A Proof Without Words: Triangles in the Triangular Grid

2022/10/31 by Kagey, Peter
#05A19 #Combinatorics (math.CO) #FOS: Mathematics #History and Overview (math.HO)

paper · doi:10.48550/arxiv.2211.00186

Abstract

This proof without words demonstrates that there are \binomn+24 equilateral triangles in the regular n-vertices-per-side triangular grid by describing a map from four-element subsets of \1,2, …, 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.

Related

• n² + 1 unit equilateral triangles cannot cover an equilateral triangle of side > n if all triangles have parallel sides
• Equilateral triangles in subsets of \Bbb R^d of large Hausdorff dimension
• Dissecting Equilateral Triangles into Non-Congruent Equilateral Triangles: A Novel Proof of Tutte's Result
• Tilings of convex polygons by equilateral triangles of many different sizes
• A characterization of all equilateral triangles in \Bbb Z³