## Guest Blogger: Geoffrey Owen Miller, I

Geoffrey was one of the two most influential artists who guided me along the path of creating digital art.  We worked at the same school a few years ago, and I sat in on many of his art classes.  Since the faculty often ate lunch together, we’d sit and have many casual chats about art and color.

What I always appreciate about Geoffery is that even though he’s not a mathematician, he is still able to understand and appreciate the mathematical aspects of what I create.  He isn’t intimidated by the mathematics, and I’m not intimidiated by his artisitic expertise.  He really helps me develop as an artist.

Geoffrey is passionate about the use of color, and thinks and writes extensively about the subject.  I’m also fascinated by color, so I thought I’d invite him to be a guest on my blog.  He had so many great things to say, though, that is soon became too much to say in one post.

So enjoy these few weeks, and learn something more about color!  If you like what you see and read, visit Geoffrey’s website at www.geoffreyowenmiller.com.  Enough from me — we’ll let Geoffrey speak for himself….

Back in 2011, I started a series of watercolors that came to be called the Disciplines of Geography. I had just been in Europe and had spent a lot of time visiting the museums and thinking about history while walking amongst all those giant oil paintings. On my return, I was asked by a friend and talented poet to help create a cover for a book of her poems. I had been using the medium of watercolor to sketch during my travels and I decided to use it to make something more substantial and sustained. I found watercolor quite difficult compared to oil paint as every mark you make on the paper remains at some level visible. It forces you to accept mistakes while you do all you can to try and minimize them. But mostly I loved the color the transparent washes of color could create despite my perpetual uncertainty and constant mistakes.

I determined to make my own version of a history painting with watercolor by focusing on the process of painting. I started thinking about how time and space were linked with the history of the European borders. To watch the ebb and flow of the borders of different countries and governments overlapping and being overlapped was similar to the way watercolor extends out from a pour of paint on wet paper. Each layer of color is effectively redrawing the boundaries, while simultaneously building a history where every subsequent state is influenced by the previous colors, values, and borders of those before them.

As these paintings are nonobjective abstractions, meaning I was not looking at anything specific to inform my color choices, I needed something to inform my decision making. Often abstractions come from found or referenced materials, like photos, or found objects, which can often help direct the choices of value, color, line, etc., that one is making. Why choose one blue instead of another? At some level we choose colors we like or look better to us, but I like working in or creating systems that push me outside of what I normally feel comfortable with. As these paintings existed in the realm of ideas and needed a similar structure to build upon, I really started looking into the color wheel as a tool to think about the relationships of the different colors. (This itself became a multi-year ongoing project.) Without going down that path too far, I wanted to share something I found interesting and helpful in the context of making these watercolors as well providing a greater understanding to how color relates in other contexts.

As a student I was shown, in conjunction with the color wheel, certain color relationships that were supposed to help us make harmonious color choices in our images. We were supposed to make color choices based off of the relationships of the colors’ locations on the color circle. Complementary colors were any two colors on opposite sides of the circle, split complements exchanged one of those two colors for the two colors on either side of it, triadic colors were three colors equally distanced from each other, and so on. But was yellow the complement of purple?  Or of blue?  Because artists use more than one color wheel, it depended on which color circle you decided to go with.

The Red Yellow Blue (RYB) primary color circle was most often used during my schooling. This was because those were the pigments we most often used, and in this case yellow did complement the mixture of red and blue, which was called purple. If you mixed all three you tended to get a dark color that, depending on your ratios, was a pretty decent neutral, which is really important in color mixing. Hardly anything in the world around us is a fully saturated color.

However, with light it is very clear, and also demonstrable in your own home with flashlights and colored films, that blue is the complement of yellow. And when they mix they create a fairly neutral white. As color is essentially about light (and pigments are a complicated world in themselves) I decided to go with the Red Green Blue (RGB) color circle. As an added bonus, those clever artists and color scientists figured out that certain other pigments can work quite nicely in this structure. Cyan, Yellow, and Magenta are the colors that Red, Green, and Blue light make when mixed, while Red, Green, and Blue are made with Cyan, Yellow, and Magenta pigments. Learning that was incredibly satisfying.

