The Mandelbrot Fractal in Pre-Calculus

Dan Anderson

Queensbury High School

NY Master Teacher - Capital District

dan@recursiveprocess.com

@dandersod

​

All resources found here: bit.ly/mandelbrotfractal

​

Motivation

• How I got started with the Mandelbrot Fractal�
• Why I use the Mandelbrot Fractal as a teaching tool in PreCalculus�

All resources found here: bit.ly/mandelbrotfractal

​

What topics are addressed?

• Complex numbers
• Arithmetic with Complex numbers
• Complex plane and Argand diagrams
• Recursive sequences
• Polar Form of Complex numbers
• Graphing using the Polar plane
• DeMoivre's Theorem

All resources found here: bit.ly/mandelbrotfractal

​

(Presenter hat) Administrivia

How do you use these materials with a class?

​

This is approximately 3-4 days of material.�I wouldn't use an accelerated presentation like this; �I'd make sure that the students are active for each step, trying out examples and doing a whole bunch of thinking/talking/calculating (in that order).

All resources found here: bit.ly/mandelbrotfractal

​

Start with the Basics - Complex Numbers

• The complex plane is a �modified Cartesian plane, �where the real part of a �complex number is graphed �on the x-axis and the �imaginary part is graphed �on the y-axis.
• What is the size (modulus) of �a+bi?

Diagram from Wikipedia

All resources found here: bit.ly/mandelbrotfractal

​

The Mandelbrot Set - Definition

• The Mandelbrot set is defined as the set of all complex numbers, c, where the following (infinite) task is bounded (the size doesn't "blow up").

��

• All points who are bounded (size < 2) are in the set, otherwise the point is out of the set.

All resources found here: bit.ly/mandelbrotfractal

​

Next step

• What points are interesting?
• Is c = 5+12i interesting in this context? �
• z_0 = 0 + 0i -> size of 0, continue
• z_1 = (z_0)^2 + c = (0+0i)^2 + (5 + 12i) -> size of 13 (out of set)

So what points do we have to check?

All resources found here: bit.ly/mandelbrotfractal

​

Let's consider the�following points:

We'll interpret each of these points as a square.

Note: (-2,1) represents �-2 + 1i on the Argand Plane

All resources found here: bit.ly/mandelbrotfractal

​

All resources found here: bit.ly/mandelbrotfractal

​

(Teacher hat) �Have the kids do some work

Assign each student a constant (there are 25 to handle).

​

If you have less than 25 students, you can assign the leftovers later.

All resources found here: bit.ly/mandelbrotfractal

​

(Presenter hat) �Your Turn

You'll be assigned a constant based on what day of the month you were born.

Yes, you will be doing some calculations!

​

There is a clicker app to keep track of the results on the resources page.

All resources found here: bit.ly/mandelbrotfractal

​

(Teacher) Work through a c together

Especially for reluctant learners, it can help to build confidence by working through an example together.

Take 1+0i and reassign the student(s) that had that as their constant.

​

All resources found here: bit.ly/mandelbrotfractal

​

1+0i

• z_0 = 0 + 0i -> Size is 0, so continue process�
• z_1 = (z_0)^2 + c = (0+0i)^2+(1+0i) = 1+0i -> Size is 1, continue process�
• z_2 = (z_1)^2 + c = (1+0i)^2+(1+0i) = 2+0i -> Size = 2 which is not <2 so process stops at the 2nd step. The "escape velocity" is 2.

​

All resources found here: bit.ly/mandelbrotfractal

​

Now it's your turn to do some math

• Take your constant and calculate z_1
• Then find the size (magnitude) of z_1.
• If it's less than 2,
• then your square is black,
• else your square is white (outside mandelbrot set).

​

(Teacher hat) For those who finish early, what shape should this have? Why?

