Colorful
images of the complex Mandelbrot set have been seen and liked by millions of
people that have no interest at all in the mathematics that defines members of
the set.

I
recently uploaded to Scratch a

*Mandelbrot Set in Scratch*project that steps back from the colorful images and computes the set in black and white. In the picture below, members of the set are those complex numbers painted white and nonmembers are in the black.
Input
the numbers in the

*Try These*list in the lower right corner and view a more detailed portion around the two o’clock position of the large, heart-shaped bulge.
Since
the complex numbers are infinitely dense, there is no limit as to how deep one
can zoom into the set. Practically, our computers have limited precision when it
comes to representing numbers and our screens have a finite resolution so we
will never see the complete Mandelbrot set. That’s why the set is sometimes
called the most

*complicated object in mathematics*.
To
view this project and play with it or to download it to your computer, click on this link.

I
did include documentation in the project notes section but nowhere near
enough information to help anyone interested in really understanding the
mathematics underlying the set and how to code the mathematics in Scratch. For
example, Scratch does not contain prebuilt blocks for computing with complex
numbers. I coded subroutines in Scratch to do the necessary complex number
arithmetic.
Something
as mathematically profound as the Mandelbrot set cannot be explained within the
134 characters of a tweet so I’m writing three documents to break up the task
of explaining the set itself and the coding. The first document in the
following list is finished and the other two will be ready within a week.

*Introduction to the Mandelbrot Set*.pdf

*The Paper and Pencil Part*.pdf

*Developing the Scratch Script for the Mandelbrot Set*.pdf

By
the way, this project was selected as a

*Featured Project*on the Scratch home page.