But since Cyan, Magenta, and Yellow pigments don’t actually create Black, Black is needed for the very darkest colors.  So this system is called the “CMYK” system, where “K” is used for Black so it’s not confused with Blue.

There’s a neat way these two systems can be combined, called the “Yurmby” wheel, which I used to create Disciplines of Geography.  That’s where we’ll start next week!

## Vienna!

Last week, I attended the Symmetry Festival 2016  in Vienna!  I’d like to share some highlights of the week.

The first was a sculpture based on a hypercube net.  One of the most famous illustrations of this geometrical object is Dali’s Crucifixion.

Without going into too many details, eight cubes may be folded into a four-dimensional cube — called a hypercube or tesseract — just as a net of six squares may be folded into a cube.  Because folding a flat net of six squares requires folding into the third dimension, folding a hypercube net requires a fourth spatial dimension.

Analogously, just as the squares are not distorted as they are folded into the third dimension, the cubes are not distorted when folded into the fourth dimension.  A little mind-blowing, but true.

Silas Drewchin created the following sculpture based on the hypercube net.

You’ll notice a few differences:  there is an extra cube added, for a total of nine cubes.  And there are three additional struts added to support the net being tilted at an angle.  Here is what he says about those design elements:

The Hypercross is a metaphor of acceptance….By giving the Hypercross an extra base cube, the mathematical purity of the net tesseract is shattered because a true tesseract only has two base cubes. To disfigure the purity of Christian symbolism embedded in the tesseract, it is tilted to an acute angle as if it is tumbling to the ground. Without this tilted adaptation the sculpture would look dogmatic. The hybrid distortions of the mathematical form and religious form actually add to the message of acceptance.

An additional feature of this work relates to the shadows it casts.  The photo speaks for itself.

A fascinating piece!  I’m a big fan of the fourth dimension, and always appreciate when it arises in mathematical and artistic contexts.

The second highlight for me was meeting someone who is interested in creating the same type of digital images I do.  I had met Paul Gailiunas before at previous conferences, but never knew of his interest.  Here is an adaptation of one of his images — you’ll notice a similarity to those I recently posted on Twitter (@cre8math, 2016 July).

What was facscinating to me was how he created images like these.  I first discussed these images on Day007, where I used the same recursive algorithm typically used to create a Koch snowflake:

F  +60  F  +240  F  +60  F.

But Paul used a different recursive algorithm, which I’ll briefly describe — it definitely has a similar flavor, but there are important differences.  Define a function R so that R(1) means move forward and make a turn of some specified angle a, and R(2) means move forward and make a turn of some angle b.  Then recursively define

R(n) = R(n – 1) R(n – 2).

For example, choosing n = 3 gives the two segments produced by

R(3) = R(2) R(1).

But to obtain R(4), we need to use recursion:

R(4) = R(3) R(2) = R(2) R(1) R(2).

This recursive process may be repeated indefinitely.

This algorithm, like the one used to generate the Koch curve, sometimes closes up and exhibits rotational symmetry.  One challenge in this case is that the center of symmetry is not the origin of the coordinate system!  A rigorous mathematical proof of which choices of angles a and b generate a closed, symmetric curve is still forthcoming….

A third highlight was the Family Day activities in front of the Karlskirche in the Karlsplatz.  At conferences like these (such as the upcoming Bridges conference in Finland), there is a day where mathematics and art activities take place in a public venue, and anyone walking by can sit down and participate.

I sat down at the table demonstrating the Poly-Universe, a system of geometrical forms created by Janos Saxon-Szasz.  They were fun to play with, and rather challenging as well!

I was happy to have solved the puzzle illustrated here — put together all 24 tiles to make a system of hexagons so that the colors and shapes all match.  There are many solutions, but none are easy to find.

