Fractals with Scientific Importance
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Mandelbrot Set
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The Mandelbrot
set is said to be the most complicated
figure in mathematics yet, mathematically,
it is very easy to describe: Z1 = Z^2 +
C.
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Julia Set
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The Mandelbrot set is an infinite catalog of determinate structures. The Julia set is defined as the set of points that border and separate basins of attraction of an attractive cycle. The Julia sets associated with the Mandelbrot set are determinate states in two dimensions. |
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The Quaternions
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A three-dimensional image of a donor set for quaternion images is the set of initial conditions in complex space that produces a deterministic structure in the quaternions.
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Superimpose
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A superimpose function allows the integration of myriad images to create a final image. Determinate structures built into a final determinate system.
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Complex Plane
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Explore the complex plane in different fractal dimensions: Z 1 =Z^exp+C
When exp=1, those points less than one iterate to zero, those points greater than one, iterate to infinity and the fractal boundary is a one-dimensional line. |
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Z1=Z^1.9+C
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Changing the exponent allows the complex plane to be explored in different fractal dimensions. The boundary between domains approaches a fractal dimension of two.
Z^3 for three-dimensional fractals. |
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Verhulst's Equation
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Verhulst's equation
is one of the best-known bifurcating equations
that lead to deterministic states of bounded
chaos.
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Period 6
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Verhulst's equation
shows the boundary between chaos and determinate
structure. Even within the chaotic
region (left side) there are density regions
formed by chaotic periodic points of the
orbits of the iterations.
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Donor Sets
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Donor sets are those sets of points that create determinate structures. The donor set of Verhulst's equation as graphed in the complex plane. Applying various coloring routines shows internal structures within a structure.
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Three-Dimensional Donor Sets
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Each donor set
in two dimensions has a three-dimensional
counterpart in complex space. Each
point within three dimensional complex space
that produces a deterministic state for Verhulst's
equation creates a three-dimensional donor
set. (The z axis is truncated)
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Electron
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Electromagnetic energy that forms a determinate state within three 42s, with a negative time vector, forms an electron. Each 42 has -1/3 electrical charge. The electron then has -3/3 electrical charge. The Mandelbrot set within a 42 is a proxy for all of the determinate states within a four-dimensional determinate system.
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Down Quark
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Each 42 within a quark carries a different atomic force. Three 42s: 42 1, -1/3 electric charge (-E42); 42 2, a Strong force 42 (S42); 42 3 , Weak force 42 (W42) combining to form a down quark. The down quark has a negative time vector. Pink for electromagnetic energy, turquoise for the weak force and purple for the strong force. [(-E42),(W42),(S42), -t]
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Up Quark
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Two of the three 42s within an up quark carry +1/3 electric charge and the third 42 carries the strong force. The up quark has a positive time vector. [2(+42),(S42),+t]
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