Celtic Knots, Part 3 takes Celtic knots into 3D, creating overlapping weaves in the plane and 3D objects covered with knotwork. I show how to wrap knots around the surfaces of some objects with interesting topologies (such as the donut above). I also show some interesting ways to take the idea of overlap and underlap and move them into 3D.

The Digital Ceraunoscope: Synthetic Thunder and Lightning, Part 1 begins the discussion of how to write a physically-based simulator that creates statistically accurate thunder and lighting. This first part focuses on lightning and its geometric structure.

The Digital Ceraunoscope: Synthetic Thunder and Lightning, Part 2 shows how to take the lightning created in the previous column, and the position of a listener, and create a sound file that represents what that person would actually hear from that particular lightning stroke.

Texturing With Symmetry shows how to create a vareity of attractive patterns using hierarchies of symmetry elements and image-processing actions. The basic idea is to create a simple prototype tile that can be placed in a variety of orientations (square tiles have eight of them; equilateral triangles have six). Then apply the tile using a symmetry transformation to create a larger tile, and apply another symmetry operator to make a still-larger tile, and so on, creating a hierarchy of tiles. The result is a pattern with a lot of structure but a lot of variation.

Soap Bubbles, Part 1 talks about the chemistry of soap films, some of their basic geometry, and how to write a soap-film shader. I talk about how soap bubbles take on their shape: basically they're trying to minimize the amount of surface area formed by the soap film while obeying the constraints of the physical environment.

Soap Bubbles, Part 2 shows the geometry behind create clusters of soap bubbles. I show how two or more bubbles combine, and how to compute the geometry of the inner walls that are formed between pairs of bubbles. It turns out that if we ignore gravity (which is a pretty safe bet for objects as light as soap bubbles) then everything is a piece of a sphere: the bubbles themselves, as well as the walls between them.

I also talk about wave interference, and how that gives rise to the beautiful colors that make up soap bubbles.