What makes these forms interesting is that each tile includes four colors and parts of four differently sized circles.  Moreover, they occur in all possible combinations — and since there are 4 sizes of circles, there are 4! = 24 possible configurations.

There are also forms made from triangles and squares, which pose a different challenge.  I moved to help a young girl and her mother try to put together a 6 x 4 rectangle from the 24 square pieces, but was not so successful there.  At some point you have to move on to see the other exhibits….

A final highlight was the Natural History museum in central Vienna.  They had an extensive array of minerals and gems — the largest collection I’ve ever seen.  The colors and textures were diverse and beautiful, and of course when looking at crystal structures, there’s a lot of geometry.

Because the rhombic dodecahedron is space-filling, it tends to occur as the basis for crystal structures, as shown here.

It is always remarkable to me how nature is constrained by geometry — or maybe geometry is derived from nature?  Regardless of how you look at it, the interaction of geometry and nature is fascinating.

The week went by pretty quickly, as they usually do when attending conferences like these. Stay tuned for further updates on my European adventure!  (I’m also tweeting daily at @cre8math.)

## Bridges: Mathematics and Art II

As I mentioned in my first post about Bridges 2016 (see Day038), one of my students, Nick, had artwork and a short paper accepted, and also received a \$1000 travel scholarship! This week, I’d like to share his work with you. His paper is called Polygon Spirals. Here is the abstract (all quotes are taken directly from Nick’s paper):

Logarithmic spirals may be classically constructed with a chain of similar triangles that share the same center of similitude. We extend this construction to chains of $n$-gons with centers on a logarithmic spiral with turning angle $\pi/n,$ and scale factors with interesting properties. Finally, polygon spirals of this kind are used to produce a variety of artistic images.

I first learned of Nick’s interest in this type of spiral when he was in my Calculus II class, where he introduced a similar idea for his Original Problem assignment (see Day013). I encouraged him to continue with this project and submit his work to Bridges 2016. Here is Nick’s motivation in his own words:

The investigation of polygon spirals began by studying curves that arise when regular polygons with an odd number of sides are strung together. When polygons are strung together into a band by gluing them together along their sides, then the choice of what subsequent edge pairs are being used will define a turning angle introduced at each joint. In this paper we focus on bands made with minimal turning angles and with a consistent turning direction. Each odd $n$-gon defines its own turning angle, $\pi / n.$ Moreover, by introducing a constant scale factor that modifies each subsequent polygon, a large variety of logarithmic spirals can be generated.

My original inspiration came from observing spirals of opposite handedness emerging from adjacent faces of the same polygon. The natural question that arose was which ratio to pick so that the two polygon chains would fall in phase, as in Figure 2. In other words, I needed to find the ratio such that every crossing point of the two logarithmic spirals coincided with the center of a polygon along each band. In this case, the two bands would share a polygon every period. I found this to be achieved when the golden ratio was applied to pentagons, which spurred a determination of the analogous ratio for generalized $n$-gons that corresponds to the sharing of every $n^{\rm th}$ polygon. The construction hinges on similar triangles whose vertices are the center of a polygon, the center of one of that polygon’s children, and the center of similitude.

Referring to Figure3 (illustrated in the case $n=5$), $\alpha=\pi/n$ and $\beta=3\pi/n,$ so that $\gamma=\pi/n.$ Now the triangle with angles labelled $\alpha,$ $\beta,$ and $\gamma$ is isosceles because $\alpha = \gamma.$ Moreover, the ratio between the longer and shorter sides is the same for all triangles since the spiral is logarithmic. If the shorter sides are of unit length, half of the base is $\cos (\pi/n),$ making the base, and thus the ratio, equal to $2\cos (\pi/n).$

Once the ratio has been found, spirals can be nested by applying a rotation of $\pi/n.$ Completing this process yields $n$ nested $n$-gon spirals.

