z4 = 52 + 1 = 26
and we see that this orbit tends to infinity, therefore 1 is not included in the Mandelbrot
Set.
If, however, the z0 valued 0 and the seed was 0, then the orbit would remain in a fixed position. If we now choose z0 = -1 and the seed is 0, then the orbit stays in a 2-period cycle, where the orbit bounces back and forth from 0 and -1:
z1 = 02 + (-1) = -1
z2 = -12 + (-1) = 0
z3 = 02 + (-1) = -1
When the absolute value is applied to the resulting numbers, we see that that |z1|, |z2|,
and |z3| are equal to 1, 0, and 1, respectively, and it stays within the boundary of 2, thus
being in the Mandelbrot Set.
We can now conclude that the iteration of z2 + z0 would either extend to infinity (escape) or would not go to infinity (stay bounded), as the Mandelbrot set just determines the record of the fate of the orbit, whether it be chaotic, cyclic, or fixed, and is determined by the value of c.
Now, in order to visualize the Mandelbrot set as a fractal, we need to shift from real numbers to imaginary numbers. Mandelbrot used a complex plane as his graph, rather than the regular Cartesian plane. The x-axis of the plane consisted of real numbers, while the y-axis contained imaginary numbers. Therefore, any point on the complex plane would have (real #, imaginary #).
For example, the orbit of constant i and seed 0 under x2 + i is given by
z0 = 0
z1 = i
z2 = -1 + i
z3 = -i
z4 = -1 + i
z5 = -i
z6 = -1 + i
and the orbit behaves in a cyclic manner, thus its inclusion in the Mandelbrot set.
The Mandelbrot set (M) consists of all the complex numbers that stay bounded,
or do not escape to infinity namely, those seeds that generate periodic sequences (cyclic, confined sequences) or generate attracted sequences.
However, it is way too tedious to require performing all of the iterations of the possible values of c, therefore we use The Escape Criterion, which states that if |c| is less than or equal to 2 and if the absolute value of zn under z2 + z0 ever lands outside of the circle of radius 2 centered at the origin, then this orbit definitely tends to infinity. Therefore, 2i does not exist in the Mandelbrot since it quickly leaves the boundary after only one iteration:
z0 = 0
z1 = 0 + 2i
Enough of whole numbers, the Mandelbrot Set requires the use of fractions for the values of the real and imaginary parts, or else we’d only be talking about a few points. For example, the iteration of .7 + 6i is not considered a point in the Mandelbrot Set only after three iterations:
z1 = 0.7 + 0.6i
z2 = 0.83 + 1.44i
z3 = -0.6847 + 2.9904i
z3 = 3.06778523531
Usually, smaller fractions for the complex and imaginary parts tend to stay in the Mandelbrot Set, even after fifty iterations! For example, when iterating .2 + .3i , it stays inside the circle of radius 2:
z1 = 0.2 +0.3i
z2 = 0.15 + 0.42i
z3 = 0.0461 + 0.426i
z4 = 0.02064921 + 0.3392772i
z5 = 0.0853173714338 + 0.314011612302i
z50 = 0.079204221383 + 0.356467487952i
z50 = 0.365160757272
In order to create the Mandelbrot Fractal, we just ask the computer to color the pixels black if the value of c is bounded; else we color the pixel white if it goes to infinity. Different colors were determined by the quickness of the equation to go until infinity, thus the presence of blue in the image below.
Fractal created by the Mandelbrot Set
French mathematician Gaston Julia had published her work at the age of 25, yet it was not recognized as an input to the mathematical world until after about sixty years, when computers were able to visualize Julia’s creation. His thought was to apply function f(z) = zn2 + c and apply that to z0.
The Julia set is a bit different than the Mandelbrot set. The Julia set contains connected points or infinite points that have no relation, whereas the Mandelbrot set contains only those complex numbers in which the Julia set is connected. In the Julia set, z represents a + bi where a and b are real numbers, while c is a complex number based on the complex plane. Similar to the Mandelbrot set, the Julia set standardizes the rules to having values of constant c that extend to infinity or stay bounded. The same Escape Criterion is used for this set as well. The interesting factor in the Julia set is that one can create fractals that are nowhere similar to each other, yet are derived from the same function. In the Julia set, the constant c can be any point on the complex point, whereas the Mandelbrot set require the constant to always be equal to 0.
The Julia set, Kc , uses the point z0 to generate the sequence of iterations using zn+1 = zn2 + c. Just as the Mandelbrot set, if the sequence doesn’t run away to infinity, then the point on the complex plane belongs to Kc, and if it does, then it does not belong in the set.
