Almost similar configurations

TL;DR

This paper determines the exact maximum number of nearly equiangular triangles in an n-point planar set, providing a precise formula and asymptotic behavior, and explores the abundance of similar triangles in such sets.

Contribution

It presents an exact formula for the maximum number of nearly equiangular triangles in planar sets and analyzes their asymptotic growth, advancing understanding of geometric configurations.

Findings

Exact formula for h(n) for maximum nearly equiangular triangles

Asymptotic growth of h(n) as n^3/24 + O(n log n)

Existence of many similar triangles exceeding n^3/15 in large sets

Abstract

Let $h(n)$ denote the maximum number of triangles with angles between $59^\circ$ and $61^\circ$ in any $n$-element planar set. Our main result is an exact formula for $h(n)$. We also prove $h(n)= n^3/24+ O(n \log n)$ as $n\to \infty$. However, there are triangles $T$ and $n$-point sets $P$ showing that the number of $\varepsilon$-similar copies of $T$ in $P$ can exceed $n^3/15$ for any $\varepsilon>0$.