

# 19 What is the dimension of the Mandelbrot Set?

The Mandelbrot Set has a dimension of 2. The Mandelbrot Set

contains and is contained in a disk. A disk has a dimension of 2, thus

so does the Mandelbrot Set.

The Koch snowflake (Hausdorff dimension 1.2619...) does not satisfy

this condition because it is a thin boundary curve, thus containing no

disk. If you add the region inside the curve then it does have

dimension of 2.

The boundary of the Mandelbrot set and the Julia set of a generic c in

M have Hausdorff dimension 2 and have topological dimension 1. The

proof is based on the study of the bifurcation of parabolic periodic

points. (Since the boundary has empty interior, the topological

dimension is less than 2, and thus is 1.) See reference above

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