With the ratio found above, it is not hard to show that the equation for one of the logarithmic spirals with this ratio, passing through the centers of the $n$-gons, has the equation:

$r = (2\cos (\pi/n)) ^{n \theta/{\pi}}.$

With most Bridges papers, there is a mix of mathematics and art. In fact, the first criterion listed on the Bridges website for art submissions is “Math content (this is a mathematically sophisticated audience.)” The second is “Esthetic appeal,” so artistry is important, too. What follows is Nick’s discussion of how he used the mathematical ideas discussed above to create digital art.

Begin with a base polygon, and consider producing spirals from every face at every iteration. Although this would create too dense a pattern, it is possible to produce an interesting subset of this set of polygons using a random algorithm. This algorithm assigns a probability that a spiral is generated from each face of the $n$-gon, and this probability is randomly altered and then inherited by each child. The colors are also inherited and altered every generation. In Figure 5, the detailed texture is actually built of many small pentagons at a deep iteration. Off to the right can be seen a randomly generated pair of pentagonal arms falling into phase.

Figure 6 is a tiling of pentagons that features nested rings of pentagons with the property that any two adjacent pentagons differ in size by a ratio of the golden ratio. Figure 7 is an overlay of nonagon spirals with ratios between 0 and 1. This image captures the vast breadth of possible spirals based on a given $n$-gon, and the fascinating way that they interact.

Figure 8 was randomly generated by the same algorithm which produced Figure 5, however with nonagons rather than pentagons. This picture illustrates the infinite detail of a fractal set based on interacting nonagon spirals.

Quite an amazing sequence of images! Nick took a Python programming class his first semester, and so was well-versed in basic coding. As a mathematics major, he had the necessary technical background, and being a double major in art as well helped him with the esthetics.

So yes, it took a lot of work! But the results are well worth it. Nick’s success illustrates what motivated undergraduates can accomplish given the appropriate encouragement and support. Let’s see more undergraduates participate in Bridges 2017!

## Mathematics and Digital Art IV

This week will complete the series devoted to a new Mathematics and Digital Art (MDA) course I’ll be teaching for the first time this Fall.  During the semester, I’ll be posting regularly about the course progression for those interested in following along.

Continuing from the previous post, Weeks 7 and 8 will be devoted to polyhedra.  While not really a topic under “digital art,” so much of the art at Bridges and similar conferences is three-dimensional that I think it’s important that students are familiar with a basic three-dimensional geometric vocabulary.

Moreover, I’ve taught laboratory-based courses on polyhedra since the mid 1990’s, and I’ve also written a textbook for such a course.  So there will be no problem coming up with ideas.  Basic topics to cover are the Platonic solids (and proofs that there are only five), Euler’s formula, and building models with paper (including unit origami) and Zometools.

There are also over 50 papers in the Bridges archive on polyhedra.  One particularly interesting one is by Reza Sarhangi about putting patterns on polyhedra (link here).  Looking at this paper will allow an interested student to combine the creation of digital art and the construction of polyhedra.

At the end of Week 7, the proposal for the Final Project will be due.  During Week 8, I’ll have one of the days be devoted to a construction project, which will give me time to go around to students individually and comment on their proposals.

This paves the way for the second half of the semester, which is largely focused on Processing and work on Final Projects.

In Weeks 9 and 10, the first two class periods will be devoted to work on Processing.  I recently completed a six-part series on making movies with Processing (see Day039–Day044), beginning with a very simple example of morphing a dot from one color to another.

These blog posts were written especially for MDA, so we’ll begin our discussion of Processing by working through those posts.  You’ll notice the significant use of IFS, which is why there were such an important emphasis during the first half of the course.  But as mentioned in the post on Day044, the students in my linear algebra course got so excited about the IFS movie project, I’m confident we’ll have a similar experience in MDA.

The third class in Weeks 9 and 10 will be devoted to work on the Final Project.  Not only does taking the class time to work on these projects emphasize their importance, but I get to monitor students’ progress.  Their proposals will include a very rough week-by-week outline of what they want to accomplish, so I’ll use that to help me gauge their progress.

