# Creating Mathematical Gems in the Wolfram Language

December 14, 2017 — Michael Gammon, Blog Administrator, Document and Media Systems

The Wolfram Community group dedicated to visual arts is abound with technically and aesthetically stunning contributions. Many of these posts come from prolific contributor Clayton Shonkwiler, who has racked up over 75 “staff pick” accolades. Recently I got the chance to interview him and learn more about the role of the Wolfram Language in his art and creative process. But first, I asked Wolfram Community’s staff lead, Vitaliy Kaurov, what makes Shonkwiler a standout among mathematical artists.

“Stereo Vision” and “Rise Up”

“Clay, I think, pays special attention to expressing a math concept behind the art,” Kaurov says. “It is there, like a hidden gem, but a layman will not recognize it behind the beautiful visual. So Clay’s art is a thing within a thing, and there is more to it than meets the eye. That mystery is intriguing once you know it is there. But it’s not easy to express something as abstract and complex as math in something as compact and striking as a piece of art perceivable in a few moments. This gap is bridged with help from the Wolfram Language, because it’s a very expressive, versatile medium inspiring the creative process.”

Shonkwiler is a mathematics professor at Colorado State University and an avid visual artist, specializing in Wolfram Language–generated GIF animations and static images based on nontrivial math. “I am interested in geometric models of physical systems. Currently I’m mostly focused on geometric approaches to studying random walks with topological constraints, which are used to model polymers,” he says.

In describing how he generates ideas, he says, “There are some exceptions, but there are two main starting points. Often I get it into my head that I should be able to make an animation from some interesting piece of mathematics. For example, in recent months I’ve made animations related to the Hopf fibration.”

“Stay Upright”

 ✕ DynamicModule[{n = 60, a = \[Pi]/4, viewpoint = {1, 1.5, 2.5}, \[Theta] = 1.19, r = 2.77, plane, cols = RGBColor /@ {"#f43530", "#e0e5da", "#00aabb", "#46454b"}}, plane = NullSpace[{viewpoint}]; Manipulate[ Graphics[{Thickness[.003], Table[{Blend[cols[[;; -2]], r/\[Pi]], InfiniteLine[ RotationMatrix[\[Theta]].plane.# & /@ {{Cot[r] Csc[a], 0, Cot[a]}, {0, Cot[r] Sec[a], -Tan[a]}}]}, {r, \[Pi]/(2 n) + s, \[Pi], 2 \[Pi]/n}]}, Background -> cols[[-1]], PlotRange -> r, ImageSize -> 540], {s, 0., 2 \[Pi]/n}]]

Like many artists, Shonkwiler draws inspiration from existing art and attempts to recreate it or improve upon it using his own process. He says, “Whether or not I actually succeed in reproducing a piece, I usually get enough of a feel for the concept to then go off in some new direction with it.”

As to the artists who inspire him, Shonkwiler says, “There’s an entire community of geometric GIF artists on social media that I find tremendously inspiring, including Charlie Deck, davidope, Saskia Freeke and especially Dave Whyte. I should also mention David Mrugala, Alberto Vacca Lepri, Justin Van Genderen and Pierre Voisin, who mostly work in still images rather than animations.” If you want to see other “math art” that has inspired Shonkwiler, check out Frank Farris, Kerry Mitchell, Henry Segerman, Craig Kaplan and Felicia Tabing.

Another artistic element in Shonkwiler’s pieces is found in the title he creates for each one. You’ll find clever descriptors, allusions to ancient literature and wordplay with mathematical concepts. He says he usually creates the title after the piece is completely done. “I post my GIFs in a bunch of places online, but Wolfram Community is usually first because I always include a description and the source code in those posts, and I like to be able to point to the source code when I post to other places. So what often happens is I’ll upload a GIF to Wolfram Community, then spend several minutes staring at the post preview, trying to come up with a title.” Although he takes title creation seriously, Shonkwiler says, “Coming up with titles is tremendously frustrating because I’m done with the piece and ready to post it and move on, but I need a title before I can do that.”

