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:
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:
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 firstname.lastname@example.org 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 email@example.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!