• Arc-triangle tilings

Every triangle tiles the plane, by 180° rotations around the midpoints of each side; some triangles have other tilings as well. But if we generalize from triangles to arc-triangles (shapes bounded by three circular arcs), it is no longer true that everything tiles. Within any large region of the plane, the lengths of bulging-outward arcs of each radius must be balanced by equal lengths of bulging-inward arcs of each radius, and the only way to achieve this with a single tile shape is to keep that same balance between convex and concave length on each tile. Counting line segments as degenerate cases of circular arcs, this gives us three possibilities:

• Congratulations, Dr. Matias!

Pedro Ascensao Ferreira Matias, one of the students working with Mike Goodrich in the UC Irvine Center for Algorithms and Theory of Computation, passed his Ph.D. defense today.

• How good is greed for the no-three-in-line problem?

The 37th European Workshop on Computational Geometry (EuroCG 2021) was earlier this month, but its book of abstracts remains online. This has an odd position in the world of academic publishing: the “abstracts” are really short papers, so it looks a lot like a published conference proceedings. However, it declares that you should really pretend that it’s not a proceedings, in order to allow the same work to go on to another conference with a published proceedings, getting around the usual prohibitions on double publication. Instead, its papers “should be considered a preprint rather than a formally reviewed paper”. But I think that doesn’t preclude citing them, with care, just as you might occasionally cite arXiv preprints. The workshop’s lack of peer review and selectivity is actually a useful feature, allowing it to act as an outlet for works that are too small or preliminary for publication elsewhere. In North America, the Canadian Conference on Computational Geometry performs much the same role, but does publish a proceedings; its submission deadline is rapidly approaching.

• Pick's shoelaces

Two important methods for computing area of polygons in the plane are Pick’s theorem and the shoelace formula. For a simple lattice polygon (a polygon with a single non-crossing boundary cycle, all of whose vertex coordinates are integers) with $$i$$ integer points in its interior and $$b$$ on the boundary, Pick’s theorem computes the area as

• Keller’s conjecture ($$\mathbb{M}$$), another new Good Article on Wikipedia. The conjecture was falsified in 1992 with all remaining cases solved by 2019, but the name stuck. It’s about tilings of $$n$$-space by unit cubes, and pairs of cubes that share $$(n-1)$$-faces. In 2d, all squares share an edge with a neighbor, but a 3d tiling derived from tetrastix has many cubes with no face-to-face neighbor. Up to 7d, some cubes must be face-to-face, but tilings in eight or more dimensions can have no face-to-face pair.
• Islands

In the neighborhood where I live, fire safety regulations require the streets to be super-wide (so wide that two fire trucks can pass even with cars parked along both sides of the street), and to have even wider turnarounds at the ends of the culs-de-sac. To break up the resulting vast expanses of pavement, we have occasional islands of green, public gardens too small to name as a park. They come in several different types: medians to separate the incoming and outgoing lanes at junctions with larger roads,