What these work days also allow for is troubleshooting and possibly revising the proposals along the way.  This is an important aspect of any project, as it is not always possible to predict one’s progress — especially when writing code is involved!  But struggling with writing and debugging code is part of the learning process, so students should learn to be comfortable with the occasional bug or syntax error.  And recall that I’ll have my student Nick as an assistant in the classroom, so there will be two of us to help students on these work days.

Week 11 will be another Presentation Week, again largely based on the Bridges archives.  However, I’ll give students more latitude to look at other sources of interest if they want.  Again, we’re looking for breadth here, so students will present papers on topics not covered in class or the first round of presentations.

I wanted to have a week here to break up the second half of the semester a bit.  Students will still include this week in their outline — they will be expected to continue working on their project as well.  But I am hoping that they find these Presentation Weeks interesting and informative.  Rather like a mini-conference in the context of the usual course.

Weeks 12 and 13 will essentially be like Weeks 9 and 10.  Again, given that most students will not have written any code before this course, getting them to make their own movies in Processing will take time.  There is always the potential that we’ll get done with the basics early — but there is no shortage of topics to go into if needed.  But I do want to make sure all students experience some measure of success with making movies in Processing.

Week 14 will be the Final Project Presentation Week.  This is the culminating week of the entire semester, where students showcase what they’ve created during the previous five weeks.  Faculty from mathematics, computer science, art, and design will be invited to these presentations as well.  I plan to have videos made of the presentations so that I can show some highlights on my blog.

Week 15 is reserved for Special Topics.  There are just two days in this last week, which is right before Final Exams.  I want to have a low-key, fun week where we still learn some new ideas about mathematics and art after the hard work is already done.

So that’s Mathematics and Digital Art!  The planning process has been very exciting, and I’m really looking forward to teaching the course this fall.

Just keep the two “big picture” ideas in mind.  First, that students see a real application of mathematics and programming.  Second, students have a positive experience of mathematics — in other words, they have fun doing projects involving mathematics and programming.

I can only hope that the course I’ve designed really does give students such a positive experience.  It really is necessary to bolster the perception of mathematics and computer science in society, and ideally Mathematics and Digital Art will do just that!

## Mathematics and Digital Art III

Now that the overall structure of the course is laid out, I’d like to describe the week-by-week sequence of topics.  Keep in mind this may change somewhat when I actually teach the course, but the progression will stay essentially the same.

Week 1 is inspired by the work of Josef Albers (which I discuss on Day002 of this blog).  Students will be introduced to the CMYK and RGB color spaces, and will begin by creating pieces like this:

We’ll use Python code in the Sage environment (a basic script will be provided), and learn about the use of random number generation to create pattern and texture.  This may be many students’ first exposure to working with code, so we’ll take it slowly.  As with many of the topics we’ll discuss, students will be asked to read the relevant blog post before class.  While we’ll still have to review in class, the idea is to free up as much class time as possible for exploration in the computer lab.

Week 2 will revolve around creating pieces like Evaporation,

which I discuss on Day011 and Day012.  Again, we’ll be in the Sage environment (with a script provided).  Here, the ideas to introduce are basic looping constructs in Python, as well as creating a color gradient.

Weeks 3–5 will be all about fractals.  This is an ambitious three weeks, so we’ll begin with iterated function systems (IFS), which I discuss extensively on my blog (see Day034, Day035, and Day036 for an introduction).

The important mathematical concept here is affine transformation, which will likely be unfamiliar to most students.  Sure, they may understand a matrix as an “array of numbers,” but likely do not see a matrix as a representation of a linear transformation.

But there is such a wealth of fascinating images which can be created using affine transformations in an IFS, I think the effort is worth it.  I’ve done something similar with a linear algebra course for computer science majors with some success.

