When I close my eyes, all I see are strings. I’ve become somewhat obsessed with the patterns made from stretching straight lines across a circle, better known as string art. Simple rules can yield beautiful patterns and reveal hidden math.

This gallery chronicles my exploration in my string art, a retelling of my twitter thread. For more mathematical details, see my paper

It all started in seventh grade math class, during our modular arithmetic unit. We drew a clock, and imagined how the hands move after waiting three hours. Addition by 3, mod 12. Connecting the starting and ending numbers with string, we get a satisfying pattern

Try this with 100 numbers (addition by 25 mod 100), and we see the strings accumulate in a smaller central circle. This manifests the underlying symmetry of modular addition: All numbers move the same amount, giving us that sweet, sweet rotational symmetry.

Next, modular multiplication. Multiply by 2 mod 100, and connect the dots for a nice heart! These classic patterns are staple examples of the beauty of math, for good reason.

7th grade me was enamored. With a little python, I made string art for every other function I could think of, modulo $n$. $x^2$ modulo $n$ lacked the elegant envelopes of multiplication, opting for a dense jungle of jumbled lines. I failed to find any functions with the beautiful symmetries of the classics. My disappointment eventually faded into memory, buried under the sands of time.

Until a month ago. I saw a my friend Fran’s video about the classics, relating string art to circles rolling around circles (check it out!) Like a windstorm through the desert of my memories, it unearthed my long buried failure. I had unfinished business.

But this time, I have a math degree! I know waaay more functions then lil baby me, I’m sure some of them look cool. I realized the “modulo n” is incidental, and only the position around the circle matters–I should look for natural maps from the circle to itself.

For example, modular addition is really rotation by a constant angle, while modular multiplication by two doubles the angle. I can make string art for any map by stretching strings between a point to its image. Here’s a sequence of maps made from smooth noise.

But really, It’s not about the strings. They concentrate along “caustics” where each string crosses its neighbors. Thin enough strings, and only the caustics become visible. You can see this above, as we add more strings and make them thinner as time goes on. The hearta appears out of the chaos. Rotation produces circular caustics, while angle doubling makes the heart-shaped cardioid.

Sidenote, “caustics” are usually light reflected off a curved surface, where straight light rays act like straight strings. Reflecting off a circle doubles the angle of light, so the light concentrates in a cardioid in the bottom of your coffee cup. For more information, see my presentation about caustics

We can also get nice caustics by changing the boundary. Here the boundary is an epicycloid, traced by a circle rolling inside another circle. The map is a time-varying rotation. If the boundary were a circle, the caustic would simply be another circle. The unusual boundary gives beautiful caustics.

I think I’ve successfully avenged my 10 year old grudge. But my ace remains firmly up my sleeve. Seventh grade me never thought about the canvas string art paints on. What if, instead of a flat disc, it was curved? That’s right: Hyperbolic sting art :)

This marks the end of my twitter thread. From now on, I will not explain everything responsibly.

One natural map from the circle to itself are mobius transforms. Think of the circle as the real line with infinity $\mathbb{R} \cup \infty$. A mobius transform is a map of the form $x\mapsto \frac{ax+b}{cx+d}$ for fixed real number $a,b,c,d$. This sends a real number to a (possibly infinite) real number, so extends to a map from the circle to itself. The video above shows the string art for this map, as the parameters $a,b,c,d$ vary.

Notice how the causitc seems to be an ellipse.

Now we take a leap of insight, and treat the inside of the circle as the hyperbolic plane. Symmetries of the hyperbolic plane act on the boundary circle through mobius transforms. We think of the “strings” in string art as geodesics through the hyperbolic plane, rendered in the Klein model. If instead we switch to the Poincare disc model, the geodesics become semicircles which meet the boundary circle orthogonally. There is a unique such geodesic connecting any two points on the boundary circle. This gives us a new way to render strings in string art.

We can visualize the symmetries of the hyperbolic plane in this picture geometrically. Here’s a video showing how the mobius transformations acting on the complex plane arise from the Riemann sphere. The symmetries of the hyperbolic plane are Mobius transforms which fix the unit circle. On the unit circle, these act by the mobius transforms described above. On the interior, the preserve the stucture of the hyperbolic plane.

Here is the mobius transform string art shown in both methods, with straight lines (left) and with circular arcs (right). Notice how the caustic in the Poincare model (right) is a circle?

Here is the string art for an “elliptic” mobius transform. These come from rotations of the hyperbolic plane. I drew the complete circle of each string, even though they leave the bounding circle. The caustic consists of two circles, one inside and one outside the bounding circle.

Here is the string art for an “hyperbolic” mobius transform. These come from translations of the hyperbolic plane. The caustic consists of two circles, which both meet the bounding circle

This table explains the string art for different classes of mobius transforms. The types of mobius transforms correspond to the number of fixed poins on the circle. The elliptic transforms arise from rotations of the hyperbolic plane, while the hyperbolic transforms arise from translations of the hyperbolic plane. The envelopes look like circles in the Poincare projection. Their corresponding shapes in the hyperbolic plane are explained in my paper

String art for a reflection of the circle. This is also a type of mobius transform.

Consider the map consisting of a mobius transform, composed with a circle doubling. The left image shows the string art for that map. The second image shows the map iterated twice, and so on. The resulting rightmost image is fractal string art

Another fractal of the same type as the last.