It’s been an exciting three weeks in Mathematics and Digital Art! We began Week 9 after our Fall break with a talk by Carlo Sequin. Here’s Carlo with one of his sculptures at the University of California, Berkeley.
I met Carlo through the Bridges conferences; he currently sits on the Board. Since he’s so close by, I thought it would be great to have him visit my class.
His talk centered around the computer graphics programs he wrote in order to design sculptures like the one you see in the picture above. Carlo included several different parameters, allowing an incredible variety of images to be generated.
Then he focused on a few of his sculptures and described the design process from conception to final sculpture. He even remarked how he got lucky once — he and his team forgot to measure the width of the door they needed to take the sculpture through, and it fit with just a few inches to spare….
Friday, Day 24, was our first Project day. Students’ ideas were still somewhat vague — they did have to write a Proposal, but still didn’t have a clear direction. We made some progress, though the next Project day helped quite a bit more.
On Monday, Day 25, we dived into working with Processing. I kept in mind the comments from students’ response papers which I mentioned in my previous post — they really wanted to learn more about code.
Now coding is a precision endeavor, as there is no room for error as far as syntax is concerned. So after working through a simple example, I presented them with the following movie.
Admittedly it’s not a blockbuster…but their lab work was to duplicate this movie as precisely as possible. I gave them a basic template (see the May 2016 archives for the start of a six-part series on Processing which I used as a basis for their lab work), but they didn’t just have to tweak numbers — they also had to add new elements.
The main mathematical tool involved was linear interpolation, which we went over in some detail in class. The next class, I had them work on recreating the following movie.
This proved a bit more challenging. Some students at first thought there was a rotation involved — but it’s just linear interpolation again, on a somewhat larger scale. Here are the prompts I gave them:
- the screen is 500 x 500 pixels;
- the dots are always 25 pixels from the edge;
- the colors are the standard red, green, blue, and orange;
- the smaller dots are 100 pixels wide, and the larger ones are 200 pixels wide.
The main challenge was working in screen space, since they needed to calculate the exact centers of the circles. Of course students progressed at different rates, and they were finally getting the main point — you can use linear interpolation to morph any aspect of an image which depends on a numerical parameter. As a mathematician, I was used to thinking along these lines all the time, but it’s a concept that takes a while to really sink in if it’s new.
Friday, Day 27, was our next project day. We worked at fleshing out more details of the students’ various projects. Two students wanted to explore image processing, so Nick continued working with them to download the appropriate packages and get Python installed on their computers. Such projects are never as easy as they sound, but Nick did get everything to work.
While he was doing that, I circulated with the other studets, discussing their projects and answering a few questions about making movies, if they had them. It was a productive day, and everyone left class with a much clearer idea of their project than they had the week before.
On Monday, we continued our work with Processing. I went through an example from my earlier blog posts — making a movie which morphs a Sierpinski triangle — in some detail, explaining it line by line. Again, the most challenging part was converting from user space to screen space. A fractal which fit in a unit square had to be scaled and moved to fit nicely into screen space.
This took just about the entire class. Although we had done work previously with iterated function systems, we did have to take some time to review certain aspects of the code we used before. Then on Day 29, I had them duplicate the following movie.
In order to focus on the coding, I began the movie with a fractal they had on a previous quiz, so they knew which affine transformations to start with. They had to figure out how to modify one of the transformations to produce the final image, and then use linear interpolation to create the movie.
This proved challenging, but everyone made good progress during the lab. They are supposed to finish by Monday.
On Friday of Week 11, we had our second guest speaker — Shirley Yap from California State University, East Bay. I met Shirley last February when Nick and I went to a regional meeting of the Mathematical Association of America. She was in charge of organizing the Art Exhibition Nick and I had pieces in.
After showing a few interactive examples from her web page, Shirley focused her discussion on the following piece she created.
She talked a lot about the challenges of making a physical piece, rather than a work of digital art. For example, she actually wanted to use glass, but is was not possible to etch in glass given the tools she had available. So she had to settle for acrylic, which is very easy to smudge if you aren’t careful. You see, the individual squares can pivot where they are screwed in, so the artwork is interactive.
There were also size requirements, since she had to be able to take in on the airplane with her to a conference. Other issues arose — in some ways working digitally is a lot easier. As Shirley remarked, once you drill a hole, you can’t undrill it…but it’s easy to change a parameter in digital work.
What was really nice was that Shirley talked about mathematical envelopes (one of my favorite topics; I’ve written about it before on my blog), and the curves she used to make her envelopes — Bernstein polynomials. She took the time to go through a few simple examples, so that students got a sense of what these curves are like. It was a nice example of yet another topic in mathematics students hadn’t seen before being used to create art. Truly, mathematics is everywhere….
Stay tuned for the next update of Mathematics and Digital Art!