## Bay Area Mathematical Artists, V.

The Spring semester is now well underway!  This means it’s time for the Bay Area Mathematical Artists to begin meeting.  This weekend, we had our first meeting of 2018 at the University of San Francisco.

As usual, we began informally at 3:00, giving everyone plenty of time to make it through traffic and park.  This time we had three speakers on the docket:  Frank A. Farris, Phil Webster, and Roger Antonsen.

Frank started off the afternoon with a brief presentation, giving us a teaser for his upcoming March talk on Vibrating Wallpaper.  Essentially, using the complex analysis of wave forms, he takes digital images and creates geometrical animations with musical accompaniment from them.  A screenshot of a representative movie is shown below:

You can click here to watch the entire movie.  More details will be forthcoming in the next installment of the Bay Area Mathematical Artists (though you can email him at ffarris@scu.edu if you have burning questions right now).  Incidentally, the next meeting will be held at Frank’s institution, Santa Clara University; he has generously offered to host one Saturday this semester as we have several participants who drive up from the San Jose area.

Our second speaker was Phil Webster, whose talk was entitled A Methodology for Creating Fractal Islamic Patterns.  Phil has been working with Islamic patterns for about five years now, and has come up with some remarkable images.

Here, you can see rings of 10 stars at various levels of magnification, all nested very carefully within each other.  While it is fairly straightforward to iterate this process to create a fractal image, a difficulty arises when the number and size of rosettes at a given level of iteration are such that they start overlapping.  At this point, a decision must be made about which rosettes to keep.

This decision involves both mathematical and artistic considerations, and is not always simple.  One remark Phil hears fairly often is that he’s actually creating a model of the hyperbolic plane, but this is in fact not the case.  Having sat down with him while he explained his methodology to me, I can attest to this fact.  His work may be visually somewhat reminiscent of the hyperbolic plane, but the mathematics certainly is not.

Moreover, in addition to creating digital prints, Phil has also experimented with laser cutting Islamic patterns, as shown in the intricate pieces below.

If you would like to learn more about Phil’s Islamic fractal patterns, feel free to email him at phil@philwebsterdesign.com.

We ended with a talk by Roger Antonsen, From Simplicity to Complexity.  Roger is giving a talk at the Museum of Mathematics in New York City next month, and wanted a chance to try out some ideas.  He casually remarked he had 377 slides prepared, and indicated he needed to perhaps trim that number for his upcoming talk….

Roger remarked that as mathematicians, we know on a hands-on basis how very simple ideas can generate enormous complexity.  But how do you communicate this idea to a general audience, many who are children?  This is his challenge.

The idea of this “tryout” was that Roger would share some of his ideas with us, and we would give him some feedback on what we thought.  One idea that was very popular with participants was a discussion of Langton’s ant.  There are several websites you can visit — but to see a quick overview, visit the Wikipedia page.

The rules are simple (as you will already know if you googled it!).  An ant starts on a grid consisting totally of white squares.  If the ant is on a white square, it turns right a quarter-turn, moves ahead one square, and the square the ant was on turns to black.  But if the ant is on a black square, it turns left a quarter-turn, moves one unit, and the square the ant was on turns to white.

It seems like a fairly simple set of rules.  As the ant starts moving around, it seems to chaotically color the squares black and white in a random sort of pattern.

The image above shows the path of the ant after 11,000 steps (with the red pixel being the last step).  Notice that the path has started to repeat, and continues to repeat forever!

Why?  No one really knows.  Yes, we can see that it actually does repeat, but only sometime after 10,000 apparently random steps.  The behavior of this system has all of a sudden become very mysterious, without a clear indication of why.

If the rules for moving the ant always resulted in just random-looking behavior, perhaps no one would have looked any further.  But there are so many surprises.  Especially since  there is no reason you have to stick to the rules above.  As suggested in the Wikipedia article, you can add more colors, more rules, and even more ants….

For example, consider the set of rules in the following image.  It should be relatively self-explanatory by now:  there are four colors; if the ant is on a black square, turn right a quarter-turn and move forward one unit, then change the color of the square the ant was on to white; then continue (where green squares becomes black, in cyclic order).

This looks like a cardiod!  And if you actually zoom in enough, you’ll see that this is the image after 500,000,000 iterations…though again, no one has the slightest idea why this happens.  Why should a simple set of rules based on 90° rotations generate a cardioid, of all things?

From the simple to the complex!  This was only one of literally dozens of topics Roger was able to elaborate on — and he illustrated each one he showed us with compelling images and animations.  For more examples, please see his web page, or feel free to email him at rantonse@ifi.uio.no.  You can also see the announcement for his MoMath talk here.

