# Center of Gravity

Robert P. Munafo, 2000 Aug 31

The "center of gravity" of the Mandelbrot Set is the unique point P for which the double integral / 2 / 2 | | | | M(a+bi) (a+bi - P) dR dI | | /a=-2 /b=-2

has value 0+0i, where M(x) is the membership function, defined as:

M(x) = 1, for all x in the Mandelbrot Set M(x) = 0, otherwise

The problem of locating this point is similar to that of calculating the area of the Mandelbrot Set, and most of the material in the area entry is relevant.

Because the Mandelbrot Set is symmetrical around the real axis, the center of gravity has to be on the real axis, so its imaginary coordinate is 0.

The pixel counting method gives the best known statistical estimate of the real coordinate, -0.2867683 +- 0.0000001. This is right about in the middle of the main cardioid, about halfway between seahorse valley and elephant valley.

It has been proposed by many people that the area might have an exact value given by some simple analytical expression, giving its value in terms of other known constants and functions. While this is still uncertain for the area, the case for the center of gravity is looking good. The expression

(ln(3) - 1/3) ^{Feig1}

where Feig1 is the first Feigenbaum constant, has the value 0.28676 82633 82935 02685 29586 ... and this is well within the error range for the current best guess of the center of gravity.

From the Mandelbrot Set Glossary and Encyclopedia, by Robert Munafo, (c) 1987-2023.

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