Regular triangle unions with maximal number of sides

TL;DR

This paper investigates the maximum number of sides in regular unions of n triangles forming simple polygons, providing bounds, realizability results, and exploring related geometric and combinatorial topics.

Contribution

It introduces triangulation shifts as a new tool to analyze regular unions and establishes realizability of certain combinatorial bounds in pseudoline geometry.

Findings

Sequence of maximum sides begins 3, 12, 22, 33, 45, 56, 67, 80, 91.

Linear lower and upper bounds on the number of sides are proven.

Realizability of bounds in pseudoline geometry is demonstrated.

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.