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. A perfect Euler brick is one whose space diagonal is also an integer, but such a brick has not yet been found.

Definition

The definition of an Euler brick in geometric terms is equivalent to a solution to the following system of Diophantine equations:

{

a

2

+

b

2

=

d

2

a

2

+

c

2

=

e

2

b

2

+

c

2

=

f

2

{\displaystyle {\begin{cases}a^{2}+b^{2}=d^{2}\\a^{2}+c^{2}=e^{2}\\b^{2}+c^{2}=f^{2}\end{cases}}}

where a, b, c are the edges and d, e, f are the diagonals.

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.

Exactly one edge and two face diagonals of a primitive Euler brick are odd.

At least two edges of an Euler brick are divisible by 3.

At least two edges of an Euler brick are divisible by 4.