Category: Pedagogy

After this course, I definitely think about math differently, because now I know how it can be used to figure out shapes and layers and colors that I can use in my art. I also think differently about art, because before this course, I had only really done traditional art, and had no idea about any digital art besides using a tablet to draw with instead of a pencil. This course has really opened my mind to what I think art can be, and definitely how it can be created in different ways.

As with most classes, I learned a lot of significant things, but this class really taught me how to push beyond my boundaries and comfort zones. Learning about fractals and affine transformations were mathematically the most difficult part for me, but without those chapters I probably wouldn’t have sharpened my basic math skills….

I felt the coding part of the course was pitched at about the right level.

It’s hard to believe that the semester has finally come to an end!  And I must say that Mathematics and Digital Art was one of most enjoyable courses I’ve ever taught.  I’ll summarize my thoughts in a later reflective post, but today I’d like to showcase my students’ Final Projects.  There really was some exceptional work — but I’ll let the images speak for themselves.

Many students built upon the work we did earlier in the semester. Safina used several different elements we explored during the course.  In addition, she researched turtle graphics in Python to incorporate additional elements (those with lines emanating from a central point).

A few students especially enjoyed the work we did with color and Josef Albers, and created projects around different ways to contrast colors.  Andrew created many variations on a triangular theme.

He took this idea further, and went so far as to combine different triangles in composite images.

Julia, on the other hand, experimented with other geometrical objects.  She played with having colors interact with each other, and created the following image.  Although it may not look like it, the two center circles are in fact exactly the same color!

Two students were interested in image processing, and worked closely with Nick to learn how to use the appropriate Python libraries in order to work with uploaded images.  Madison’s work focused on sampling pixels in an image and replacing them with larger circles to create an impressionistic effect.  She found that using a gray background gave the best results.

Lucas also worked with image processing.  He began by choosing a fairly limited color palette.  Then for each pixel in his image, he found the color on his palette “nearest” the color of that pixel (using the usual Euclidean distance on the RGB values), and then adjusted the brightness.  The image below is one example illustrating this approach.

Two students worked with textures we experimented with earlier in the semester to create pieces like Evaporation.  Maddie wanted to recreate a skyline of Dubai (photo from the Creative Commons).

She picked out particular elements to emphasize in her work.  You can clearly see the tall buildings and lit windows in a screenshot of her movie.

Sharon was interested in working with some of Dali’s paintings, creating impressionistic images by mimicking Dali’s color palette.  Here is her interpretation of The Elephants.

Although we only touched on L-systems briefly in class, Ella wanted to focus on them in her project.  Nick also worked closely with her to create different effects.  Here is a screenshot from one of her movies.

She was able to create some interesting effects by just slightly altering the parameters to L-systems which created trees and superimposing the new L-system on top of the original.  This gave some depth to the trees in her forest.

And one student decided to revisit Iterated Function Systems, but this time using nonlinear transformations in a variety of different ways.  Here is one of her images.

So you can see the wide range of projects students undertook!  This typically happens when you give students the freedom to choose their own direction.  I was also inspired by the enthusiasm of some of the students’ presentations.  They really got into their work.

The last assignment was to write a final Response Paper about the course.  The prompt was the same as that for the Response Paper I assigned midway through the course, so I’m curious to see if their attitudes have changed.  I’ll talk about what I learn in next week’s post!

The last three weeks have been very intense!  The main focus has been on laboratory work, both with Processing and Final Projects.

Week 12 began with another round of Presentations on papers from past Bridges conferences.  This proved to be successful again — with topics ranging from geometrical furniture to zippergons to maps of Thomas More’s Utopia.  We all learned something new!

The time not spent on Presentations that week was devoted to working on projects.  Students were finding their stride, and their ideas were really beginning to take shape.

Two students were interested in image processing, so Nick has been working with them.  The initial problem was that doing anything in Sage involving image processing was just too slow — you’re code is essentially sent to a remote server, executed, and the results sent back.  Since we’re using the free version, this means performance is unpredictable and far too slow for computation-heavy tasks.

