Euler brick
In mathematics, an Euler brick, named after Leonhard Euler, is a rectangular cuboid whose edges and face diagonals all have integer lengths. A primitive Euler brick is an Euler brick whose edge lengths are relatively prime.
Definition
The definition of an Euler brick in geometric terms is equivalent to a solution to the following system of Diophantine equations:
where a, b, c are the edges and d, e, f are the diagonals. Euler found at least two parametric solutions to the problem, but neither gives all solutions.
Properties
If (a, b, c) is a solution, then (ka, kb, kc) is also a solution for any k. Consequently, the solutions in rational numbers are all rescalings of integer solutions.
Given an Euler brick with edge-lengths (a, b, c), the triple (bc, ac, ab) constitutes an Euler brick as well.
At least two edges of an Euler brick are divisible by 3.
At least two edges of an Euler brick are divisible by 4.
At least one edge of an Euler brick is divisible by 11.
Generating formula
An infinitude of Euler bricks can be generated with the following parametric formula. Let (u, v, w) be a Pythagorean triple (that is, u2 + v2 = w2.) Then the edges
