Fractals: The mapping of Numbers....for infinity
"Did you know that amazing, beautiful shapes have been built into numbers? Believe it or not, numbers like 1, 2, 3, etc., contain a “secret code”—a hidden beauty embedded within them.
...... a sort of “map.”
In mathematics, the term “set” refers to a group of numbers that have a common property. For example, there is the set of positive numbers (4 and 7 belong to this set; -3 and 0 do not).
A few decades ago, researchers discovered a very strange and interesting set called “the Mandelbrot set.”...a “baby” Mandelbrot set is built into the tail of the “parent.” This new, smaller Mandelbrot set also has a tail containing a miniature version of itself, which has a miniature version of itself, etc.—all the way to infinity. The Mandelbrot set is called a “fractal” since it has an infinite number of its own shape built into itself.
The formula for the Mandelbrot set is zn+1 = zn2 + c. In this formula, c is the number being evaluated, and z is a sequence of numbers (z0, z1, z2, z3…) generated by the formula. The first number z0 is set to zero; the other numbers will depend on the value of c. If the sequence of zn stays small (zn ≤ 2 for all n), c is then classified as being part of the Mandelbrot set. For example, let’s evaluate the point c = 1. Then the sequence
of zn is 0, 1, 2, 5, 26, 677… . Clearly this sequence is not staying small, so the number 1 is not part of the Mandelbrot set. The different shades/colors in the figures indicate how quickly the z sequence grows when c is not a part of the Mandelbrot set.
The complex numbers are also evaluated. Complex numbers contain a “real” part and an “imaginary” part. The real part is either positive or negative (or zero), and the imaginary part is the square-root of a negative number. By convention, the real part of the complex number (RE[c]) is the x-coordinate of the point, and the imaginary part (IM[c]) is the y-coordinate. So, every complex number is represented as a point on a plane. Many other formulae could be substituted and would reveal similar shapes.
Evolution cannot account for fractals. These shapes have existed since creation and cannot have evolved since numbers cannot be changed." AnswersInGenesis
Such knowledge is too wonderful for me;
it is high,
I cannot attain unto it.
Psalm 139:6
Gap between the "head" and the "body",
also called the "seahorse valley" |
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Figure 1 is a plot—a graph that shows which numbers are part of the Mandelbrot set. Points that are black represent numbers that are part of the set. So, the numbers, -1, -1/2, and 0 are part of the Mandelbrot set. Points that are colored (red and yellow) are numbers that do not belong to the Mandelbrot set, such as the number 1/2. Although the formula that defines the Mandelbrot set is extremely simple, the plotted shape is extremely complex and interesting. When we zoom in on this shape, we see that it contains beautiful spirals and streamers of infinite complexity.
The Mandelbrot set (Figure 1) is infinitely detailed. In Figure 2, we have zoomed in on the “tail” of the Mandelbrot set. And what should we find but another (smaller) version of the original. This new, smaller Mandelbrot set also has a tail containing a miniature version of itself, which has a miniature version of itself, etc.—all the way to infinity.
The central endpoint of the "seahorse tail" is also a Misiurewicz point.
Part of the "tail" — there is only one path consisting of the thin structures that lead through the whole "tail". This zigzag path passes the "hubs" of the large objects with 25 "spokes" at the inner and outer border of the "tail"; thus the Mandelbrot set is a simply connected set
Satellite. The two "seahorse tails" are the beginning of a series of concentric crowns with the satellite in the center.
Each of these crowns consists of similar "seahorse tails"; their number increases with powers of 2, a typical phenomenon in the environment of satellites. The unique path to the spiral center passes the satellite from the groove of the cardioid to the top of the "antenna" on the "head".
"Antenna" of the satellite. Several satellites of second order may be recognized.
The "seahorse valley" of the satellite. All the structures from the start of the zoom reappear.
Double-spirals and "seahorses" - unlike the 2nd image from the start they have appendices consisting of structures like "seahorse tails"; this demonstrates the typical linking of n+1 different structures in the environment of satellites of the order n, here for the simplest case n=1.
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