• Linkage with many Wikipedia Good Articles

There are two reasons for the large number of Good Articles in this set. First, I had previously been trying to keep my nominations and reviews in balance, but there were too few nominations to review on topics of interest to me, and the inability to find things to review was preventing me from nominating other articles when they were ready. So I started nominating more often. And second, the Wikipedia Good Articles editors are having a drive this month to clean out old nominations, as they tend to do a couple of times per year.

• Angles of arc-triangles

Piecewise-circular curves or, if you like, arc-polygons are a very old topic in mathematics. Archimedes and Pappus studied the arbelos, a curved triangle formed from three semicircles, and Hippocrates of Chios found that the lune of Hippocrates, a two-sided figure bounded by a semicircle and a quarter-circle, has the same area as an isosceles right triangle stretched between the same two points. The history of the Reuleaux triangle, bounded by three sixths of circles, stretches back well past Reuleaux to the shapes of of Gothic church windows and its use by Leonardo da Vinci for fortress floor plans and world map projections. But despite their long history and frequent use (for instance in the design of machined parts), there are some basic properties of arc-polygons that seem to have been unexplored so far.

• Spanners for unit ball graphs in doubling spaces

My student Hadi Khodabandeh had a paper with me on spanners earlier this year at SoCG, in which we showed that the greedy spanner algorithm for points in the Euclidean plane produces graphs with few crossings and small separators. Now we have another spanner preprint: “Optimal spanners for unit ball graphs in doubling metrics”, arXiv:2106.15234.

• Flow lines ($$\mathbb{M}$$). Web gadget editable open source code thingy to draw streamlines of mathematical formulas, in svg format, by Maksim Surguy.
• Greedy orderings with transposition

I’m a big fan of using antimatroids to model vertex-ordering processes in graphs such as the construction of topological orderings in directed acyclic graphs and perfect elimination orderings in chordal graphs. In each case a vertex can be removed from the graph and added to the order when it obeys a local condition: its remaining neighbors are all outgoing for topological orderings, or all adjacent for perfect elimination orderings. Once this condition becomes true of a vertex it remains true until the vertex is added to the order, the defining property of an antimatroid. Because of this property, a greedy algorithm for finding these orderings can never make a mistake: if there exists an ordering of all of the vertices, it is always a safe choice to add any vertex that can be added.

• What is a natural question? ($$\mathbb{M}$$) Gasarch on distinguishing notions of interestingness of mathematical problems based on ability to answer them, versus whether they lead to deeper mathematics.
• Carrying as chip-firing for the Zeckendorf representation

You may have heard of the Zeckendorf representation according to which any positive integer can be represented as a sum of non-consecutive Fibonacci numbers. Its uses include the optimal strategy in the game of Fibonacci nim. But did you know that it’s possible to efficiently add and subtract Zeckendorf representations?1 The algorithm from the paper linked above takes three passes over the input digit sequences using finite state automata, much like binary number addition can be performed by a single pass of a finite state automaton. I thought it might be interesting to describe an alternative path to the same result, using chip-firing games.

1. Connor Ahlbach, Jeremy Usatine, Christiane Frougny, and Nicholas Pippenger (2013), “Efficient algorithms for Zeckendorf arithmetic”, Fibonacci Quarterly 51 (3): 249–255, arXiv:1207.4497, MR3093678