As usual, we went our for dinner afterwards, this time for Thai.  It seems that no one wanted to leave — but some of the participants had a 90-minute drive ahead of them, so eventually we had to head home.  Stay tuned for the summary of next month’s meeting, which will be at Santa Clara University!

## Bay Area Mathematical Artists, IV.

We had our last meeting of the Bay Area Mathematical Artists in 2017 this weekend!  We had a somewhat lower turnout than usual since we’re moving into the holiday season.  But it really wasn’t possible to move the seminar a week earlier, since many of us affiliated with universities were in the middle of Final Exams.

As we had been doing before, we began with a social half hour while waiting for everyone to show up.  We then moved on to the more formal part of the afternoon.

There were three speakers originally slated to give presentations, but one had to cancel due to illness.  Still, we had two very interesting talks.

The first talk, Squircular Calculations, was given by Chamberlain Fong.  Chamberlain did speak at the inaugural September meeting, but wanted a chance to practice a new talk he will be giving at the Joint Mathematics Meetings in San Diego this upcoming January.

So what is a squircle?  Let’s start with a well-known family of curves parameterized by p > 0:

$|x|^p+|y|^p=1.$

When p = 2, this gives the usual equation for a circle of radius 1 centered at the origin.  As p increases, this curve more and more closely approaches a square, and it is often said that “p = ∞” is in fact a square.

However, in Chamberlain’s opinion, the algebra becomes a bit unwieldy with this way of moving from a circle to a square.  He prefers the following parameterization:

$x^2+y^2-s^2x^2y^2=1,$

where s = 0 gives a circle, and the central portion of the curve when s = 1 is a square.  As s varies continuously from 0 to 1, the central portion of this curve continuously transforms from a circle to a square.  This parameterization was created by Manuel Fernandez Guasti; you can read his original paper here.

Chamberlain’s talk was about extending this idea in various ways into three dimensions.  He showed images of squircular cylinders, squircular cones, etc., and also gave equations in three-dimensional Cartesian coordinates for all these surfaces.  You can see some of the images in the title page of his presentation above.  It was quite fascinating, and there were lots of questions for Chamberlain when his talk was finished.  Feel free to email him at chamb3rlain@yahoo.com if you have further questions about squircles.

The second talk was given by Dan Bach (also a speaker at our inaugural meeting), entitled Making Curfaces with Mathematica.  Yes, “curfaces,” not “surfaces”!

Dan took us through a tour of his very extensive library of Mathematica-generated images.  He is fond of describing curves using parameters, and then changing the parameters over and over again to generate new images.

This is easy to do in Mathematica using the “Manipulate” command; below is a screen shot from Mathematica’s online documentation showing an example.

The parameter n is used in plotting a simple sine function — as you move the slider, the graph changes dynamically.  Note that any numerical parameter may be experimented with in this way.  Simply make a slider and watch how your image changes with the varying parameter.

So what are “curfaces”?  Dan uses the term for images create by a family of closely related curves which, when graphed together, suggest a surface.  As we see in the example above, the family of curves suggests a spiraling ribbon in which several brightly colored balls are nestled.  Dan showed several more examples of this and discussed the process he used to create them.  To see more examples, you can visit his website www.dansmath.com or email him at art@dansmath.com.

Once the talks were over, we had some time for puzzles!  Earlier in the week, when I knew we were not going to have an overabundance of talks, I asked participants to bring some of their favorite puzzles so we could all have some fun after the talks.  We were all intrigued with the wide variety of puzzles participants brought.

My dissection puzzle was actually quite popular — that is, until a few of the participants solved it!

You might recognize this from my recent blog post on geometrical dissections.  The pieces above are arranged to make a square, but they may also be rearranged to make an irregular dodecagon.  Some asked if I had any more copies of this puzzle, but unfortunately, I didn’t.  Maybe I’ll have to start making some….

As has been our tradition, many of us went out to dinner afterwards.  We went to our favorite nearby Indian buffet, and engaged in animated conversation.  Interestingly, after talking a bit about mathematics and art, Chamberlain began entertaining us with his wide repertoire of word puzzles.

To give just one example, he asked us to come up with what he calls “mismisnomers.”  Usually, the prefix “mis-” means to incorrectly take an action, as in “misspell.”  But some words, like “misnomer,” begin with “mis-,” while the remainder of the word, “nomer” is not even a word!  How many mismisnomers can you think of?  This and similar amusing puzzles kept us going for quite a while, until it was finally time to head home for the evening.

