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- Roy Simpson
- Cosumnes River College

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The picture on the left below is a photo of a fern leaf. The picture on the right is a computer-generated image called the Barnsley fern. It is named after the British mathematician Michael Barnsley, who developed it in the 1980’s.

The Barnsley fern is an example of a **fractal**. A fractal is an infinitely complex pattern created by iteration: repeating a process over and over in an ongoing feedback loop. You can watch a Barnsley fern being created at

https://www.youtube.com/watch?v=iGMGVpLMtMs

One of the most famous and intriguing fractals is the Mandelbrot set, named after the French mathematician Benoît Mandelbrot, who lived from 1924 to 2010. The Mandelbrot set looks like a cardioid, or heart-shaped figure, studded with circles all around its boundary.

Like the Barnsley fern, we can create the Mandelbrot set by repeatedly evaluating a simple equation over and over. This equation uses complex numbers.

The plane of complex numbers is similar to a two dimensional coordinate system. Every point on the plane is represented by a complex number \(c = x + iy\), and we perform the following steps for every point:

Starting with \(z_1 = 0\), we create an infinite sequence of complex numbers \(z_n\) according to the rule

\(z_{n+1} = (z_n)^2 + c\)

If this sequence increases to infinity (it diverges), we color the point \(c\) white. If the sequence does not increase beyond a certain limit (it is bounded), we color the point black.

We create a sequence for every point \(c\) in the plane. The collection of all the black points is the Mandelbrot set. It sits in the portion of the plane where \(x\) is between −1 and 2, and \(y\) is between −1.5 and 1.5. Outside the set, all the values of \(c\) cause the sequence to go to infinity, and the color of the point is proportional to the speed at which the sequence diverges.

Because the calculations must be performed thousands or millions of times, we need computers to study them. The higher the number of iterations, the better the quality of the image produced. Not coincidentally, the Mandelbrot set was discovered in 1980, shortly after the invention of the personal computer.

One of the intriguing features of fractal images is **self-similarity**. If we zoom in on a fractal, we see the same pattern repeated again and again, sometimes with interesting variations. Here is a portion of the boundary of the Mandelbrot set, blown up to many times its original size. You can see the copy of whole set embedded in the image.

You can find a ”Mandelbrot Zoomer” at

https://mathigon.org/course/fractals/introduction

Here are some more fractals, called Julia sets:

Fractal patterns seem familiar because the laws that govern the creation of fractals are found throughout the natural world. Tree branches, rivers, ice crystals, and seashells all form in fractal shapes. You can see some examples at

https://www.treehugger.com/earth-matters/wilderness-resources/blogs

- 9.1: Polar Coordinates
- This section introduces polar coordinates, explaining the relationship between polar and rectangular coordinates, and how to convert between them. It covers plotting points using polar coordinates, interpreting angles and radii, and understanding the use of polar equations. The section also provides examples and exercises to help solidify the concepts and explore applications in graphing and real-world contexts.

- 9.2: Polar Graphs
- This section covers polar graphs, focusing on how to plot equations in the polar coordinate system. It explains common polar graph shapes, such as circles, limaçons, rose curves, and lemniscates, and provides techniques for sketching these graphs. The section also includes methods for converting polar equations to rectangular form and vice versa, along with examples to demonstrate these processes. Practical applications and exercises help reinforce understanding.

- 9.3: Complex Numbers
- This section introduces complex numbers, covering their standard form 𝑎 + 𝑏 𝑖, where 𝑖 is the imaginary unit. It explains operations with complex numbers, including addition, subtraction, multiplication, and division. The section also covers how to represent complex numbers graphically in the complex plane and discusses the polar form of complex numbers, including how to convert between rectangular and polar forms. Practical examples and exercises reinforce these concepts.

- 9.4: Polar Form of Complex Numbers
- This section covers the polar form of complex numbers, explaining how to represent complex numbers using magnitude and angle. It details the conversion between rectangular and polar forms, and the use of De Moivre's Theorem for powers and roots of complex numbers. The section includes examples of multiplying and dividing complex numbers in polar form and provides exercises to practice these concepts.