/sci/ - Science & Math

This thread will be ALL ABOUT THE EULER BRICK, and related areas of number theory, particularly Pythagorean triangles (triples), and Diophantine equations.

As a refresher, using my previously given definition of an Euler brick, "An Euler brick is a cuboid with three natural edges, and three natural side diagonals." So, in our cheesy 90's graphics OP image, the yellow and green lines all have natural (positive whole number) lengths. Since the figure is a cuboid, the edges (a,b,c) and side diagonals (d,e,f) satisfy the coupled Diophantine equations

<span class="math"> \displaystyle a^2 + b^2 = d^2 [/spoiler]

<span class="math"> \displaystyle a^2 + c^2 = e^2 [/spoiler]

<span class="math"> \displaystyle b^2 + c^2 = f^2 [/spoiler]

This is going to be a /long/ thread, as the author has some new observations to make, unproven conjectures to shepherd, and material to review. You, kind anon, are welcome and wanted to comment, ask questions, give insight and, if you can, BTFO of anything that I may say in the course of this thread.