So that’s all for the Bay Area Mathematical Artists in 2017.  Stay tuned in 2018…our first meeting next year will be at the end of January, and I’ll be sure to let you know how it goes!

## Bay Area Mathematical Artists, III

Another successful meeting of the Bay Area Mathematical Artists took place yesterday at the University of San Francisco!  It was our largest group yet — seventeen participants, include three new faces.  We’re gathering momentum….

Like last time, we began with a social half hour from 3:00–3:30.  This gave people plenty of time to make their way to campus.  I didn’t have the pleasure of participating, since the campus buildings require a card swipe on the weekends; I waited by the front door to let people in.  But I did get to chat with everyone as they arrived.

We had a full agenda — four presenters took us right up to 5:00.  The first speaker was Frank A. Farris of Santa Clara University, who gave a talk entitled Fibonacci Wallpaper Spirals.

He took inspiration from John Edmark’s talk on spirals at Bridges 2017 in Waterloo, which I wrote about in my blog last August (click here to read more).  But Frank’s approach is rather different, since he works with functions in the complex plane.

He didn’t dive deeply into the mathematics in his talk, but he did want to let us know that he worked with students at Bowdoin College to create open-source software which will allow anyone to create amazing wallpaper patterns.  You can download the software here.

Where do the Fibonacci numbers come in?  Frank used the usual definition for the Fibonacci numbers, but used initial values which involved complex numbers instead of integers.  This allowed him to create some unusually striking images.  For more details, feel free to contact him at ffarris@scu.edu.

Next was our first student talk of the series, My Experience of Learning Math & Digital Art, given by Sepid Ebrahimi.  Sepid is a student in my Mathematics and Digital Art course; she is a computer science major and is really enjoying learning to code in Processing.
First, Sepid mentioned wanting to incorporate elements into her work beyond simple points, circles, and rectangles.  Her first project was to recreate an image of Rick and Morty, the two main characters in the eponymous cartoon series.  She talked about moving from simple blocks to bezier curves in order to create smooth outlines.
Sepid then discussed her second project, which she is using for her Final Project in Mathematics and Digital Art.  In order to incorporate sound into her work, she learned to program in Java to take advantage of already-existing libraries.  She is creating a “live audio” program which takes sound input in real time, and based on the frequencies of the sound, changes the features of various geometrical objects in the video.  Her demo was very fascinating, and all the more remarkable since she just started learning Processing a few months ago.  For more information, you can contact Sepid at sepiiid.ebra@gmail.com.
The third talk, Conics from Polygons: the Chord Ratio Construction, was given by Scott Vorthmann.  He is spreading the word about vZome, an open-source virtual environment where you can play with Zometools.
The basis of Scott’s talk was a simple chord ratio construction, which he is working on with David Hall.  (Here is the GeoGebra worksheet if you would like to play with it.)  The essential idea is illustrated below.
Begin with two segments, the red and green ones along the coordinate axes.  Choose a ratio r.  Now add a chord parallel to the second segment and r times as long — this gives the thick green segment at y = 1.  Connect the dots to create the third segment, the thin green segment sloping up to the right at x = 1.  Now iterate — take the second and third segments, draw a chord parallel to the second segment and r times as long (which is not shown in the figure), and connect the dots to form the fourth segment (the thin green segment sloping to the left).
Scott then proceeded to show us how this very simple construction, when iterated over and over with multiple starting segments, can produce some remarkable images.
Even though this is created using a two-dimensional algorithm, it really does look three-dimensional!  Conic sections play a fundamental role in the geometry of the points generated at various iterations.  Quadric surfaces in three dimensions also come into play as the two-dimensional images look like projections of quadric surfaces on the plane.  Here is the GeoGebra worksheet which produced the graphic above.  For more information, you can contact Scott at scott@vorthmann.org.
The final presenter was Stacy Speyer, who is currently an artist-in-residence at Planet Labs.  (Click here to read more about art at Planet Labs.)  She didn’t give a slideshow presentation, but rather brought with her several models she was working on as examples of Infinite Polyhedra Experiments with Planet’s Satellite Imagery.
One ongoing project at Planet Labs is planetary imaging.  So Stacy is taking high-resolution topographical images and using them to create nets for polyhedra.  She is particularly interested in “infinite polyhedra” (just google it!).  As you can see in the image above, six squares meet at each vertex, and the polyhedron can be extended arbitrarily far in all directions.
One interesting feature of infinite polyhedra (as you will notice above) is that since you cannot actually create the entire polyhedron, you’ve got to stop somewhere.  This means that you can actually see both sides of all the faces in this particular model.  This adds a further dimension to artistic creativity.  Feel free to contact Stacy at cubesandthings@gmail.com for more information!
We’ll have one more meeting this year.  I am excited to see that we’re making so much progress in relatively little time.  Presentations next time will include talks being prepared for the Joint Mathematics Meetings in San Diego this coming January, so stay tuned!

