Pretty triangles on ugly grids
L. de Hoop (TU Delft - Electrical Engineering, Mathematics and Computer Science)
N.D. Verhulst – Mentor (TU Delft - Discrete Mathematics and Optimization)
B. van den Dries – Graduation committee member (TU Delft - Analysis)
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Abstract
In this paper, we provide a method to find all equable triangles on a given grid. Equable triangles are triangles that have the same perimeter and area. The amount of equable triangles was already found for the integer and the Eisenstein lattice. We adapt the proof by Aebi and Cairns for the Eisenstein lattice to work on general lattices. This is done in three steps:
1. Find a constraint on the side lengths of equable triangles on a given grid.
2. Find all equable triangles subject to this constraint.
3. Check for each triangle found if it can be placed on the grid.
Using this method, we can find all equable triangles on a given grid. On most grids, no equable triangles can be placed. When it is possible, they often had integer side lengths.