Physical Process Modeling

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    If there really is a unifying formula that explains the creation of everything, then perhaps we can use our understanding of it to support our health. To explore this idea further, we need to look at the basic mathematics behind these principles. Our starting point must be a suitable fractal equation, one that is easy to use but is clear enough for our body's intelligence to understand the intent behind its formation.
    Some fractals are based on the square root of negative numbers, which takes us further into the conceptual world than we need to travel. For this reason, the Fibonacci sequence is the perfect equation. It uses 'real' numbers, which are easier for our bodies to interpret.
    This is the core design behind the equations in this manual. I have occasionally used a Julia Set sequence in some areas, but only when working with concepts outside the basic functions of the human body, like blocking electromagnetic stress, or undoing the trauma of dental work.
    So, if we are going to use the Fibonacci sequence as our primary building block, we need to start looking at how it works.
    Take any two numbers you wish, say 2 and 5. Add them together and you get 7. Add the 5 and the 7 together and you get 12. 12 and 7 makes 19. Keep adding the last two numbers together and before long you have the following sequence:

2, 5, 7, 12, 19, 31, 50, 81, 131, 212, 343, 555, 898, 1453, 2351, 3804 etc

    As the numbers grow, you move increasingly closer to the 'golden ratio' of approximately 1 to 1.618. In other words, every subsequent number is 1.618 times the previous one. I'll explain the significance of this in a minute.
    Lets work through another sequence first - the true Fibonacci progression which starts with 0 and 1. (I like the undertones here; first there was nothing, then there was the first act of creation...)

0 + 1 = 1 1 + 1 = 2 1 + 2 = 3 2 + 3 = 5 3 + 5 = 8 ... and off it runs... ...0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597 etc

    The mathematical representation of this sequence of additions looks like this:



All this is saying is that the third number in the sequence (n+2) is the addition of the previous two, (n+1) and (n). I have added infinity signs - <-> + to show that the fractal should be computed in both directions, as in nature where there is no beginning or end to the depth of the calculations.

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    The term "fractal" was coined by Benoit Mandelbrot in 1975. It comes from the Latin fractus, meaning an irregular surface like that of a broken stone. Fractals are non-regular geometric shapes that have the same degree of non-regularity on all scales. Just as a stone at the base of a foothill can resemble in miniature the mountain from which it originally tumbled down, so are fractals self-similar whether you view them from close up or very far away.
    Fractals are the kind of shapes we see in nature. We can describe a right triangle by the Pythagorean theorem, but finding a right triangle in nature is a different matter altogether. We find trees, mountains, rocks and cloud formations in nature, but what is the geometrical formula for a cloud? How can we determine the shape of a dollop of cream in a cup of coffee? Fractal geometry, chaos theory, and complex mathematics attempt to answer questions like these. Science continues to discover an amazingly consistent order behind the universe's most seemingly chaotic phenomena.
    Mathematicians have attempted to describe fractal shapes for over one hundred years, but with the processing power and imaging abilities of modern computers, fractals have enjoyed a new popularity because they can be digitally rendered and explored in all of their fascinating beauty. Fractals are being used in schools as a visual aid to teaching math, and also in our popular culture as computer-generated surfaces for landscapes and planetary surfaces in the movie industry.

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Mira's Model


    The coordinates of the points on the Mira curve are generated iteratively through the following system of nonlinear difference equations:


where:


    Here, a=–0.99, and we consider the cases b=1 and b=0.98. The starting point coordinates are (4, 0). This case can be viewed by editing and executing the following script M-file (MATLAB):


for n=1:12000
a=-0.99;b1=1;b2=0.98;
x1(1)=4;y1(1)=0;x2(1)=4;y2(1)=0;
x1(n+1)=b1*y1(n)+a*x1(n)+2*(1-a)*(x1(n))^2/(1+(x1(n)^2));
y1(n+1)=-x1(n)+a*x1(n+1)+2*(1-a)*(x1(n+1)^2)/(1+(x1(n+1)^2));
x2(n+1)=b2*y2(n)+a*x2(n)+2*(1-a)*(x2(n))^2/(1+(x2(n)^2));
y2(n+1)=-x2(n)+a*x2(n+1)+2*(1-a)*(x2(n+1)^2)/(1+(x2(n+1)^2));
end
subplot(2,1,1); plot(x1,y1,'.')
title('a=-0.99 b=1')
subplot(2,1,2); plot(x2,y2,'.')
title('a=-0.99 b=0.98')


    Manifest the computer artist inside yourself. Generate new geometrical morphologies, in Mira's model, by new choices of the parameters (-1 a 1 and b=1) and of the starting point. You can start with:

ab1b2(x1,y1)
-0,4810,93(4,0)
-0,2510,99(3,0)
0,110,99(3,0)
0,510,9998(3,0)
0,9910,9998(0,12)






Henon’s Model


    The coordinates of the Henon's orbits are generated iteratively through the following system of nonlinear difference equations:



x1(1)=0.5696;y1(1)=0.1622;
x2(1)=0.5650;y2(1)=0.1650;
for n=1:120
x1(n+1)=a*x1(n)-b*(y1(n)-(x1(n))^2);
y1(n+1)=b*x1(n)+a*(y1(n)-(x1(n))^2);
x2(n+1)=a*x2(n)-b*(y2(n)-(x2(n))^2);
y2(n+1)=b*x2(n)+a*(y2(n)-(x2(n))^2);
end
plot(x1,y1,'ro',x2,y2,'bx')


 




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