## Bay Area Mathematical Artists, II

Yesterday was the second meeting of the Bay Area Mathematical Artists!  We had a somewhat different group — some who came last time were unable to make it, but there were some new faces among the enthusiasts as well.  Fourteen showed up for the afternoon, and two others joined us for dinner afterwards.

After the last meeting, I received a few emails offering suggestions about different ways to organize the gatherings.  One suggestion was to have a social period for the first part of the meeting, where participants could meet those they didn’t already know, or perhaps bring artistic items for show-and-tell.

This seemed like a good idea, since in addition to letting participants get to know each other or just catch up on the previous month, it gave them a 30-minute buffer to arrive on campus.  It turns out this part of the afternoon went very well, with just about everyone arriving by 3:30.  Dan Bach was giving away old copies of mathematics journals he had collected over the years, while Colin Liotta was giving away laser-etched wooden pendants.

The only scheduled talk of the afternoon was given by Roger Antonsen, entitled Mathematical explorations and visualizations.

Roger talked about several of his extensive library of Processing animations. He takes his inspiration from many different places – but his Processing mantra is sometimes “make it move.” He takes an image which fascinates him, recreates it as best as he can, and then varies different parameters to make the image move.

Some of his most intriguing examples involved optical illusions, and are based on the work of Pinna; see his web page on Art to see some examples.

He also emphasized the need to create balance in his work – when you parameterize features of an image, you need to decide the range the parameter takes. Too much motion, and the animation looks too distorted, but too little, and the animation is too static. Of course the need to create balance in artwork is not restricted to creating animations, but it is constantly in Roger’s mind as he creates.

After Roger’s talk, we moved on to a discussion about the nature of mathematical and digital art.  There are other similar terms in use, such as algorithmic art and generative art.  Typically, generative art is described as art generated by some autonomous system.  This means that algorithmic art — that is, art created by use of a computer program — is a subset of generative art.  But what do all these terms mean to us, individually, as artists?

I had originally thought to break into smaller groups at first, but the group wanted to have just one, larger discussion.

There were many and varied opinions expressed — from commentary about the art community as a whole, to those who thought the question “What is mathematical art?” is nonsensical to ask because there is really no answer, and it doesn’t impact the creative process at all.

Two points I made relate to my teaching Mathematics and Digital Art.  First, every time you settle on the value for a parameter in some work you’re doing, you are making an aesthetic choice.  You can tweak all you like, but then all of a sudden you just say to yourself, “OK, that’s it!”  Of course there are times you just stop because you can’t find precisely what you want and just need to move on, but I think you get the gist of what I’m saying.  You’re making artistic choices all the time.

And second, I have students write short narratives about their work, describing the parameter choices they make.  I want to them to think about when a digital image becomes a piece of digital art.  Of course (as alluded to above) there really is no definitive answer to this question, but for students just beginning to dive into the world of digital and mathematical art, it is a useful question to consider.

This took us right to our designated ending time, 5:00.  One of the participants suggested a Thai place nearby, and so eleven of us made the short trek just north of campus.  It was actually quite good, according to all accounts.  And as I mentioned earlier, two participants who couldn’t make it earlier in the day joined us at the restaurant.  The discussions were lively, covering a broad range of mathematics, art, and other topics as well.

So again, a good time was had by all!  To round out today’s post, I’d like to say a few words about the new Digital Art Club I’m advising at the University of San Francisco.  It seems each week, one or two more students become interested, mostly by word of mouth.  There are new faces all the time!

It turns out this is a great opportunity for mathematics or science majors to learn about digital art.  The Mathematics and Digital Art course I’m teaching is not at a high enough level to count towards a mathematics major, and many students in the sciences (like computer science or physics) have required mathematics courses, also at a higher level.

We’re focusing on learning Processing at the moment, since not only are students very interested in learning the software, but I have written quite a bit about Processing on my blog.  So I can easily get students started by pointing them to an appropriate post.

Not only that, we’ve got a few students willing to take leadership positions and secure status as an officially recognized student club.  This means there are funds available for field trips and other activities.  We’re hoping to make a trip to the Pace Gallery in Palo Alto.  I’ll occasionally update on our progress in future posts.  Stay tuned!