So Nick helped with installing Python, finding image processing packages, etc.  This process always takes a lot longer than you imagine, but eventually all issues were resolved.

While Nick was working on image processing, I was helping others.  One student was really interested in using L-systems — one of my favorite geometrical topics recently!   But all my work with Thomas involved code written in Postscript and Mathematica.  Which meant I had to rewrite it in Python.

This proved to be quite a bit trickier than I thought.  The student was looking at The Algorithmic Beauty of Plants, and was interested in modeling increasingly complex L-systems.  First, the simplest type, like that used to generate the Koch curve.  Next came bracketed L-systems, and then bracketed L-systems with multiple rules.  I did finally get all these to work; I intend to clean up the code and share it in a future post.

Week 13 was devoted entirely to Processing and project work.  I started the week with introducing a few new ideas.  What really inspired the class were the mouseX and the mouseY variables.  When your mouse is in screen space, the mouseX and mouseY variables contain the location of your mouse.  So you can put an object where the mouse is, or have the x-coordinate of the mouse in some way determine the color or the size of the object.  The possibilities are virtually endless.

Here’s a sample movie made with this technique.

Since the code is just a few lines long, it won’t take long to explain.  Here is the complete program:

The drawPoint function will draw a point centered at the position of the mouse.  But in addition to those coordinates, the function takes an addition argument:  float(mouseX)/500.  This argument is 0 at the left edge of the screen, and 1 at the right edge.  (Recall the need for “float,” since otherwise Python will perform integer division and give 0 for any number less than 500.)

So the stroke command uses the parameter p to determine how much red is in the color specification.  When p is 0, there is no red, so the dot is black.  And when p is 1, the dot is red.  I used “p**0.4” as an illustration that interpolation need not be linear — the exponent of p determines how quickly or slowly the dot gets brighter as you move the mouse to the right.  Of course the dot also gets larger as you move your mouse to the right, as is clear by the strokeWeight function call.

I showed this example as I introduced the movie project.  Their assignment was to make a movie — anything they wanted to try.  The complete prompt is given on Day 34 of the course website, but I’ll give the gist of it here.  The main goal was to have students use linear interpolation in at least four different ways in their movie.  Of course they could use nonlinear interpolation if they wanted to, but it wasn’t required.

There was no length requirement — it’s easy to make a movie longer by adding more frames.  Just be creative!  Interpolation is a very useful tool in making movies and animations, and has a mathematical basis as well.  So I wanted that to be the focus of the project.  I also had them write a brief narrative about their use of interpolation.

This is Andrew’s movie.  You can clearly see the use of interpolation here in different ways.  It was also nice to see the use of trigonometry to calculate the centers of the dots.

Ella was interested in L-systems, and so Nick spent some time working with her on Python code during his weekly office hours.  Here is what she created.

Lucas wanted to use some interaction with the mouse, and he also had the idea of the sun setting as you moved the mouse down the screen.  Watch how the fractal clouds move with the location of the mouse as well.

So you can see how varied the use of interpolation was!  The students really enjoyed having control over the possible special effects, and created a wide range of interesting features in their movies.

That takes us to Week 14.  On Monday, we had a guest speaker come in, Chamberlain Fong.  I met him in Finland at Bridges this summer, only to find out he lives the next neighborhood over in San Francisco!  He gave a very interesting talk about taking pictures with a spherical or hemispherical camera lens, and the issues involved in printing the pictures.

The problem is essentially the same as creating a map of the globe — making a sphere flat.  There will always be distortion, but you have some control over what type of distortion.  You can keep angles the same, or areas the same — or some combination of the two.  And as these cameras keep getting cheaper, there will be a growing interest in making your spherical photos look realistic.

And although we had class on Wednesday, some students had already left for Thanksgiving Break.  So Nick and I were available to help on an open lab day.  Most of the students actually showed up, and we had a productive day working on movies (which were due Wednesday) and projects.

The next (and final!) post on Mathematics and Digital Art will survey the students’ Final Projects, so stay tuned!

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!

The last two weeks were focused on a study of polyhedra.  While not strictly a digital art topic, I thought it important for students to develop a basic three-dimensional vocabulary in the event they wanted to do further study in computer graphics.

