Mathematics and Digital Art I

Last week I discussed a movie project I had my linear algebra students do which involved the animation of fractals generated by  iterated function systems.  This week, I’d like to discuss a new classroom project — a Mathematics and Digital Art course I’ll be teaching this Fall at the University of San Francisco!

The idea came to me during the Fall 2015 semester when we were asked to list courses we’d like to teach for the Fall 2016 semester.  I noticed that one of my colleagues had taught a First-Year Seminar course  — that is, a course with a small enrollment (capped at 16) focused on a topic of special interest to the faculty member teaching the course.  The idea is for each first-year student to get to know one faculty member fairly well, and get acclimated to university life.

So I thought, Why not teach a course on mathematics and art?  My department chair urged me to go for it, and so I drafted a syllabus and started the process going.  Here’s the course description:

I have two”big picture” motivations in mind.  First, I want the course to show real applications of mathematics and programming.  Too many students have experienced mathematics as completing sets of problems in a textbook.  In this course, students will use mathematics to help design digital images.  I’ll say more about this in later posts.

And second, I want students to have a  positive experience of mathematics.  This course might be the only math course they have to take in college, and I want them to enjoy it!  Given prevailing attitudes about mathematics in general, I think it is completely legitimate to have “students will begin to enjoy mathematics” as a course goal.

I also think that every student should learn some programming during their college career.  Granted, students will start by tweaking Python code I give to them, just like with the movie project.  Some students won’t progress much beyond this, but I am hopeful that others will.  Given the type of course this is, I really can’t have any prerequisites, so I’m assuming I will have students who either haven’t taken a math course in a year or two, and/or have never written a line of code before.

I’ll go into greater detail in the next few posts about content and course flow, but today I’ll share three project ideas which will drive much of the mathematics and programming content.  The first revolves around the piece Evaporation, which I discuss on Day011 and Day012 of my blog.

Creating a piece like this involves learning the basics of representing colors digitally, as well as basic programming ideas like variables and loops.

The second project revolves around the algorithm which produces the Koch curve, which I discuss in some detail on Day007, Day008, Day009, and Day027 of my blog.

By varying the usual angles in the Koch curve algorithm, a variety of interesting images may be produced.  Many exhibit chaotic behavior, but some, like the image above, actually “close up” and are beautifully symmetric.

It turns out that entire families of images which close up may be generated by choosing pairs of angles which are solutions to a particular linear Diophantine equation.  So I’ll introduce some elementary number theory so we can look at several families of solutions.

The third (and largest) project revolves around creating animated movies of iterated function systems, as I described in the last six posts.

This involves learning about linear and affine transformations in two dimensions, and how fractals may be described by iterated function systems.  The mathematics is at a bit higher level here, but students can still play with the algorithms to generate fractal images without having completely mastered the linear algebra.

But I think it’s worth it, so students can learn to create movies of fractals.  In addition, fractals are just cool.  I think using IFS is a good way not only to show students an interesting application of mathematics and programming, but also to foster an enjoyment of mathematics and programming as well.  I had great success with my linear algebra students in this regard.

I’d like to end this post with a few words on the process of creating a course like Mathematics and Digital Art at USF.  Some of these points might be obvious, others not — and some may not even be relevant at your particular school.

• Start early!  In my case, the course needed to be first approved by the Dean, then next by a curriculum committee in order to receive a Core mathematics designation, and then finally by the First-Year Seminar committee.  The approval process took four months.
• Consider having your course in a computer lab.  At USF, I could not require students to bring a laptop to class, since it could be the case that some students do not have their own personal computer.  I hadn’t anticipated this wrinkle.
• Don’t reinvent the wheel!  One reason I’m writing about Mathematics and Digital Art on my blog is to make it easier for others to design a similar course.  I’ll be talking more about content and course flow in the next few posts, so feel free to use whatever might be useful.  And if would help, here is my course syllabus.

As I mentioned, next week’s post will focus more on the actual content of the course.  Stay tuned!