## Bay Area Mathematical Artists, I

Yesterday was the first meeting of the Bay Area Mathematical Artists at the University of San Francisco!

It all began one balmy Friday evening in Waterloo, Ontario, Canada at the Bridges 2017 conference….the Mendlers and I hosted a pot luck dinner at our AirBnB, and we realized how many of us were from the Bay area.  In fact, we remarked upon the fact that nine of us were actually on the same flight from San Francisco to Toronto for the conference!

Bridges participants do really form a community.  There is a spirit of sharing and mutual appreciation for each others’ work.  We really do cherish those few days each year when we can all come together.  The only drawback is that Bridges comes around just once a year.

So throughout the evening, between chowing down on grilled fare and sipping a glass of beer or wine, the idea of informally gathering now and then kept cropping up.

But as we all know, ideas do not automatically become reality.  They have a tendency to wither if not watered and fertilized….so I decided to take up gardening.

I had the advantage of being associated with a University, so I could arrange a meeting space.  Location was also somewhat convenient — some of us were to the northeast in Oakland and Berkeley, and others were to the southwest in Santa Clara and Scotts Valley.  It might be nice to move around occasionally so not everyone has to drive as far all the time.  But since the meetings are on Saturdays, at least traffic is not so much of a bother.

And then come the emails!  Yes, lots of them….  The main decision to be made was deciding on a format.  I thought informal was best — I sent out a call for speakers, and put them on the docket on a first-come, first-served basis.  I wanted to take away the stress of competing for time; if there were more speakers than we had time for, we’d just start where we left off the last time.

The other reason for this is that I wanted to encourage students from my Mathematics and Digital Art class, as well as members of the newly formed Digital Art Club, to participate as well.  I think it is important to let mathematical artists of all levels have a place to share ideas and get feedback on their work.

So for our inaugural meeting, we had three speakers:  Chamberlain Fong, Karl Schaffer,  and Dan Bach.

Chamberlain’s talk was entitled The Conformal Hyperbolic Square and Its Ilk.  He discussed different ways to transform circular hyperbolic tilings (particularly those of Escher) to square images.  Chamberlain did give a version of this talk at Bridges in 2016, but included more recent results as well.  For more information, you can contact him at chamberlain@yahoo.com.

Karl Schaffer’s talk was entitled Dance’s Center of Attention Mass.  Inspired by Joseph Thie’s Rhythm and Dance Mathematics and Kasia Williams’ idea of “Center of Attention Mass,” Karl is interested in graphically showing where the center of attention is by weighting the position of each dancer on stage.  He went so far as to contact Thie — now in his 80’s — and they are actively collaborating together.

Karl is also giving the lecture/demonstration Calculated Movements at the Montalvo Art Center next March.  There is more information here.  You can reach Karl at karl_schaffer@yahoo.com.

Finally, Dan Bach’s talk was entitled 3D Math Art and iBooks Author.  Dan is keen on creating highly interactive math books which engage students of all ages.  He gave a practical talk demonstrating the software he uses, including examples of converting graphics to various different formats since it is not always a simple task to take a 3D image created by one software package and import it into another.  You can reach Dan at dan@dansmath.com.

After the talks — which included ample room for Q&A — we had a brief discussion on the future of the group.  I wanted to make it clear that while I am willing to keep things going in the current format, it is really up to the group to decide how to run our meetings.  We opted to keep things going the same way for next month — but suggestions for the future included workshops, or perhaps themed sessions, like a series of talks on polyhedra.  Participants were encouraged to think of other ways to use our time together as a topic of discussion for the next meeting.  Keeping it informal means lessening the pressure of submitting talks/papers for conferences, etc.

Then dinner!  Most of us were available for a meal afterwards.  There were two nice options nearby — a cafe with sandwiches and salads, and an Indian restaurant with a buffet.  I went with the group who preferred Indian food — and truly, a good time was had by all!  We left for dinner at about 5:30, and I finally had to break things up shortly before 8:00, since some of us had a ways to drive home.  We could clearly have kept talking for quite a while….

So our first meeting of the (tentatively named) Bay Area Mathematical Artists was a success!  There were a total of 15 of us present, including three students from USF — a very respectable number for a first time event.  We plan to meet approximately monthly, modulo the University schedule of classes and holidays.

I’ll post summaries each month of our meetings, including a brief synopsis of the talks, workshop(s), or whatever other form the meetings take.  Feel free to contact the speakers for more information about the talks they gave this weekend, and don’t hesitate to spread the word to others who might be interested!