In Imagifractalous! 1, I talked about varying parameters in the usual algorithm for creating the Koch curve to produce a variety of images, and casually mentioned p-adic valuations.  What happened was about a year after I began exploring these interesting images, I did a Google search (as one periodically does when researching new and exciting topics), and stumbled upon the paper Arithmetic Self-Similarity of Infinite Sequences.

It was there that I recognized the sequence of 0’s and 1’s I’d been using all along, and this sequence was called a 2-adic valuation (mod 2).

Here’s the definition:  the p-adic valuation is the sequence such that the nth term in the sequence is the highest power of p which divides n. So the 2-adic valuation begins

0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 0, 1, 0, 4, ….

Of course all terms with odd n are 0, and for the even n, you just look at the highest power of 2 dividing that even number.  And you can take this sequence (mod 2), or mod anything you like.

I naturally did what any right-thinking mathematician would do — looked up p-adic sequences in the The On-Line Encyclopedia of Integer Sequences.  Now it’s not that p-adic valuations are all that obscure, it’s just I had never encountered them before.

Of course there it was:  A096268.  And if you read through the comments, you’ll see one which states that this sequence can be used to determine the angle you need to turn to create the Koch snowflake.

I wasn’t particularly discouraged — this sort of thing happens all the time when exploring new mathematical ideas.  You just get used to it.  But something unexpected happened.

I starting experimenting with other p-adic valuations mod 2 (since I had two choices of angles), and found similar interesting behavior.  Not only that, p didn’t have to be prime — any integer would do.

But most papers about p-adic valuations assumed p was prime.  Why is that?  If $\nu_p$ is a p-adic valuation and p is prime, it’s not hard to show that

$\nu_p(mn)=\nu_p(m)+\nu_p(n),$

an often-used property in developing a theory of p-adic valuations.  But it only takes a moment to see that

$\nu_6(4)=0,\quad \nu_6(9)=0,\quad \nu_6(4\cdot9)\ne\nu_6(4)+\nu_6(9).$

Bad news for the p-adic theorists, but the fractal images couldn’t seem to care whether p was prime or not….

So I didn’t plunge into researching p-adic valuations, since I needed a treatment which included p composite, which didn’t seem to be out there.

But here’s the neat part.  Most of the work I’d done to prove something already known — that a 2-adic valuation (mod 2) could be used to produce a Koch curve — could be used to study generic p.  So I was able to make fairly good progress in a relatively short amount of time, since I’d already thought about it a lot before.  I suspect it would have taken me quite a bit longer if I’d just casually read about the 2-adic result rather than prove it myself.

The progress made was similar to that of the 2-adic case — number of segments in each arm in the symmetric case, how many solutions given a particular order of symmetry, and so forth.

Now my fractal brain was truly revved up!  I enjoyed creating images using various p-adic valuations — the variety seemed another dimension of endless.  So I started brainstorming about ways to even further diversify my repertoire of fractal images.

The first two weeks of November were particularly fruitful.  Two ideas seemed to coalesce.  The first revolved around an old unexplored question:  what happened when you didn’t only change the angles in the algorithm to produce the Koch snowflake, but you also changed the lengths?

Of course this seemed to make the parameter space impossibly large, but I was in an adventurous mood, and the only thing at stake was a few moments with Mathematica generating uninteresting images.

But what I found was that if an image closed up with a particular symmetry, then as long as the sequence of edge lengths was appropriately periodic, the image with different edge lengths also closed up with the same order of symmetry!

This was truly mind-boggling at first.  But after looking at lots of images and diving into the algorithm, it’s not all that improbable.  You can observe that in the image above, segments of the same length occur in six different orientations which are 60 degrees apart, and so will ultimately “cancel out” in any final vector sum and take you right back to the origin.

Now I don’t have the precise characterization of “appropriately periodic” as yet, but I know it’s out there.  Just a matter of time.

The second big idea at the beginning of November involved skimming through the paper on arithmetic self-similarity mentioned above.  Some results discussed adding one sequence to another, and so I wondered:  what if you added two p-adic sequences before taking their sum (mod 2)?

Well, preliminary results were promising only when the p-adic valuations involved were of the same power of some p, like in the image above (which also involves different edge lengths).

These ideas are only in a very preliminary stage — it’s the perennial problem of, well, waiting…..  It may look liking adding 2-adic and 3-adic valuations doesn’t get you anywhere, but maybe that’s just because you need so many more iterations to see what’s actually happening.  So there’s lots more to explore here.

So as the parameter space gets larger — mutliple different lengths, adding several different p-adic valuations — the variety becomes infinitely diverse, and the analysis becomes that much more involved.

