MAA – Creativity in Mathematics

Art Exhibition: Golden Section 2018

The Regional Meeting of the Golden Section of the Mathematical Association of America was held at California State University, East Bay. The local organizer was Shirley Yap, fellow mathematical artist, who deserves kudos for the monumental amount of work it takes to organize a conference like this! I helped out by organizing this year’s Art Exhibition.

This year, I thought I’d give you a virtual tour of the exhibit! So I asked contributing artists to submit their own personal statement about their work and/or mathematical art in general, as well as an image of one of their displayed artworks. I’ll let the artists speak for themselves…. (By the way, the order the artists are presented in is the order in which they sent me their information. There is no ranking implicit in the order.)

Shirley Yap (shirley.yap@csueastbay.edu)

Day134test7

I created this image out of golden spirals. While working on a math demonstration for my students, I unpacked a roll of netting. During the unraveling process, I had a vision of a two-dimensional golden spiral unravelling, which I tried to recreate with code. I wanted to viewer to not just witness the unraveling, but also be inside the web of the fabric. I often create code that has a lot of randomness in it, so that it captures a moment in time that can never be recreated.

Frank A. Farris (ffarris@scu.edu and math.scu.edu/~ffarris)

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Mathematicians can feel lonely to find ourselves face to face with the most beautiful thoughts humans have ever known, only to realize that communicating our experience is unreasonably difficult. I have found comfort in visual art, digitally computed images that are the best I can do (short of giving an hour lecture) to say, ‘This is the beauty of mathematics.’

David Honda (snaporigami@gmail.com and snaporigami.weebly.com)

Day134p1019930_orig

I’m primarily a middle school mathematics teacher with one of my hobbies being Origami and other paper crafts. Some years back I became interested in the work of Heinz Strobl which uses joined, folded strips of paper to create various structures. Much like unit origami, the structures are held together solely by the folds, no adhesives. My interest soon became an obsession and I’ve been neck-deep in little strips of paper ever since. Lately I’ve been exploring concepts in Topology. This particular work is my attempt at a Klein Bottle.

Dan Bach (www.dansmath.com, art@dansmath.com, and @dansmath)

Day134Primebeadssmall — Prime Bead Spirals

“I’m a career college mathematics teacher, now an interactive book author and 3D math artist. I like to illustrate number theory and vector calculus principles with surprising and colorful images, using a software palette of Mathematica, Cheetah3D, and iBooks Author. My math art encourages viewers to think, notice, and wonder. And hopefully say, ‘That’s cool! That’s math?’ “

Roger Antonsen (http://rantonse.no/ and rantonse@ifi.uio.no)

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This is a visualization of seven repeated perfect out-shuffles of a deck with 128 cards. The horizontal lines represent the particular orders of the cards throughout the shuffling, and the vertical curves represent the path each card takes from start to finish. The curves are colored from black to white in order to show the mechanics of the shuffling. The dots are colored from black to red to black in order to show that each perfect out-shuffle preserves the so-called “stay-stack principle”. Notice that the order of cards returns to the original order after seven shuffles.

Linda Beverly (lbeverly@horizon.csueastbay.edu and https://mathcsresearch.wordpress.com/)

Day134watercolor_frame_17_k128-8_by_10_webpage_quality — Watercolors

This image is an embedding of a photograph of a set of watercolors. The embedding was performed using Locally Linear Embedding (LLE), a nonlinear dimensionality reduction technique, introduced by Saul & Roweis in 2000. LLE is an unsupervised machine learning algorithm.

I am pursuing a double major in Mathematics and Computer Science. I am currently working on research in geometric dimensionality reduction and unsupervised machine learning. This work will be extended into neural networks and deep learning. I enjoy seeking out interesting intersections between mathematics, computer science, and art.

Jason Herschel (jherschel@gmail.com)

Day134JasonHerschel-100pingpongballswLEDs

Undulating organic light show with minimal code generated by a 16 MHz processor calculating color and brightness values through a Perlin noise algorithm. Blobs appear to grow, shrink, and drift relaxingly over the LED grid. 100 ping pong balls covering 100 individually addressable LEDs on poster board with Arduino Nano v.3 controller and battery pack.

Paul Gonzalez-Becerra (pgonzbecer@gmail.com and http://www.pgonzbecer.com/)

Programming is my art. I might not be a good “designer”, but I am a good developer where I am able to take a structured approach on art. I specialize in computer graphics, thus my understanding of the mathematics behind geometry, 3D models, and 2D sprites are better than my ability to free-form draw them.

Phil Webster (phil@philwebsterdesign.com, http://www.philwebsterdesign.com/ )

Day134Islamic 8-fold Fractal Flower Median 900x900

All of my work stems from one core impulse: to celebrate the inherent beauty of mathematical forms. Since traveling to India in 2012, I have been particularly focused on blending traditional Islamic motifs with polyhedra and fractals. The results are distinctly Islamic in flavor but with a modern twist.

