Triangle unions with maximal number of sides

Abstract

Given an integer n>=1. Suppose, a simple polygon is a union of n triangles. What is the maximal number of sides it can have? This is a sequence A375986, a recent entry into the OEIS. In this paper we prove that it starts as 3, 12, 22, 33, 44, 55, 67, 79, and satisfies simple linear lower and upper bounds. The proof of the latter is combinatoric and is valid for segments of pseudolines instead of lines, too. It is still unknown whether such optimal combinatoric configuration is stretchable for larger n.

Figures (13)

• Figure 1: $I=mn-m-n+3$ number of intersections of $n$-gon and $m$-gon for odd $m,n$ ($m=9$, $m=7$, $I=50$ here).
• Figure 2: Union of $n=1,2$ and $3$$\triangle$
• Figure 3: The unique solution to Problem B in case $n=4$; here $\vartheta=0.9709631$
• Figure 4: $10$ lines forming $25$ non-overlapping $\triangle$
• Figure 5: Three ways to draw 4 circles. Configuration $\mathbf{c}$ has $12$ boundary arcs.