I’ll start with the well-known Sierpinski triangle, and ask students to think about the self-similar nature of this fractal.  While the self-similarity may be simple to explain in words, how would you explain it mathematically?  This (and similar examples) will be used to motivate the need for affine transformations.

In parallel with this, we’ll look at a Python script for creating an IFS.  There is a bit more to this algorithm than the others encountered so far, so we’ll need to look at it carefully, and see where the affine transformations fit in.  I’ll create a “dictionary” of affine transformations for the class, so they can see and learn how the entries of a matrix influence the linear/affine transformations.

Having students understand IFS in these three weeks is the highest priority, since they form the basis of our work with Processing later on in the semester.  As with any course like this, so much depends on the students who are in the course, and their mathematical background knowledge.

With this being said, it may be that most of these weeks will be devoted to affine transformations and IFS.  With whatever time is left over, I’ll be discussing fractal images based on the same algorithm used to produce the Koch curve/snowflake (which I discuss on Day007, Day008, Day009, and Day027).

The initial challenge is to get students to understand a recursive algorithm, which is always a challenging new idea, even for computer science majors.  Hopefully the geometric nature of the recursion will help in that regard.

If there is time, we’ll take a brief excursion into number theory.  Without going into too many details (see the blog posts mentioned above for more), choosing angles which allow the algorithm to close up and draw a centrally symmetric figure depends on solving a linear diophantine equation like

$ax+by\equiv c\quad ({\rm mod}\ m).$

It turns out that the relevant equation may be solved explicitly, yielding whole families of values which produce intricate images.  Here is one I just created last week for a presentation on this topic I’ll be giving at the Symmetry Festival 2016 in Vienna this July:

There is quite a bit of number theory which goes into setting up and solving this equation, but all at the elementary level.  We’ll just go as far as we have time to.

Week 6 will be the first in a series of three Presentation Weeks.  This week will be devoted to having students select and present a paper or two from the Bridges archive.  This archive contains over 1000 papers given at the Bridges conferences since 1998, and is searchable.

The idea is to expose students to the breadth of the relationship between mathematics and art.  Because of the need to explain both the mathematics and programming behind the images we’ll create in class, there necessarily will be some sacrifice in the breadth of the course content.   Hopefully these brief presentations will remedy this to some extent.

With three 65-minute class periods and 13 students, it shouldn’t be difficult to allow everyone a 10-minute presentation during this week.  It is not expected that a student will understand every detail of a particular paper, but at least communicate the main points.

Presentations will be both peer-evaluated and evaluated by me.  As these are first-year students, it is understood that they may not have given many presentations of this type before.  It is expected that they will improve as the semester progresses.

I realize that some of these ideas are repeated from last week’s post, but I did want to make these two posts covering the week-by-week sequence of topics self-contained.  I also wanted to give enough detail so that anyone considering offering a similar course has a clear idea of what I have in mind.  Next week, we’ll finish the outline, so stay tuned!

## Mathematics and Digital Art II

This week, I want to talk more about the overall structure of the Mathematics and Digital Art (MDA) course I’ll be teaching in the fall.  I won’t have time to address specifics about content today, but I’ll begin with that next week.

As I mentioned last week, because I can’t require students to bring a laptop to class, MDA will meet in a computer laboratory.  Here is my actual classroom:

Each day, there will be some time in class — usually at least half the 65-minute period — for students to work at their comptuers.  This is a typical 16-week course meeting three times a week.  (Though courses at USF are four credits, hence the longer class time each day.)

Because the course is project-based, there are homework assignments and projects due, but no exams.  There may be an occasional homework quiz on the mathematics, where I let students use their notes.  I prefer this method to collecting homework, since there are always issues of too much copying.  Because I typically change the numbers in homework quiz problems, it is difficult to do well on this type of quiz if all you do is copy your homework from someone else.

Instead of a Final Exam, there is a major project due at the end of the course.  So the first half of the semester — roughly eight weeks — covers a breadth of topics so that students have lots of options when writing a proposal for their Final Project.

