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.
The definition of an Euler brick in geometric terms is equivalent to a solution to the following system of Diophantine equations:
\begin{cases}a2+b2=d2\ a2+c2=e2\ b2+c2=f2\end{cases}
where are the edges and are the diagonals.
The smallest Euler brick, discovered by Paul Halcke in 1719, has edges .[2] Some other small primitive solutions, given as edges — face diagonals, are below:
| ( | 85, | 132, | 720 | ) — ( | 157, | 725, | 732 | ) | |
| ( | 140, | 480, | 693 | ) — ( | 500, | 707, | 843 | ) | |
| ( | 160, | 231, | 792 | ) — ( | 281, | 808, | 825 | ) | |
| ( | 187, | 1020, | 1584 | ) — ( | 1037, | 1595, | 1884 | ) | |
| ( | 195, | 748, | 6336 | ) — ( | 773, | 6339, | 6380 | ) | |
| ( | 240, | 252, | 275 | ) — ( | 348, | 365, | 373 | ) | |
| ( | 429, | 880, | 2340 | ) — ( | 979, | 2379, | 2500 | ) | |
| ( | 495, | 4888, | 8160 | ) — ( | 4913, | 8175, | 9512 | ) | |
| ( | 528, | 5796, | 6325 | ) — ( | 5820, | 6347, | 8579 | ) |
Euler found at least two parametric solutions to the problem, but neither gives all solutions.
An infinitude of Euler bricks can be generated with Saunderson's[3] parametric formula. Let be a Pythagorean triple (that is, .) Then[1] the edges
a=u|4v2-w2|, b=v|4u2-w2|, c=4uvw
give face diagonals
d=w3, e=u(4v2+w2), f=v(4u2+w2).
There are many Euler bricks which are not parametrized as above, for instance the Euler brick with edges