Diagrams based on the Hexagonal and Triangular Grids

Abstract

In this study we show that sometimes it is worth to consider a non-traditional regular grid in diagrams and proofs by diagrams instead of the traditional rectangular arrangement. We refer to the hexagonal and triangular grids as non-traditional regular grids. Particularly, we show that not only triangular numbers can be arranged in triangular shape, but also, it is natural to represent the square numbers via triangles on the triangular grid. On the other hand, we show that hexagonal arrangement is useful to represent binomial coefficients, and also to solve jug puzzles. The triangular and hexagonal grids are dual of each other and share many properties including symmetric properties. Based on our study, we propose to consider them more frequently in education and diagrammatic applications as well. This would be fruitful also for critical thinking, design and problem solving in mathematics, engineering and informatics including artificial and computational intelligence.