Constants

Robert P. Munafo, 2012 Dec 4.

Φ (phi), 1.608033...

This constant does not show up physically in the shape of the Mandelbrot Set, but there are various golden ratio places that you can find in the Mandelbrot Set's filament structure. For example, the mu-units R2.1/2, R2.1/3, R2.2/5, R2.3/8, R2.5/13, R2.8/21, ... converge on a special point that is sometimes thought of as the "golden ratio point" on the boundary of R2a. See Fibonacci Series.

2 is the most important integer constant in the Mandelbrot set, because of the various types of bifurcation in multiples and powers of 2, which includes period doubling and rotational symmetry around islands. Jonathan Leavitt turned it around, finding examples of reverse bifurcation in powers of 2.

e, 2.718281...

This constant does not show up physically in the shape of the Mandelbrot Set, but it has been pointed out that e is involved in any work with complex numbers because (for example) e^{pi i}=-1. That's sort of a trivial relation, it's like saying that all circles are related to the number 7 because either a circle's radius or its area (or both) contain a 7 somewhere in their decimal expansion.

π (pi), 3.141592...

It is fairly surprising to many that Pi shows up in the spacing of the dwell bands in cusps; see the last portion of the entry R2.C(1/2) for a complete description.

The Feigenbaum constant, 4.669201...

This shows up as the ratio between each successive mu-atom in the series R2a, R2.1/2a, R2.1/2.1/2a, R2.1/2.1/2.1/2a, ... For more information see the separate Feigenbaum constant heading.

revisions: 20030519 oldest on record; 20121204 Add 2 and link to Fibonacci Series.

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