Sculpture of a cusp caustic. A two-parameter family of lines in ℝ³ created by sewing thread through sheets of vinyl. These lines concentrate along a cusp, tracing a line through 3D space (the central line of the image).

Geolocation

This picture was the front cover of the 2024 berkeley math department newsletter.

Here is another string art sculpture. I started with a circular rim, with 24 hooks at equal intervals, bent into the shape of a pringle. I threaded each string through a hook, then skipped a fixed number of hooks, then repeat. I colored the strings according to how many hooks they skip each time. This would ordinarily produce sting art with circular caustics. Because the rim is deformed into 3D, we get a whole host of other caustics when we look from different angles. Here is a shot from the side:

I made this commemorative interactive handout. Creasing the paper perpendicular to the boundary ellipse, the lines of creases concentrate along two cusps.

Animation of the swallowtail catastrophe. On the top, a drawing of the singular surface of the swallowtail. On the bottom, a series of slices of the swallowtail. In the background, a visualization of a map from the plane to itself exhibiting the swallowtail catastrophe. At the start of the video, the plane is folded over itself along the bright line. As time goes on, the fold develops two cusps (as depicted in the comic on the bottom).

In a funny confluence of history, Rene Thom (the forerunner of catatrophe theory) was good friends with Salvador Dali. The swallowtail catastrophe was the subject of Salvador Dali’s last painting.

An animation of the elliptic umbilic catastrophe, following the same setup as the swallowtail. The elliptic umbilic catastrophe occours when three cusps come together into a point and invert.

Here is a 2-dimensional family of lines exhibiting an elliptic umbilic catastrophe:

An animation of the elliptic umbilic catastrophe, also known as the purse catastrophe, following the same setup as the last two. A cusp and a fold coallesce and trade places.

A shadow of a ruled surface. The shadow consists of a bunch of lines, which concentrate around a cusp catastrophe. The darkest part occours on the cusp itself.

A cusp appearing in sunlight refracting through a glass of water. We can think of the light from the sun as a surface moving through three dimensional space. Passing through the glass, this surface folds over itself. When this sheet of light hits the table, the light concentrates along two folds and a cusp. Geometrically, the cusp appears for the same reason as the shadow from the last picture.

Some beautiful caustics I saw reflected from windows across the street.

Here’s a picture of me giving this talk :) I’m talking about how raindrops make caustics, which become rainbows. I’m shining an array of lazer dots into a glass sphere. Each beam bounces off the back of the sphere, and ends up on the whiteboard behind me. After refraction, the grid of beams seems to fold around a circle. This is the caustic.

I bought this green lazer for cheap on amazon, thinking it would be 5mw like always. It was 100 mw. I should probably be wearing safety goggles in this picture.