But that makes it all the more intriguing.  And it will be all the more rewarding when I finally figure everything out….

## Digital Art VIII: The End.

Yes, Mathematics and Digital Art is officially over.  Personally, this has been one of the most enjoyable courses I’ve taught, but also one of the more challenging.  From the initial course proposal — begun in November 2015! — to the Final Projects, it seems there was always something unexpected popping up.  But in a good way, since these surprises often involved such things as writing code for a student’s project, or helping a student incorporate a creative aspect into a digital artwork which I had not previously considered.

As I mentioned last week, the last assignment was to write a final Response Paper about the course.  Students had many good things to say, but also made some suggestions for improvement.

I feel like now, though, especially through the talks we were given from other artists, that my scope of the word “art” has broadened. The realm of digital art is so much wider than I could have imagined and includes a lot more mediums than I would have thought.

After completing this course, I realize that the world of art is even vaster, with so many areas still left unexplored or underexplored.

Wow!  I was so glad to see how the course broadened students’ perspectives.  Some students also mentioned the presentations on Bridges papers in this regard — how they were able to learn about many diverse topics in a brief amount of time.  They got a small taste of what it’s like to go to a Bridges conference….

As I had hoped, many students’ perspectives on mathematics changed during the course.  I’ll let the students speak for themselves.

Overall, I really enjoyed the class. Through high school, math was complicated and boring, but this class made me appreciate math in a different way, and I enjoyed learning about coding and digital art.

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.

I soon realized that even though I had no background in code the material was explained and taught so that anyone could understand it.

But some  students commented that they would like to go into more depth as far as programming is concerned.

And one student even decided to minor in computer science!

Best of all, this class is part of the reason why I decided to declare a minor in computer science. It is something I have been considering as I have always had an interest in the subject, but I feel this class had really helped fuel that interest and give me the final nudge I needed.

Most students remarked about how much they loved learning to make movies in Processing, and how the small class size really helped them in terms of their personal learning experience.  The class was just nine students, and I had Nick to help me out — so I felt I really got to know the students.  Not a luxury I’ll always have….

Aside from focusing more on code, some students commented on how we didn’t really use the few weeks on polyhedra anywhere else in the course.  Yes, I wanted to give them some exposure to three-dimensional geometry without having to spend the time developing the mathematics of a three-dimensional Cartesian coordinate system.  But it seems this was just too disjointed from the natural flow.

I think a good substitute would be to discuss L-systems for these two weeks instead.  There are two advantages here.  First, L-systems are another really neat way to create fractals, and the class responded very positively when I gave my Bridges talk on L-systems and Koch curves.  And second, this would give a few more weeks when we could discuss coding, especially recursion.  In general, recursion is a difficult topic to teach — but teaching recursion in the context of computer graphics might really help the learning process.

I also asked Nick if he’d say a few words about his experience with the course.

I was very excited to work with Professor Matsko on Math and Digital Art, I think we both caught on a while back that the great flexibility I’ve found within the math department to support creative interests can be shared with other students. Also the instant gratification that we were finding from programming was really picking up. When communication from generated images was profoundly more efficient than any attempt to explain with words, it was clear that we had to invite more people to the conversation.

What I think has been so powerful following my discovery of programming – and what I hope I left with the students – is the ability to paste mathematical notation for very specific thoughts directly into the computer so that I can just look at what those thoughts literally mean. This continues to be the best way that I’ve found to meet and greet interesting new patterns and behaviors. Ultimately I think this is extremely natural and that the students caught on quite well: curious how a fractal might react? Poke it and find out!

Some areas definitely saw unexpected challenges, but once we got their mathematical comfort zones lined up with the curriculum the enthusiasm was excellent. My favorite part was definitely helping students let their imaginations fill the newly available parameter space. It was really great that we had a small class size, too – the two of us walking around made the perfect environment for any question to be asked during open days when they worked on self-directed projects. And it really felt like success to observe students becoming fixed on a single idea of what they wanted to create, whether or not they knew anything about how they would create it.

Overall I may have learned more than the students and was very surprised by the deeper understanding that begins to build after explaining to the masses. I would be very interested in assisting a course again!

As long as enough students enroll, I’ll be teaching Mathematics and Digital Art again next semester!  I won’t be reporting as frequently as I did this first semester, but expect updates every month or so….  Who knows what creative ideas next semester’s students will come up with?

## Digital Art VII: Final Projects!

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!

## Imagifractalous! 1: How it all began.

September 2, 2015 — that was the day Thomas sent an email to the faculty in the math department asking if anyone was willing to help him learn something about fractals.  And that was when it all began.

