Euler brick [SEP] named after
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 are the edges and are the diagonals.
Properties.
- If is a solution, then is also a solution for any . Consequently,
the solutions in rational numbers are all rescalings of integer solutions. Given an Euler brick with edge-lengths , the triple 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.
Examples.
The smallest Euler brick, discovered by Paul Halcke in 1719, has edges and face diagonals . Some other small