Papers and Notes

I am also a research mathematician, mainly working in discrete geometry, aperiodic tilings, and some other odds and ends. My later papers are on the ArXiv. Here are some worth posting:

My main interest has been in how local matching rules — the way shapes may fit together — can force non-periodic structure in the plane. Really this an aspect of a larger widespread phenomenon, in which local conditions force emergent, intractable large-scale structure — that this is in general possible, even immediate, is itself an immediate consequence of Turing’s Theory of Computation.

For context, Can’t Decide? Undecide! shows many examples of this in action, even in simple recreational mathematics: It’s amazingly easy for irreducible complexity to arise from the interaction of a collection of elementary gadgets. Like tiles.

Tilings

Tilings are a great example of how local rules can force incredible global structure, as Dave Smith’s discovery of aperiodic monotiles really highlights. Neighbors fitting together, one next to another, somehow FORCE structure across all scales, one foot, one mile, one universe (that’s just getting started to a mathematician), across the entire infinite plane.

Early on I wrote extensively on hierarchical aperiodic tilings, proving that every (*) substitution tiling system can be enforced by matching rules. Matching Rules and Substitution Tilings, Annals of Mathematics 147 (1998), 181–223. Here is an introduction to that paper, and an example of the construction for the Sphinx tile. More on this will be coming.

(*) (that satisfies the simple condition that when a tile is subdivided, its vertices are among the vertices of the children, and that neighboring children’s meet “vertex-to-vertex”; I don’t know of a meaningful example that doesn’t satisfy these conditions, but they can be loosened further!)

Here is an early survey Aperiodic Hierarchical Tiling that sketches out the idea of “addressing”.