Now I’ve told this story in various forms over the past year or so on my blog, but I want to begin a thread along a different direction. I’d like to give a brief chronological history of my work with fractals, with an eye toward the fellow mathematician interested in really understanding what is going on.

I’ll be formal enough to state fairly precisely the mathematical nature of what I’m observing, so as to give the curious reader an idea of the nature of the mathematics involved.  But I’ll skip the proofs, so that the discussion doesn’t get bogged down in details.  And of course provide lots of pictures….

You might want to go back to read my post of Day007 — there is a more detailed discussion there.  But the gist of the post is the question posed by Thomas at one of our early meetings:  what happens when you change the angles in the algorithm which produces the Koch curve?

What sometimes happens is that the curve actually closes up and exhibits rotational symmetry.  As an example, when the recursive routine

F   +40   F   +60   F   +40   F

is implemented, the following sequence of segments is drawn.

Now this doesn’t look recursive at all, but rather iterative.  And you certainly will have noticed something curious — some sequences of eight segments, which I call arms, are traversed twice.

All of this behavior can be precisely described — all that’s involved is some fairly elementary number theory.  The crux of the analysis revolves around the following way of describing the algorithm:  after drawing the nth segment, turn +40 degrees if the highest power of 2 dividing n is even, and turn +60 degrees if it’s odd.  Then move forward.

I would learn later about p-adic valuations (this is just a 2-adic valuation), but that’s jumping a little ahead in the story.  I’ll just continue on with what I observed.

What is also true is that the kth arm (in this case, the kth sequence of eight segments) is retraced precisely when the highest power of power of 2 dividing k is odd.  This implies the following curious fact:  the curve in the video is traced over and over again as the recursion deepens, but never periodically.  This is because the 2-adic valuation of the positive integers isn’t periodic.

So I’ve written up some results and submitted them to a math journal.  What I essentially do is find large families of angle pairs (like +40 and +60) which close up, and I can describe in detail how the curves are drawn, what the symmetry is, etc.  I can use this information to create fractal images with a desired symmetry, and I discuss several examples on a page of my mathematical art website.

As one example, for my talks in Europe this past summer, I wanted to create images with 64 segments in each arm and 42-fold symmetry.  I also chose to divide the circle in 336 parts — subdivision into degrees is arbitrary.  Or another way of looking at it is that I want my angles to be rational multiples of $\pi,$ and I’m specifying the denominator of that fraction.

Why 336?  First, I needed to make sure 42 divided evenly into 336, since each arm is then 8/336 away from the previous one (although they are not necessarily drawn that way).  And second, I wanted there to be enough angle pairs so I’d have some choices.  I knew there would be 96 distinct images from the work I’d already done, so it seemed a reasonable choice.  Below is one of my favorites.  The angles used here are 160 and 168 of the 336 parts the circle is divided into.

Now my drawing routine has the origin at the center of the squares in the previous two images, so that the rotational symmetry is with respect to the origin.  If you think about it for a moment, you’ll realize that if both angles are the same rational multiple of $\pi,$ you’ll get an image like the following, where one of the vertices of the figure is at the origin, but the center of symmetry is not at the origin.

Of course it could be (and usually is!) the case that the curve does not close — for example, if one angle is a rational multiple of $\pi,$ and the other is an irrational multiple of $\pi.$  Even when they do close, some appealing results are obtained by cropping the images.

So where does this bring me?  I’d like a Theorem like this:  For the recursive Koch algorithm described by

F   $\alpha_0$   F   $\alpha_1$   F   $\alpha_0$   F,

the curve closes up precisely when (insert condition on $\alpha_0$ and $\alpha_1$), and has the following symmetry (insert description of the symmetry here).

I’m fairly confident I can handle the cases where the center of symmetry is at the origin, but how to address the case when the center of symmetry is not the origin is still baffling.  The only cases I’ve found so far are when $\alpha_0=\alpha_1,$ but that does not preclude the possibility of there being others, of course.

At this point, I was really enjoying creating digital artwork with these Koch-like images, and was also busy working on understanding the underlying mathematics.  I was also happy that I could specify certain aspects of the images (like the number of segments in each arm and the symmetry) and find parameters which produced images with these features.  By stark contrast, when I began this process, I would just randomly choose angles and hope for the best!  I’d consider myself lucky to stumble on something nice….

So it took about a year to move from blindly wandering around in a two-dimensional parameter space (the two angles to be specified) to being able to precisely engineer certain features of fractal images.  The coveted “if-and-only-if” Theorem which says it all is still yet to be formulated, but significant progress toward that end has been made.

But then my fractal world explodes by moving from 2-adic to p-adic.  That for next time….