The delta-lambda configurations in tiling the square
The authors begin with the problem of tiling an integer sided square with integer sided triangles. There are essentially four possible configurations consisting of four triangles, which they call the nu, delta, chi, and kappa configurations. The main purpose of this note is a fairly thorough discussion of the delta configuration. A number x is called a rectangular number if \(x^ 2+1\) is a perfect square. The authors show that the delta configurations are characterized by solutions of the equation \(1-Y=M(1-X)\) with X,Y,M all rectangular numbers. This quickly leads to the problem of finding all rational points on a certain surface. After several transformations, the authors exhibit this surface (birationally) as an elliptic surface E over \({\mathbb{P}}^ 1\) given by the Weierstrass equation \[ y^ 2=x[x+2(\lambda +1)^ 2][x+2\lambda^ 2(\lambda -1)^ 2]. \] They perform a standard 2-descent to show that E has rank 1 over \({\mathbb{Q}}(\lambda)\). (However, they leave open the question of the rank of E over (\({\mathbb{C}}(\lambda).)\). This leads to an infinite family of parametrized solutions to the delta configuration. Next they substitute specific rational values for \(\lambda\). If the resulting elliptic curve has rank 1 over \({\mathbb{Q}}\), they essentially obtain all delta configurations for that \(\lambda\). However, it can happen that the rank jumps. They perform a descent to find an upper bound for the rank. Finally, they find many values of \(\lambda\) for which the rank is 2, including a certain one-parameter family, and also one value for which the rank is 3.
scientific article; zbMATH DE number 4110706