## 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 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!

## Bridges: Mathematics and Art I

Just registered for Bridges 2016 last week!

Simply put, Bridges is the best mathematics conference ever.  You meet people from all around the world who are interested in the interplay between mathematics and art.

Not just M. C. Escher, either (though many are interested in his work).  Some Bridges attendees (shall we call them Bridgers?) are artists by profession, but others are mathematicians, computer scientists, physicists — you name it.  All are artists by vocation.

Interests span not only art in a more usual sense — watercolor, acrylic, oil, pastel, drawing — but also digital art, sculpture in almost any medium you can think of, poetry, architecture, music, fiber arts, dance, digital animations and movies, fashion, origami, and likely there will be some new art form introduced this summer as well!

The art exhibition is amazing.  You can see a few examples above.  You see the wooden spiral?  Each inlaid rectangle is a different piece of wood!  The craftsmanship is really superb.

One neat aspect is that most of the artists also attend Bridges.  That means if you see something you really like, you can just look for the right name tag and start up a conversation.  As you would expect, all the artists are eager to discuss their work.

Be ready for some surprises, too.  I met my friend Phil Webster at the conference – we starting talking because I was from San Francisco, and he also lives in the Bay area.  So we’ve met up a few times since the conference to discuss mathematics, art, and programming.  He even gave a talk in our Mathematics Colloquium at the University of San Francisco.  Of course, his talk was great….

Even if you don’t go to the conference, you can still appreciate all the art.  You can visit the Bridges 2015 online gallery and see all the art that was exhibited.  Not only are there descriptions of all the works by the artists themselves, but there’s also contact information so you can get in touch if you’d like.  Please do!

The Bridges 2016 gallery is not online yet, but I’ve got two pieces accepted for this year’s exhibition.  This is my favorite.

Then there are the talks.  You learn so much just by going to them.  The range of topics is incredibly diverse — look back at the list above!  Last summer, I gave a talk about Random Walks on Vertices of Archimedean Tilings.  My favorite work discussed in the paper is Bear.  You can read the paper to learn how it was made, if you’re interested.  The first print of Bear is hanging in my friend Cory’s house in Florida.  Hi, Cory!

As you’ll see if you click on the link to my paper, there is an archive of all papers — over 1000! — given at Bridges conferences since 1998.  What’s nice is that you can actually search for specific topics, so it’s easy to use.  No shortage of reading material on mathematics and art….

In addition to the exhibition and all the presentations, there are also dance performances, poetry readings, theatre performances, movie showings, a music night — any number of interesting activities relating mathematics and art.  If you want to learn more, just go to the Bridges 2016 website.  There’s complete information on the upcoming conference there.

This year, the conference is being held at the University of Jyväskylä in Jyväskylä, Finland.  I’ve never been to Finland before, so I’m looking forward to an exciting trip!  What’s also nice about the conference is that in the evenings, you can just take a stroll with other Bridgers, looking for some interesting place to have dinner.  I always love exploring new countries, and especially like trying new cuisines!

But even though Bridges 2016 is in July, I actually starting preparing last November.  Since there was a January deadline for submitting papers to the conference, and since I knew I’d be doing a lot of traveling over our Winter Break, I wanted to get an early start.  The papers are all reviewed by three referees, so your mathematics should be sound.  Usually they have comments to make, and so you often need to make some revisions a few months later before a final submission.

My paper is on fractals this year.  A lot of what I wrote in that paper you’ve already seen on my blog — but I’ll be sure to give a link when I write a follow-up post on Bridges 2016 later on in the summer.  Here’s one of my favorite images discussed in the paper.

There are deadlines to submit artwork as well, so it’s important to be organized.  For both papers and artwork, the online submission system is actually really easy to use.  I just wanted to let you know something about the process so you can submit something to next year’s conference….

Last Fall, I received an email about a new addition to the Bridges menu — student scholarships.  And in my calculus class, I had a student Nick who is a double major in mathematics and art.

Turns out Nick was really interested in trying to submit to Bridges, so we worked out a one-credit directed study course just for that purpose.  As of this moment, I’m happy to say that two of Nick’s artworks were accepted!  And we just submitted the final revisions to his paper, and are waiting to hear back.  We should know about the scholarship soon — I’ll update this post when I have more information.  One of my favorite images from Nick’s paper is this one.

You can read the paper to see how he creates it….link to follow.

So think about including Bridges in your future travel!  Many artists bring their families and make a summer vacation out of the conference.  It’s quite an experience.

And if you’re a student, consider submitting as well!  Maybe you’ll earn a scholarship to attend:  here’s more information on the Student Travel Scholarship.  Preference is given to those student who submit papers, artwork, or movies.

You will need a letter from one of your teachers or professors — so ask someone to be your mentor.  If you can’t find someone, well, just ask me.  I’ll be glad to help out (as long as I don’t get too many requests!).

Later on in the summer, I’ll tell you all about the experience.  Hope to see you at some Bridges conference soon!

P.S. (10 July 2106):  Nick did receive the travel scholarship.  Congratulations!