# Tilings of convex polygons by equilateral triangles of many different sizes

## TL;DR

This paper completely solves the problem of tiling equilateral triangles with many different sizes and extends the analysis to convex polygons with equilateral triangle dissections, including special subclasses.

## Contribution

It provides a complete solution to the maximum number of different sizes in equilateral triangle tilings and explores analogous problems for convex polygons and specific subclasses.

## Key findings

- Solved the maximum number of sizes in equilateral triangle tilings
- Extended results to convex polygons with equilateral triangle dissections
- Analyzed tilings with non-translational tiles

## Abstract

An equilateral triangle cannot be dissected into finitely many mutually incongruent equilateral triangles [Tutte 1948]. Therefore Tuza [Tuza 1991] asked for the largest number $s=s(n)$ such that there is a tiling of an equilateral triangle by $n$ equilateral triangles of $s(n)$ different sizes. We solve that problem completely and consider the analogous questions for dissections of convex $k$-gons into equilateral triangles, $k=4,5,6$. Moreover, we discuss all these questions for the subclass of tilings such that no two tiles are translates of each other.