Expt 1.docx

Department of Information Technology, KBTCOE

Experiment No: 01

Date:

1.1 Aim: To Study Mandelbrot & Julia Sets of Complex Number and Observe Different Patterns 1.2 Objectives: 1.To understands the Mandelbrot & Julia Sets Different Patterns. 2.To understand the concept of Fractals. 1.3 Programming Languages Used: C++ Programming (OpenGL) 1.4. Theory: 1.4.1 Mandelbrot Sets The Mandelbrot is the larger set and all the Julia sets are interconnected to the Mandelbrot sets. The term Mandelbrot set is used to refer both to a general class of fractal sets and to a particular instance of such a set. In general, a Mandelbrot set marks the set of points in the complex plane such that the corresponding Julia set is connected and not computable. "The" Mandelbrot set is the set obtained from the quadratic recurrence equation.

With

, where points

in the complex plane for which the orbit of

tend to infinity are in the set. Setting

does not

equal to any point in the set that is not a

periodic point gives the same result. The Mandelbrot set was originally called a molecule by Mandelbrot. J. Hubbard and A. Douay proved that the Mandelbrot set is connected.

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A plot of the Mandelbrot set is shown above in which values of

in the complex

plane are colored according to the number of steps required to reach

. The

kidney bean-shaped portion of the Mandelbrot set turns out to be bordered by a cardioid with equations

1.4.2 Julia Sets The Julia set is the smaller set Compare to the Mandelbrot set.A set of complex numbers which do not converge to any limit when a given mapping is repeatedly applied to them. In some cases the result is a connected fractal set. A Julia set plot showing Julia sets for different values of c; it resembles the Mandelbrot set. The parameter plane of quadratic polynomials - that is, the plane of possible c-values gives rise to the famous Mandelbrot set. For a Julia set, for each pixel apply an iterated complex function. This function is newz = oldz² + c, with z and c both being complex numbers. Z is initially the coordinates of the pixel, and will then constantly be updated through every iteration: each iteration, the "newz" of the previous iteration is now used as "oldz". The color value of the pixel will then become the number of times we had to iterate the function before the distance of z to the origin got larger than 2. The constant c in the formula can be anything really, as long as it's also inside the circle with radius 2. Different values of c give different Julia Sets. Some Julia Sets are connected, others aren't. The Mandelbrot Set is the collection of all points c that generate a connected Julia Set.

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Julia arrays are 1-indexed, which can really trip you up sometimes when you're used to Python, C/++, Java, etc. Julia list comprehensions lack the ability to use conditionals, unlike Python. One can do this with for loops and

if/else, though, as

normally done. Julia matrices are accessed column-major, whereas Python Numpy matrices are accessed row-order. This can affect some design decisions of how to iterate over matrices effectively in memory. Julia dictionaries are hashed differently than Python dictionaries, which can make them slower in many cases. 1.5 Conclusion: Hence we have Studied and Implemented Mandelbrot & Julia Sets of Complex Number and observed its different Patterns. 1.6 Questions: Q 1) Prove that Mandelbrot set is Closed Set? Q 2) Difference between Mandelbrot set and Julia set.

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