All resources found here: bit.ly/mandelbrotfractal

​

Next Step

• Note: If your c is out already, pick a different square and start verifying people's answers. �
• Calculate z_2 (from your z_1 and your c). �Size? In Mandelbrot set?
• Calculate z_3 (from z_2 and your c). �Size? In Mandelbrot set?

All resources found here: bit.ly/mandelbrotfractal

​

Step 4+

• Calculate |z_4|
• z_5?
• Infinite process right? Are we getting a better picture?

All resources found here: bit.ly/mandelbrotfractal

​

(Teacher hat) �How do we make this better?

How can we improve the picture?

All resources found here: bit.ly/mandelbrotfractal

​

How do we make this automatic?

• The computer programming languages don't know about complex numbers. Can you teach them how to square a complex number?
• What are the Real and Imaginary portions of (a+bi)^2?
• You tell me! Expand and separate.

All resources found here: bit.ly/mandelbrotfractal

​

Automatic - Mandelblocks Program

The Mandelblocks program is linked on the resources page.

• First jump into the code to show where the (a+bi)^2 code is.
• Talk about coloring mode
• How can we do even better?

All resources found here: bit.ly/mandelbrotfractal

​

Mandelbrot Program

Treat each pixel as a coordinate on the Complex Plane.

The Mandelbrot program is linked on the resources page.

• Step 1. A circle? Why??
• Symmetry?

All resources found here: bit.ly/mandelbrotfractal

​

Mandelbrot Zoom

We can do better, let's zoom in and see the detail.

The Mandelbrot Zoom program is linked on the resources page.

• Is there a limit to how far we can zoom in?
• Note the window width as we zoom.
• Note the resolution required as we zoom in.

​

All resources found here: bit.ly/mandelbrotfractal

​

What's next?

• How can we expand on this Mandelbrot set?
• What if we consider z_(n+1) = z_n^3 + c? How will cubing the number change the picture of the set?
• Time for you to get to work and expand (a+bi)^3 and separate the real and imaginary parts.

Let's put in the code and see the fractal!

​

​

All resources found here: bit.ly/mandelbrotfractal

​

What's next continued?

What about ?

Fifth power?

Sixth power?

Tenth power? (Binomial Expansion right?)�3/2 power? What does that even mean in this context?

All resources found here: bit.ly/mandelbrotfractal

​

Polar to the rescue!

If we convert from rectangular coordinates to polar coordinates then we can find the general solution for any power of z_n!

​

​

All resources found here: bit.ly/mandelbrotfractal

​

Polar Form

All resources found here: bit.ly/mandelbrotfractal

​

Why Polar Form?

deMoivre's Theorem!

​

The math is so much easier, operations with real numbers instead of binomial expansion. And n doesn't have to be an integer!

So:

• Convert from rectangular to polar
• Use deMoivre's Theorem
• Convert back to rectangular

​

All resources found here: bit.ly/mandelbrotfractal

​

Mandelbrot Family Interactive

The Mandelbrot Family program is linked on the resources page.

All resources found here: bit.ly/mandelbrotfractal

​

Extensions: Julia Set

Let's consider the following rule (Julia Set).

Start with a complex constant j.

​

All resources found here: bit.ly/mandelbrotfractal

​

Julia Set Interactive

The Julia Set program is linked on the resources page.

​

How are the Julia and Mandelbrot Sets related? drawMandelbrotAndJulia program is also linked on resources page.

All resources found here: bit.ly/mandelbrotfractal

​

Experimental

What if you consider the following rule?

​

​

​

The Experimental program is linked on the resources page.

​

All resources found here: bit.ly/mandelbrotfractal

​

Questions? and Thanks!

All resources found here: bit.ly/mandelbrotfractal

​

Dan Anderson

Queensbury High School

Master Teacher - Capital District

dan@recursiveprocess.com

@dandersod

​

All resources found here: bit.ly/mandelbrotfractal

​

Source: https://www.youtube.com/watch?v=MVzGyAAtHiU

All resources found here: bit.ly/mandelbrotfractal

​