The “normalized radical” of the ℳ-set

  • Benoit B. Mandelbrot


A “normalized radical” ℛ of the ℳ-set is defined as the shape that satisfies exactly all the self-similarity properties that hold approximately for the molecules of the ℳ-set of the quadratic map. Explicit constructions show that the complement of ℛ is a σ-lune, and prove that the ℛ-set does not self-overlap. The fractal dimension D of the boundary of ℛ is shown to satisfy \( \sum\nolimits_2^\infty {\Phi (n){n^{ - 2D}} = 1} \), where Φ(n) is Euler’s number-theoretic function.

A rough first approximation is the solution D = 1.239375 of the equation \( \sum\nolimits_2^\infty {{n^{1 - 2D}} = \zeta (2D - 1) - 1 = {\pi ^2}/6} \), where ζ is the Riemann zeta function. A less elegant but doubtless closer second approximation is D=1.234802. The same D applies to the ℳ-sets of other maps in the same class of universality.

Interesting “rank-size” probability distributions are introduced.


Fractal Dimension Generation Atom Hausdorff Measure Riemann Zeta Function Closed Disc 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Benoit B. Mandelbrot 2004

Authors and Affiliations

  • Benoit B. Mandelbrot
    • 1
    • 2
  1. 1.Mathematics DepartmentYale UniversityNew HavenUSA
  2. 2.IBM T.J. Watson Research CenterYorktown HeightsUSA

Personalised recommendations