We began with the Platonic solids, naturally, looking at enumerating them geometrically and algebraically.  The algebraic enumeration involved solving the usual

This proved challenging, especially when I gave some additional, similar Diophantine equations for homework.  We also took time to build a dodecahedron and an icosahedron.  This occupied us on Day 17, Day 18, and part of Day 19.

Since the first half of the semester was nearing its end, it was time to begin thinking about Final Projects.  So I took the rest of Day 19 to help students individually choose a topic, and assigned the Project Proposal over the weekend.

We had a brief discussion of graph theory on Day 20, which involved looking at the adjacency graphs of the vertices of the Platonic solids, such as the one for the dodecahedron.

I introduced much of the basic vocabulary, using the chapter I wrote in my polyhedra textbook as a guide.  Of course there is only so much progress to be made during a single class, but I did want to indicate how two apparently different areas in mathematics are related.

The homework involved untangling adjacency graphs, such as the one below.

This is just a triangular prism, although drawn a little unconventionally.  This assignment again proved more difficult than I thought, even with Euler’s formula to help in calculating the number of faces.  So we spent extra time on Day 21 going over the homework, followed by a very brief discussion of duality.  And as students were having difficulty narrowing their focus for their project proposals, I spend the rest of Day 21 talking individually with them as they started building a few rhombic dodecahedra.

Over half of Day 22 was taken up by a quiz on their homework; I didn’t want there to be much time pressure.  The last part of this class was spent creating an in-class sculpture with rhombic dodecahedra.  I chose this dual model for them to build as it is space-filling.

I was surprised at how much they really got into it!  I do hope we have time for a similar project at the end of the semester.  We were in a bit of a rush for time, but still managed to create something intriguing.

Last time I mentioned I assigned a short response paper getting feedback from the students about how the course was going so far, and I said I’d share some of their comments.  All (anonymous) quotes are from student papers.

I really like how hands-on the course is, and how there is a good balance between lecture and lab time.

This was a common opinion — and validates a major feature of the course design.  I am glad students appreciate it!  Another student made a similar remark about the lab time.

I enjoy the lab assignments that we get because I like the designs I create.  It allows us to put to practice what we have learned with each lesson.

I specifically asked about how students’ attitudes about mathematics changed.  I got some encouraging responses.

I was used to thinking of math as just something I had to do, that would probably be useful later in life, but wouldn’t really pertain to whatever I wanted to do with art.  I’ve realized that I would actually really like to use this kind of math in my art in the future, because I never realized what kind of things I could make with this medium.

As someone who wasn’t the best at geometry in high school (I’m more of an algebra person), I think this class has given me a practical use for all the things I learned in high school that I found difficult to grasp or uninteresting.

I was pleased to read responses like this, since again, this reflects an overarching purpose of the course — see how mathematics can actually be used in practice.  One student even went so far as to say,

I would also like to mention something, though this might not be considered significant, I never thought I would have to use matrices ever again in my life.

A few students remarked on using mathematics in the creative process.

At first, I was a little unsure of the role that mathematics took in graphic design, but as soon as we started playing with Sage, I noticed that it affects almost every aspect brought forth in the image.

Making “rules” for you art was something very different for me…..combining the left brain and the right brain creates incredible pieces of work….

Some students made more specific comments.  One student liked the presentations the best.

Looking through the different papers in the Bridges archive and hearing everyone’s presentations really made me realize the extent that mathematics is related to so many other topics.

In addition, I asked for specific suggestions for improvement.  By far the most common remark was that students want to learn more about how the Python code in Sage works.  I was really encouraged when I read those comments!  We will start to learn Processing in a few weeks, and I’ll make sure we discuss the code in more detail.

One student really liked the discussion board I set up for a few of the assignments — it is not difficult to create discussion boards for future assignments, so we’ll try that again.  Another remarked that it would be good to learn about the printing process — and I certainly agree!  But as I remarked in previous posts, I thought the logistics of this challenging endeavor would be too difficult to implement.  It is certainly a future goal.

