Gaston Maurice Julia was born in the 19th century, and he studied iterated polynomials and rational functions. He published a paper about his work and became famous for his discoveries. However, when he died, everyone forgot about him. Then in 1970, Benoit Mandelbrot praised Julia's work once again and used it to generate Julia sets using a computer.
History
Gaston Julia and Pierre Fatou studied iterative functions and their work led to the idea of attractors, they act as magnets and bring other points towards them, and repellers, points that push others away. Repellers and attractors are important in chaos theory and Mandelbrot used them in his studies of fractals. Julia sets are the boundaries of different basins of attraction, or a region where as a point is put through the function, it reaches its attractor. Also, Julia was interested in a paper called “The Newton-Fourier Imaginary Problem” which tried to find roots using the equation f(z) = z^3 +c = 0 and the Newton-Fourier iterative method. This inspired his work with the Julia set.
In 1918, Julia wrote a paper about his work with iterative functions and the Julia set. His paper is what won him the Grand Prix of the Académie des Sciences. He became famous for winning this award and for his work with iterated polynomials and rational functions. However, Julia lost his fame when he died.
Julia Sets
Julia sets are generated using the same equation as the Mandelbrot set, except the variables mean different things. With the Julia set, the n is still the iteration. However, c is a constant, fixed parameter and z represents a point in the plane. Also, Julia sets can be found within the Mandelbrot set. To generate different kinds of Julia sets, change the constant or zoom into a different place on the Mandelbrot set. The equation works just like it does with the Mandelbrot, though, with the result becoming the new z in every iteration and everything reverts back to its original value when a new point is run through the function.
It can also be made using this function f(z) = z^2 + c where z is a point in the plane and c is a constant.
Julia sets act as strange attractors because they attract points in the set to be in a certain range. The points either escape the set towards infinity as they are put through many iterations of the function, or they will always remain inside the bounded amount and be a part of the set. More interesting Julia sets tend to be bounded by 2 or numbers less than 2 from the origin (2 is like a radius of the point, 2 to -2 is the boundary).
Here is a video showing how a circle can be turned into the Julia set.
Julia sets act as strange attractors because they attract points in the set to be in a certain range. The points either escape the set towards infinity as they are put through many iterations of the function, or they will always remain inside the bounded amount and be a part of the set. More interesting Julia sets tend to be bounded by 2 or numbers less than 2 from the origin (2 is like a radius of the point, 2 to -2 is the boundary).
Here is a video showing how a circle can be turned into the Julia set.
Properties of Julia Sets
Julia sets can either be connected or separated. If it is a special case, the set can be a dendrite where the set contains continuous lines and the removal of any point would split the line in two parts. Disconnected Julia sets are called “Fatou dust” or “Cantor dust.” Cantor dust is even its own fractal where a line is split into thirds and the center third is cut out. This continues for each remaining segment forever.
To determine whether a Julia set is connected or not, this function is used where i is the iteration and x is the point:
Every result becomes the new x for each iteration and the sequence starts again at 0 when a new point is run through the equation. If the sequence grows to infinity, then it is disconnected.
Some Julia sets have reflection symmetry and others have rotational symmetry. They type of symmetry a Julia set has depends on its boundary amount. If it is purely real (any real number + 0i) , then it is reflection symmetry. If it is complex (any real number + any number multiplied by i), then it has rotational symmetry.
Some Julia sets have reflection symmetry and others have rotational symmetry. They type of symmetry a Julia set has depends on its boundary amount. If it is purely real (any real number + 0i) , then it is reflection symmetry. If it is complex (any real number + any number multiplied by i), then it has rotational symmetry.
Julia Set vs. Mandelbrot set
Julia sets are directly related to the Mandelbrot set. The Mandelbrot set and Julia sets view the same set of points on different planes. The Julia set views the points along the the z axis while the Mandelbrot set views points along the c axis through the origin.
They both use the same equation except, when generating a Julia set, the c amount stays constant and the z represents the point. With the Mandelbrot set, c is the point. Also, points in a Julia set are only linked together, or connected, if the points are related to the same point within the Mandelbrot set. To find the Julia set in the Mandelbrot set, zoom in on the edge of the set.
The julia set in the mandelbrot set
Here is a picture showing six Julia sets that can be found within the Mandelbrot Set.