A Proof Without Words: Triangles in the Triangular Grid

TL;DR

This paper presents a visual proof establishing a precise count of equilateral triangles in a regular triangular grid using a bijection from four-element subsets, providing an elegant combinatorial insight.

Contribution

It introduces a proof without words that directly correlates four-element subsets with equilateral triangles in the grid, offering a novel combinatorial visualization.

Findings

Number of equilateral triangles is inom{n+2}{4} in the grid.

Provides a bijective map between subsets and triangles.

Visual illustration of the bijection for n=10.

Abstract

This proof without words demonstrates that there are $\binom{n+2}{4}$ equilateral triangles in the regular $n$-vertices-per-side triangular grid by describing a map from four-element subsets of $\{1,2, \dots, n+2\}$ into the set of equilateral triangles in this grid. Specifically, we illustrate the triangle that corresponds to the subset $\{4,5,8,11\}$ under this bijection when $n = 10$.