This piece has global and local 8-fold rotational symmetry around each gold star. Star centers occupy the nodes of 8 fractal quaternary trees, which are pruned at the octant boundaries. The original central star is replaced by an inward progression of the same fractal diminution.

Vince Matsko (vince.matsko@gmail.com, www.vincematsko.com)

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This piece is based on fractal binary trees. The usual way of creating a binary tree is to move forward, then branch to the left and right some fixed angle as well as shrink, and repeat recursively. Recent work involves specifying the branching by arbitrary affine transformations. In this piece, the affine transformations were chosen so that as the tree grows, nodes are repeatedly visited. The nodes are covered by disks which become smaller with each iteration, accounting for the overlapping circles. The research needed to produce these images was undertaken jointly with Nick Mendler.

I hope you enjoyed this virtual tour of the Art Exhibition at the recent MAA Regional Conference. The upcoming conference is next March; stay tuned for another virtual tour in about a year!

More CrossNumber Puzzles

Last fall, I mentioned that while looking at the Puzzle Page of the FOCUS magazine published by the Mathematical Association of America, I thought to myself, “Hey, I write lots of puzzles. Maybe some of mine can get published!” So I submitted a few Number Search puzzles to the editor, and to my delight, she included them in the December/January issue. Here’s the proof….

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Incidentally, these puzzles are the same ones I wrote about almost two years ago — hard to believe I’ve been blogging that long! So if you want to try them, you can look at Number Searches I and Number Searches II.

Since I had success with one round of puzzles, I thought I’d try again. This time, I wanted to try a few CrossNumber puzzles (which I wrote about on my third blog post). But as my audience was professional mathematicians and mathematics teachers, I wanted to try to come up with something a little more interesting than the puzzles in that post.

To my delight again, my new trio of puzzles was also accepted for publication! So I thought I’d share them with you. (And for those wondering, the editor does know I’m also blogging about these puzzles; very few of my followers are members of the MAA….)

Here is the first puzzle.

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Answers are entered in the usual way, with the first digit of the number in the corresponding square, then going across or down as indicated. In the completed puzzle, every square must be filled.

I thought this was an interesting twist, since every answer is a different power of an integer. I included this as the “warmup” puzzle. It is not terribly difficult if you have some software (like Mathematica) where you can just print out all the different powers and see which ones fit. There are very few options, for example, for 3 Down.

The next puzzle is rather more challenging!

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All the answers in this puzzle are perfect cubes with either three or four digits, and there are no empty squares in the completed puzzle. But you might be wondering — where are the Across and Down clues? Well, there aren’t any….

In this puzzle, the number of the clue tells you where the first digit of the number goes — or maybe the last digit. And there’s more — the number can be written either horizontally or vertically — that’s for you to decide! So, for example, if the answer to Clue 5 were “216,” there would be six different ways you could put it in the grid: the “2” can go in the square labelled 5, and the number can be written up, down, or to the left. Or the “6” can go in the square labelled 5, again with the same three options.

This makes for a more challenging puzzle. If you want to try it, here is some help. Let me give you a list of all the three- and four-digit cubes, along with their digit sum in parentheses: 125(8), 216(9), 343(10), 512(8), 729(18), 1000(1), 1331(8), 1728(18), 2197(19), 2744(17), 3375(18), 4096(19), 4913(17), 5832(18), 6859(28), 8000(8), 9261(18). And in case you’re wondering, a number which is a palindrome reads the same forwards and backwards, like 343 or 1331.

The third puzzle is a bit open-ended.

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To solve it, you have to fill each square with a digit so that you can circle (word search style) as many two- and three-digit perfect squares as possible. In the example above, you would count both 144 and 441, but you would only count 49 once. You could also count the 25 as well as the 625.

I don’t actually know the solution to this puzzle. The best I could do was fill in the grid so I could circle 24 out of the 28 eligible perfect squares between 16 and 961. In my submission to MAA FOCUS, I ask if any solver can do better. Can you fit more than 24 perfect squares in the five-by-five grid? I’d like to know!

I’m very excited about my puzzles appearing in a magazine for mathematicians. I’m hoping to become a regular contributor to the Puzzle Page. It is fortunate that the editor likes the style of my puzzles — when the magazine gets a new editor, things may change. But until then, I’ll need to sharpen my wits to keep coming up with new puzzles!

Art Exhibition: Golden Section 2017

Yesterday, artists from the Golden Section of the Mathematical Association of America contributed to yet another art exhibition! Each Spring, members of the MAA from Northern California, Nevada, and Hawaii attend a regional conference — this year, at Santa Clara University.

rectangletriangle — *Woven,* by Nick Mendler.

Last year the event was held at the University of California, Davis, and Shirley Yap from California State University, East Bay organized a highly successful exhibit — what we believe to be the first art exhibition ever to be a part of a sectional MAA meeting. I asked Shirley to say a few words about what motivated her to take on this task.

