II. The Trigon: the anti-fractal of the classic M-Set
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The complex universe is actually the impossible mixture of two worlds that will never meet: real and imaginary. It exists only on the ploted 2D plane where the real and imaginary parts of the complex values are represented by the X and Y axes accordingly and here is where the fractals are rendered.
I rendered the T-Set fractals by using the following programming technique: each fractal equation is actually divided into 2 sub equations: a Real equation holding the real segments of the formula and an Imaginary equation, holding the imaginary parts. The classic square M-Set 'z = z^2 + C is created by the 2 following simple formulas: Real 'Z = rz2 - iz2 + a Imag 'Z = 2 * rz * iz + b These are the equations used to render the M-Set fractal in picture 1. Now I did a simple switch: I substituted these two Real and Imaginary components with each other. This was all that was needed to create the Anti Fractals Set: Imag 'Z = rz2 - iz2 + a Real 'Z = 2 * rz * iz + b This process was the one that created the Trigon Anti Fractal in picture 2 above.
Indeed the Trigon fractal above was described by Robert L. Devaney in his book "Chaos, Fractals, and Dynamics" (Addison-Wesley, 1990) where it was referred to by the name "Tricorn" due to it's resemblance to a tricornered hat. Though Tricorn is a worthy artistic title it relates only to a single member shape and its accidental resemblance to a familiar object. I prefer the close sounding name Trigon that relates to the characteristics of the math shape and is in accordance with all members of the T-Set. Anyway, the tricorn was plotted by using the complex conjugate of Z instead of Z in the Mandelbrot equation: Z = sqr(conj(Z))+C but was not regarded as the Anti Fractal Set of the M-Set. The body of the T-Set members has a clear polygonal shape. The number of the polygon sides is one more then the power of the equivalent fractal in the M-Set. Thus the classic squared Mandelbrot with one primary bulb is represented in the T-Set by the Trigon with 3 Ovoids. The ratio between the number of bulbs and the number of ovoids in the same power of the M-Set and the T-Set could be calculated by this equation: n(T-Set) = exp(M-Set) + 1 where: n(T-Set) is the number of the ovoids in the T-Set polygon and exp(M-Set) is the power of both M-Set and T-Set. This is the same for the number of the polygon sides, as each ovoid emerges from a vertex of the polygon. This equation is kept along the whole family. At each vertex of the polygonal bodies of the T-Set members emerges an Ovoid body, the equivalent of the M-Set cardioid. Roughly speaking the ovoids are an elongated stretched equivalent of the Cardioid. Looking at the morphology of the M-Set we see, that the number of cardioids equals the power of the given member minus 1. This derives another interesting equation: P = {n(M-Set) + n(T-Set)} / 2 where: n(M-Set) is the number of cardioids of a member of the M-Set, n(T-Set) is the number of ovoids in the member of the T-Set and P is power of both members. This means that the mean value of any Fractal + Anti Fractal will give the order (power) of the members.
Following is a short description of the structures and morphology of this interesting family of fractals. Author: Dr. Joseph Trotsky |
