Description

In mathematics, an Euler Brick, named after the famous mathematician Leonhard Euler, is a cuboid with integer edges and also integer face diagonals.

Alternatively stated an Euler Brick is a solution to the following diophantine problem

a² + b² = d²

b² + c² = e²

a² + c² = f²

Euler found at least 2 parametric solutions to the problem.

Of Interest

A Perfect Cuboid (also called a Perfect Box) is an Euler Brick whose body diagonal is also an integer.

In other words the following equation is added to the above diophantine problem

a² + b² + c² = g²

Some interesting facts about a Perfect Cuboid.

• 2 sides must be even and 1 side must be odd.
• 1 side must be divisible by 4 and 1 side must be divisible by 16
• 1 side must be divisible by 3 and 1 side must be divisible by 9
• 1 side must be divisible by 5
• 1 side must be divisible by 11

History

The smallest Euler brick has sides

(a,b,c) = (240,117,44) and face polyhedron diagonals 267, 244, and 125 and was first discovered by Paul Halcke in 1719.

As of March 14 2005 no example of a Perfect Cuboid has been found and no one has proven that it cannot exist. Exhaustive computer searches have proven that the smallest side of the perfect box is at least 4.3 billion.

External Links

• Information at MathWorld