Their proposals are due mid-semester, so I have time to evaluate and discuss them, as well as make suggestions.  I try to make sure each project is appropriate for each student — enough to challenge them, but not frustrate them.  Of course there is flexibility for projects to undergo changes along the way, but the proposal allows for a very concrete starting point.

In the second half of the semester, most weeks will include one day for working on Final Projects.  Not only does this emphasize the importance of the projects, but it also lets me see their progress and perhaps alter the direction they’re going if necessary.

The other main focus of the second half of the semester is the use of Processing to make movies.  Because most students will not have studied programming before, I need to make sure there is plenty of time for them to be successful.  We’ll need to take it slowly.

Of course this means that students will not be able to include the use of Processing in their course proposals, but that doesn’t mean they can’t adapt their project along the way to include the use of Processing if they want to.  This is a necessary trade-off, however, since front-loading the course with a discussion of Processing would mean sacrificing the breadth of topics covered.  I like the students to see as much as possible before they write their Final Project proposals.

This is the broad structure of the course.  There are a few other aspects of MDA which also deserve mention.  Three weeks of the course are devoted to presentations.  The idea here is twofold.  First, there is the clear benefit of developing students’ public speaking abilities.

Second, because students will be giving presentations on papers from the Bridges archive (the archive of all papers presented in the Bridges conferences since 1998), they will need to find a paper on a topic of interest to themselves at a level they can understand.  As there are over 1000 papers here, along with an ability to search using keywords, this should not pose a siginificant problem.  Of course should a student have another source about mathematics and art they are keen to share, this would be acceptable as well.

Because the class size is small (13 students), it will feasible to have all students present in each of the three weeks.  The first Presentation Week on Bridges papers will be about the sixth week of the semester, and the second will be about the eleventh week.

The third Presentation Week will be at the fourteenth week of the semester, but this time will be focused on Final Projects.  I will invite mathematics, computer science, and art/design faculty to these presentations as well, and of course will let the students know this in advance.  All presentations will be both peer-evaluated and evaluated by me.

There is also a plan to bring guest speakers from the Bay area into the classroom.  I know a handful of mathematical artists in the area, so bringing in two or three speakers over the course of a semester would be feasible.  This is one of the design features of the First-Year Seminar, incidentally — expose students to the larger San Francicso/Bay area community.

In addition, I can have a student assistant in the classroom as well.  Nick, my student who is also going to the Bridges conference in Finland this year, will serve in that role.  We’ve spent a semester in a directed study to prepare for the Bridges 2016 conference, so he has unique qualifications.  I’ll talk more about Nick in a future post.

When teaching a programming course with a laboratory component, it is difficult to be able to get around to help all students in any given class period.  Certainly some questions students ask have simple answers (as in a syntax fix), but others will require sitting down with a student for several minutes.

So it will be great to have Nick as an assistant, since that will allow two of us to circulate around the classroom during the laboratory part of the class.  The benefit to students will be obvious, and with the small class size, I’m confident they’ll get the attention they need.

Finally, I left the last week (just two class periods) open for special topics.  Given all the demands of a first-semester student just before Final Exam week, I thought it would be nice for them to have a short breather.  I’ll take suggestions for topics from the students, with the Bridges papers they presented on as a good starting point.

So that’s what the course looks like, broadly.  Next week, I’ll begin a week-by-week discussion of the mathematical/artistic content of the course.  I also intend to post weekly or biweekly while the course is going on — course design is a lot easier in theory than in practice, and I’ll be able to share pitfalls and triumphs in real time!

## 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:

What is digital art? It is easy to make a digital image, but what gives it artistic value? This question will be explored in a practical, hands-on way by having students learn how to create their own digital images and movies in a laboratory-style classroom. We will focus on the Sage/Python environment, and learn to use Processing as well. There will be an emphasis on using the computer to create various types of fractal images. No previous programming experience is necessary.

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!