It’s been about a month since my first update, so it’s time for another status report on my second semester teaching Mathematics and Digital Art. It really has been a wonderful semester so far!
Later we’ll look at some student work (like Collette’s iterated function system),
but first, I’d like to talk about course content.
The main difference from last semester in terms of topics covered was including a unit on L-systems instead of polyhedra. You might recall the reasons for this: first, students didn’t really see a connection between the polyhedra unit and the rest of the course, and second, the little bit of exposure to L-systems (by way of project work) was well-received.
I’ve talked a lot about L-systems on my blog, but as a brief refresher, here is the prototypical L-system, the Koch curve. The scheme is to recursively follow the sequence of turtle graphics instructions
F +60 F +240 F +60 F.
There is also an excellent pdf available online, The Algorithmic Beauty of Plants. This is where I first learned about L-systems. It is a beautifully illustrated book, and I am fortunate enough to own a physical copy which I bought several years ago.
Talking about L-systems is also a great way to introduce Processing, since I have routines for creating L-systems written in Python. Up to this point, we’ve just explored changing parameters in the usual algorithm, but there will a deeper investigation later.
One main focus, however, was just seeing the fractal produced by the algorithm. When working in the Sage environment, the system automatically produced a graphic with axes labeled, enabling you to see what fractal image you created.
In Processing, though, you need to specify your screen space ahead of time. So if your image is drawn off-screen, well, you just won’t see it. You have to do your own scaling and translating, which is sometimes not a trivial undertaking.
I also decided to introduce both finite and infinite geometric series in conjunction with L-systems. This had two main applications.
First, we looked at the Sierpinski triangle. Begin with any triangle, and take out the triangle formed by joining the midpoints of the sides. Then repeat recursively, creating the Sierpinski triangle.
Now assume your original triangle had an area of 1, and calculate the area of all the triangles you removed. Since the process is repeated infinitely, this sum is just an infinite geometric series. Interestingly, the sum of this series is 1, meaning, in some sense, you’ve taken away all the area — but the Sierpinski triangle is still left over! This illustrates an idea not usually encountered by students before: infinite sets of points with no area. Makes for a nice discussion.
Second, we looked at the Koch curve (and similarly defined curves). Using a geometric sequence, you can look at the length of any iteration of the polygonal path drawn by the recursive algorithm. And, as expected, these paths get longer each time, and their lengths tend to infinity as the number of iterations increases. This is another nice way to involve geometric sequences and series.
We’ll be doing more with L-systems in the next few weeks, so I’ll finish this discussion on my next update.
A highlight of the past month was a visit by artist Stacy Speyer.
Having worked with weaving and textiles for some time, Stacy has moved on to an investigation of polyhedral forms.
Stacy’s talk provided a wonderful insight into integrating mathematics and art in ways we did not study in class. One of the goals of the Bridges papers presentations and the guest speakers is to do precisely this
I’m now on a mission to share the fun of making geometric forms with others; I designed Cubes and Things, a 3D coloring book. These easy-to-make paper constructions have patterns that can be colored which emphasize different kinds of symmetric properties of the polyhedra. I bring this fun activity to schools and other groups in the form of Polyhedra Parties. And whenever possible, I still work on making more geometric art and learning more about math.
Visit Stacy’s website to take a look at her book, and view many more examples of her stunning work!
Now we’ll take a look at a few more examples of student artwork. These pieces were submitted for the assignment on iterated function systems. Karla created a piece which reminded her of icicles or twinkling lights.
Lainey thought her piece looked like a bolt lightning coming out of a wizard’s staff.
And Peyton’s piece reminder her of flowers.
Finally, as I did last semester, I asked students for some mid-semester comments on how the course was going. You can see the complete prompt on Day 19 of the course website. Here are a few of the comments:
I like how it takes a subject that we are all required to take and creates a real, palpable output. Rather than some types of math, where everything is theoretical, it creates a clear chain of events with an even clearer consequence.
[A]fter seeing the kinds of art works there are that involve the kind of math and programming we use, it opened up a new world of artistic possibilities.
What I enjoy most about this course aside from it being small and very interactive in terms of doing labs and having all of our questions answered, is the fact that I would never thought I would be able to create images using programming or math let alone enjoying the satisfaction of the final product.
I was pleased to read these responses, as they suggest the course is fulfilling its intended purpose. But there were also suggestions for improvement — there was a consensus that the math moved a bit too quickly. When we start the discussion on number theory for analyzing the Koch curve next week, I’ll make sure to keep an eye on the pace. I’ll let you know how it goes in my next update in April!