Geometry
Recall that, for every rational number, where p and q are relatively prime, there is a hyperbolic component of period q bifurcating from the main cardioid. The part of the Mandelbrot set connected to the main cardioid at this bifurcation point is called the p/q-limb. Computer experiments suggest that the diameter of the limb tends to zero like . The best current estimate known is the famous Yoccoz-inequality, which states that the size tends to zero like .
A period-q limb will have q − 1 "antennae" at the top of its limb. We can thus determine the period of a given bulb by counting these antennas.
In an attempt to demonstrate that the thickness of the p/q-limb is zero, David Boll carried out a computer experiment in 1991, where he computed the number of iterations required for the series to converge for z = ( being the location thereof). As the series doesn't converge for the exact value of z =, the number of iterations required increases with a small ε. It turns out that multiplying the value of ε with the number of iterations required yields an approximation of π that becomes better the smaller ε. For example, for ε = 0.0000001 the number of iterations is 31415928 and the product is 3.1415928.
Read more about this topic: Mandelbrot Set
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