“Interlock”

 ✕ Stereo[{x1_, y1_, x2_, y2_}] := {x1/(1 - y2), y1/(1 - y2), x2/(1 - y2)}; With[{n = 30, m = 22, viewpoint = 5 {1, 0, 0}, cols = RGBColor /@ {"#2292CA", "#EEEEEE", "#222831"}}, Manipulate[ Graphics3D[{cols[[1]], Table[Tube[ Table[Stereo[ RotationTransform[-s, {{1, 0, 0, 0}, {0, 0, 0, 1}}][ 1/Sqrt[2] {Cos[\[Theta]], Sin[\[Theta]], Cos[\[Theta] + t], Sin[\[Theta] + t]}]], {\[Theta], 0., 2 \[Pi], 2 \[Pi]/n}]], {t, 0., 2 \[Pi], 2 \[Pi]/m}]}, ViewPoint -> viewpoint, Boxed -> False, Background -> cols[[-1]], ImageSize -> 500, PlotRange -> 10, ViewAngle -> \[Pi]/50, Lighting -> {{"Point", cols[[1]], {0, -1, 0}}, {"Point", cols[[2]], {0, 1, 0}}, {"Ambient", RGBColor["#ff463e"], viewpoint}}], {s, 0, \[Pi]}]]

(This code plots the curves with fewer points so as to increase the responsiveness of the Manipulate.)

Other Wolfram Community members have complimented Shonkwiler on the layers of color he gives his geometric animations. Likewise, his use of shading often enhances the shapes within his art. But interestingly, his work usually begins monochromatically. “Usually I start in black and white when I’m working on the geometric form and trying to make the animation work properly. That stuff is usually pretty nailed down before I start thinking about colors. I’m terrible at looking at a bunch of color swatches and envisioning how they will look in an actual composition, so usually I have to try a lot of different color combinations before I find one I like.”

Shonkwiler says that the Wolfram Language makes testing out color schemes a quick process. “If you look at the code for most of my animations, you’ll find a variable called cols so that I can easily change colors just by changing that one variable.”

“Magic Carpet” and “Square Up”

I asked Shonkwiler if he conceives the visual outcome before he starts his work, or if he plays with the math and code until he finds something he decides to keep. He said it could go either way, or it might be a combination. “‘Magic Carpet’ started as a modification of ‘Square Up,’ which was colored according to the z coordinate from the very earliest versions, so that’s definitely a case where I had something in my head that turned out to require some extra fiddling to implement. But often I’m just playing around until I find something that grabs me in some way, so it’s very much an exploration.”

“Renewable Resources” and “Inner Light”

Shonkwiler actually has a lot of pieces that are related to each other mathematically. Regarding the two above, “They’re both visualizations of the same Möbius transformation. A Möbius transformation of a sphere is just a map from the sphere to itself that preserves the angles everywhere. They’re important in complex analysis, algebraic geometry, hyperbolic geometry and various other places, which means there are lots of interesting ways to think about them. They come up in my research in the guise of automorphisms of the projective line and as isometries of the hyperbolic plane, so they’re often on my mind.”

“To make ‘Inner Light,’ I took a bunch of concentric circles in the plane and just started scaling the plane by more and more, so that each individual circle is getting bigger and bigger. Then I inverse-stereographically project up to the sphere, where the circles become circles of latitude and I make a tube around each one. ‘Renewable Resource’ is basically the same thing, except I just have individual points on each circle and I’m only showing half of the sphere in the final image rather than the whole sphere.”

When I asked Shonkwiler about his philosophy on the relationship between math and aesthetics, he said, “Part of becoming a mathematician is developing a very particular kind of aesthetic sense that tells you whether an argument or a theory is beautiful or ugly, but this has probably been overemphasized to the point of cliché.”

However, Shonkwiler continued to mull the question. “I do think that when you make a visualization of a piece of interesting mathematics, it is often the case that it is visually compelling on some deep level, even if not exactly beautiful in a traditional sense. That might just be confirmation bias on my part, so there’s definitely an empirical question as to whether that’s really true and, if it is, you could probably have a metaphysical or epistemological debate about why that might be. But in any case, I think it’s an interesting challenge to find those visually compelling fragments of mathematics and then to try to present them in a way that also incorporates some more traditional aesthetic considerations. That’s something I feel like I’ve gotten marginally better at over the years, but I’m definitely still learning.”

Here is Shonkwiler with one of his GIFs at MediaLive x Ello: International GIF Competition in Boulder, Colorado:

Check out Clayton Shonkwiler’s Wolfram Community contributions. To explore his work further, visit his blog and his website. Of course, if you have Wolfram Language–based art, post it on Wolfram Community to strike up a conversation with other art and Wolfram Language enthusiasts. It’s easy and free to sign up for a Community account.

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