But overall, I was very glad to read how students were enjoying the course — and also pleased about their suggestions for improvement.  So I’ll work hard at making the second half of the course even better than the first!

Recall that at the end of Week 4, we had just begun a lab on affine transformations and iterated function systems.  At the beginning of Week 5 (Day 11), students were supposed to finish the exercise they began on Friday, and try another.  This was the new prompt:  Create a fractal using two affine transformations. For the first, rotate by 60 degrees, then scale the x by 0.6 and the y by 0.5. For the second transformation, rotate 60 degrees clockwise, scale the x by 0.5 and the y by 0.6, and then move to the right 1.  I provided a link to the fractal which should be produced, shown below.

Again, this proved to be a challenge!  The reason is that the computer is somewhat unforgiving.  To produce the image above, every calculation has to be correct.  Here were some common issues:

1. There was difficulty interpreting the statements as geometric transformations.  In particular, getting the order of the matrix multiplication correct, and interpreting a clockwise rotation as a rotation by -60 degrees.
2. Performing the matrix multiplication correctly.  Some students were using online calculators, and some had trouble with converting degrees to radians.
3. Translating the results into Python code; the affine transformations needed to be converted to the form

I also gave them an exercise to reproduce one of the fractals they needed to analyze for homework the previous week.

This occupied us for all of Week 5, since I also wanted to give them time to work on their upcoming assignment.

Week 6 was our first Presentation Week.  Since we’d only need two days for the presentations given the class size, I spent the first day of the week giving my Bridges talk on producing Koch-like fractal images.

The class was interested to learn more about this algorithm, and I had hoped to spend a week devoted to the topic.  But because of the Processing work we’ll be doing later — animation of fractals using iterated function systems — I didn’t want to rush through the preliminary work on affine transformations and IFS.  Any misunderstandings now would surely be problematic later.

The presentations were to be on papers from the Bridges archives.  All papers since the Bridges conferences began (which was in 1998) are archived online.  The assignment was simple:  find a paper in the archive that’s at least four pages long, and give a five-minute presentation on it to the class.  I did create a discussion board where they posted the title/author of their selections so there wouldn’t be any duplicate presentations.

These presentations went very well.  Most students’ presentations were close to ten minutes long, and the enthusiasm for their presentations was quite evident.  I must admit that I learned a lot, too — I had not read most of the papers they selected.

I had students do simple peer evaluations, and included the item “I would like to learn more about this topic after hearing the presentation.”  They selected a number from 1–5, with 5 meaning “most interested.”  The overall average score was about 3.9 — meaning the papers piqued their curiosity.  In addition, I wanted to get additional ideas about what to include in the few days I set aside for special topics at the end of the semester, or what new ideas to incorporate into next semester’s class.

The most popular paper (4.75 average) was On Generating Dot Paintings in the Style of Howard Arkley, which Madison said reminded her of some of the textures we created during a few of our lab sessions.  The next highest average was 4.25, so this was the clear favorite.  I hope we’ll have time to explore it further later on.

And that took us to the end of Week 6!  I had hoped to have some time in the lab to have students find interesting polyhedra nets they’d like to build, since the next two weeks will be devoted to polyhedra.  No, it’s not really a digital art topic — but it is an area of expertise.  I wanted students to have some exposure to three-dimensional geometry since if they continue to study computer graphics, they will certainly move from two into three dimensions.

Now it’s time to look at some student work!   I’ll focus on one fractal from the assignment on iterated functions systems.  Here is the prompt:  Create a morphed Sierpinski triangle, based on the code in the Sage worksheet. The idea is to have your fractal look like it was derived from a Sierpinski triangle, but just barely. Someone looking at it should wonder about it, and maybe after 30 seconds or so, say “Hey, that looks like a Sierpinski triangle!”

One student created the following image, and wrote:

I think this fractal looks like many sets of pine trees, and I like the way that it shears out. I made the whole thing the same color (gray) to make the fact that it was derived from the Sierpinski triangle less obvious.

Julia creating the following fractal image, and remarked that at one point, she went “too far” and had to backtrack a bit.

Safina went a different direction, and created this.

She wrote,