I exhibited an art piece at the Joint Mathematics Meetings in 2016. It was an interactive piece and I wanted to see how people would experiment with it. So I just hung around the exhibit for a while and not only saw how people played with my piece, but how they observed other pieces. The kind of delight that came from people’s faces convinced me that the art was really drawing them to math in a way that was different from how I had seen before. Perhaps because one is expected to sit in front of art for a long time to contemplate it, people felt relaxed enough to enjoy it. Whatever it was I saw, I knew that I wanted to share the experience with others outside of the JMM.

When we put a call for artists out on our Golden Section website, we didn’t get any responses. So I went through years of JMM art exhibit catalogs and looked up each artist to see if they lived in our section. Then I just started emailing them individually to ask if they were interested in showing their work at a local exhibition.

This year, I offered to help Shirley with organizing the exhibition. Given what was involved in the second year, I have a new appreciation for Shirley’s dedication to spreading the word about mathematical art. Such events do not organize themselves — and we are all grateful Shirley took on this huge task to start a new Golden Section tradition.

redmandalalores — *Red Mandala,* by Frank Farris.

We didn’t have as many artists participate this year — but that’s part of the ebb and flow of yearly events like these. But the quality has high, as was the enthusiasm of the artists. Two of the artists this year were undergraduates — Nick Mendler from the Univiersity of San Francisco, and Juli Odomo from Santa Clara University. I think of them as future organizers of sectional MAA art exhibits….

In the morning, we had the usual opening remarks and a series of excellent speakers. The art exhibit took place in parallel with the Student Poster Sessions, which took place after lunch from 1:00-2:30. This was followed by another series of talks. You can see the full program here.

phil — *Islamic 8-fold Fractal Flower (Median),* by Phil Webster.

I asked the artists to say a few words about their experience about creating or exhibiting mathematical art. Here a few remarks.

Frank Farris (see artwork above):

I love the idea that we’re entering a golden age of mathematical art. New tools become available all the time and a growing community is finding new creative ways to use them. Can’t wait to see what the next years will bring.

I believe the sentiment in Gwen’s quote resonates very strongly with many mathematical artists.

Gwen Fisher:

The thing that keeps bringing me back to bead weaving is mathematics. Of course, I love colors of glass beads and the way they sparkle, but mostly, I keep returning to my seed beads because I keep finding new ways to use and represent mathematical structures with them.

gwenfisher — *Pixel Painting Number VI “Sunnyvale Boogie Woogie”* by Gwen Fisher.

Nick Mendler (see artwork above):

Since my first sectional meeting last Spring, I’ve continued research into the questions that generated my first mathematical artwork over a year ago.
Recognizing that my projects and thoughts are the most rewarding when realized through an aesthetic process has been not only productive, but has been a fascinating source of guidance to new questions. That focusing on more elegant images brings about more elegant mathematics has been only too clear from the sessions I’ve attended so far; I’m looking forward to seeing and learning from more art pieces!

Interested in organizing an art exhibit in your section? Since I helped Shirley with the organizational details this year, I can say a bit about what’s involved in putting together an exhibition.

imag5546 — Intense discussion about mathematical art at the exhibition.

The first step is, clearly, finding artists who want to show their work. It would be easier to get a student worker to do the search Shirley undertook — but don’t forget about the exhibitions at the Bridges conferences! Here is a link to both JMM and Bridges galleries. You can also contact the SIGMAA-ARTS and request that an email blast be sent to members.

koch336_19_107 — *Symmetric Koch Curve I,* by Vince Matsko.

As far as the submission process goes, that’s pretty standard. While it is always nice to accept every submission, sometimes it just isn’t possible. The works should have some real mathematical content, and be of good quality.

Since not all artists necessarily have business cards (especially student artists), I had the idea of making nametags for those who wanted one. You can download this nametag template in LaTeX if you would like, then edit and print onto cardstock. (Note: WordPress would not let me upload a .tex document, so I saved it as an Open Office document.)

It is a good idea to have an assistant or a student helper in the exhibition venue during the conference. Not all artists attended the meeting, and so brought in their work at various times during the day.

Shirley had the wonderful idea of arranging a dinner for contributing artists after the conference. Last year we went to an excellent Thai restaurant, and this year, Frank Farris generously offered to host a pot luck dinner (he provided the lasagna) at his house. These have been very wonderful events, and give artists the opportunity to get to know each other a little better.

A subset of the artists at the celebratory post-exhibition pot-luck. (Photo by Frank Farris.)

Finally, I wanted to mention that I am in the middle of my second time teaching Mathematics and Digital Art at the University of San Francisco. I say this in the event you are interested in offering such a course at your university. I have written extensively about this experience on my blog, and also have all course materials as well as a day-by-day outline available on the Fall 2016 course website. I would be happy to help you get such a course off the ground if you’re interested.

If you would like more information, or want to get in touch with any of the artists whose work is shown above, please make a comment and I’ll get back to you. I hope this is just the beginning of a long tradition of having mathematical art exhibits at sectional MAA meetings!