First, take coordinates that lie between -1 and 1. We will choose the coordinates of (0.5, 0.7), therefore z0 = 0.5 + 0.7i. Then, we choose our constant; in this case, the constant is (- 0.25, 0.25), or written as c = -0.25 + 0.25i.
As you can see, the fourth iteration does not stay inside the circle. Here is a better
illustration of why this point is not in the Julia set:
Here is another example of a point that is in the Julia set with the seed equal to 0 and the constant equal to 0 - .5i :
Since this starting seed stays bounded, it is marked in the Julia set. This starting
seed may seem very familiar … because it is used in the same way that Mandelbrot had
made his set, and as we defined before, the Mandelbrot set consists of the points that are
connected in the Julia set.
This simple recursive formula provides the creation of these types of complex fractals:
Created at www.easyfractalgenerator.com with constant -0.80102 - 0.10772i
The Sierpińksi Triangle is a preeminent type of fractal that does not require the use of the computer if you just want the basic levels of the fractal. The Sierpińksi triangle can be based off of Pascal’s triangle. By coloring all the placements of odd numbers on Pascal’s triangle black, we can make the Sierpińksi Triangle very easily. As the Pascal’s triangle unfolds further down, it is found true that the resulting images will become the self-similar patterning characteristic of a fractal.
There is, however, another way to make the Sierpińksi Triangle by using the Lindenmayer System. Developed by biologist Aristied Lindenmayer in 1968, The L-system is a process of modeling the growth of natural life, such as organisms and plants. However, it can also be used to form simple self-similar fractals. The L-System consists of symbols that can be used as strings, an initial starting point that is a string, and a mechanism for translating the generated strings into geometric structures such as fractals. In order to create the Sierpińksi Triangle, we need to set variables. Variable A and B both mean to go forward, variable + means to turn left by an angle of sixty degrees, while variable – means to turn right by sixty degrees. Our production rules are (A → B−A−B), (B → A+B+A). The arrows specify that the variable is to be substituted with these set of rules. These production rules are followed over and over again until the specified amount of iterations. For the aforementioned production rules, we need ten iterations to create the image below.
With initial axiom A, the first iteration (G0) leads us to: B – A – B.
Our next iteration (G1) yields: A+ B + A + B – A – B + A + B + A.
(G2) outputs :
B – A – B + A + B + A + B – A – B + A + B + A + B – A –B + A + B + A + B – A – B + A + B + A + B – A – B.
The iterations tend to get really big; however, you can find the outputs of every iteration over here: http://www.michaelnorris.info/software/l-system-generator.html . On the computer, you can simulate the creation of fractals using the L-system by assigning these rules to a turtle-based programming language (logo programs, Netlogo) and the turtle will draw out the fractal by drawing its trail. To see a Netlogo-based L-system generator, go to: http://ccl.northwestern.edu/netlogo/models/run.cgi?L-SystemFractals.722.481 .
Another field in fractals research is the theory of fractal dimensions. We know that the first dimension consists of a line, the second consists of a plane, and the third consists of a cube. However, we want to figure out the dimension of the Sierpińksi Triangle.
When trying to explain why the first, second, and third dimensions are recognized as a line, plane, and cube, many fail to know the real reason. People say it is determined by length, width, and height, or the self-similarity in a line and a plane. In general, we can break a line segment into N1 self-similar pieces, each with magnification factor N. The two-dimensional square can be broken down into N2 self-similar pieces, with a magnification factor of N. However, we can break the cube into N3 self-similar pieces, each of which has magnification factor of N. Therefore, to find out the dimension of anything, say a curve or a fractal, one may use the exponent of the N self-similar pieces of magnification factor of N.
To find the exponent of a fractal, however, we need to include logarithms. Logarithms allow us to figure out the value of a variable that is an exponent. For example, in order to figure out 5x = 12, we take the log of both sides: log5x = log12. Now, using the law of exponents for logarithms, we can bring down the x, so that: x *log5= log12. Now we get that x = log5log12. Likewise, we can find the variable of the self-similar pieces.
In order to find the dimension of any self-similar fractal, we use this formula:
fractal dimension = log (self similar pieces)log (magnification factor)
The dimension of a square is two, and is proven below:
dimension = log (self similar pieces)log (magnification factor)
= logN2log N
= 2 log Nlog N = 2
Similarly, to find the dimension of a fractal, we use the same formula. The Sierpińksi Triangle consists of three self-similar pieces under the magnification of two.
Therefore:
dimension=log3log2 ≅ 1.585
The fractal dimension of the Sierpińksi triangle is approximately 1.58, less than a square yet more than a line. Fractal dimension is the measure of how "complicated" a self-similar figure is and how large it is. The Sierpińksi triangle is more complicated than the Koch Curve (1.26) and the Cantor Dust (.63), thus it is larger than them.
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