From Osher Doctorow Ph.D.

The Mandelbrot Set from a computational viewpoint involves following orbits of 0 like 0, c, c^2 + c, (c^2 + c)^2, ..., where c is a complex number of form c1 + ic2 where i = sqrt(-1) and only those c orbits which never exceed 2 in modulus or absolute value are in the Mandelbrot set. However, the Mandelbrot set is from another viewpoint a bunch of cardioids that are smaller and smaller, where a cardoid is a heart-shaped object which looks like a circle but has a sharp bend on "top" (or on its side depending on whether it is sideways) where the heart bends inward. The equation of a cardioid is simplest in polar coordinates, where any of the Mandelbrot cardioids in particular have equations of form: 1) r = a(1 - cos(u)) where r is the distance from the origin to the cardioid (the radius or length of radial vector) and u is the angle that r makes with the positive x axis or "pole". The pair (r, u), often written (r, theta) where theta looks like a zero with a horizontal slash through its center, is called the polar coordinates of a point on the cardioid, and there is also a Cartesian or rectangular coordinates given by: 2) (x, y) = (rcos(u), rsin(u)) Let's multiply both sides of (1) by r, obtaining: 3) r^2 = ar - arcos(u) It is always the case that r^2 = x^2 + y^2 and rcos(u) = x, so we can write this as: 4) r^2 = ar - ax Let's calculate the partial derivative of r with respect to x, Dx(r), with u held constant. Actually, there is no explicit u in (4), but just in case we use the notation Dx(r) which is the "derivative of r with respect to x holding u constant". Those who are familiar with calculus can follow the following derivation, while others can just accept the final result below. We get (by implicit differentiation): 5) 2rDx(r) = aDx(r) - a and algebraically this gives us the final result: 6) Dx(r) = a/(a - 2r) Although this is not one of the usual growth equations dy/dt = ky, dy/dt = ky (1 - y), dy/dz = f(z) + g(z)y + h(z)y^2, it has the form dy/dz = a/(a - 2y) if u is held constant. By comparison with the other equations, it is recognizable as a somewhat rarer type of Growth Equation. If we had 2r = a or r = a/2, the denominator of (6) would be 0, which is not allowed in mathematics, but which would signify infinitely large value of Dx(r). Thus, the Mandelbrot set's cardioid(s) are the first growth equations which we have examined which are potentially "explosive" in a sense. Of course, the presence of z^2 in the orbits is another clue, as is the fact that replacing r^2 by x^2 + y^2 in (4) results in: 7) x^2 + y^2 = ar - ax which can be used to solve for r as a quadratic or bilinear/affine polynomial in x and y. The type of Existence expressed by the right-hand-side of (6) can be distinguished from ordinary Geometric Existence by referring to it as Potentially Infinite/Explosive Geometric Existence, or more simply by Chaotic- Fractal Existence or an example of the latter. Osher Doctorow Ph.D.