Regular triangle unions with maximal number of sides

Abstract

Fix an integer n>=1. Suppose that a simple polygon is the union of n triangles whose vertices along the common boundary are arranged cyclically. How many sides can such a union -- to be called regular -- have at most? This gives OEIS sequence A375986, a recent entry. It will be shown here that the sequence begins 3, 12, 22, 33, 45, 56, 67, 80, 91, and satisfies linear lower and upper bounds. The latter is not merely an estimate: it is realizable combinatorially. This leads to two further questions: can the same combinatorics be realized in pseudoline geometry, and if so, can such a realization be stretched? The paper is largely expository, with excursions into neighboring topics (union complexity, the Zone Theorem, stretchability, the Kobon triangle problem, Davenport-Schinzel sequences, lower envelopes of line segments). However, it adds a new tool tailored for studying regular unions; namely, triangulation shifts. In essence, this is a method to represent any such n-union by a triangulation of a regular (n+1)-gon and its dynamical mutation.