Euler brick

In mathematics, an Euler brick, named after Leonhard Euler, is a cuboid whose edges and face diagonals all have integer lengths. A primitive Euler brick is an Euler brick whose edge lengths are relatively prime.

1 Properties
2 Examples
3 Perfect cuboid
4 Perfect parallelepiped
5 Notes
6 References

Properties

Alternatively stated, an Euler brick is a solution to the following system of Diophantine equations:

$\begin{cases} a^2 %2B b^2 = d^2\\ b^2 %2B c^2 = e^2\\ a^2 %2B c^2 = f^2\end{cases}$

Euler found at least two parametric solutions to the problem, but neither give all solutions.^[1]

Given an Euler brick with edges (a, b, c), the triple (bc, ac, ab) constitutes an Euler brick as well.

Examples

The smallest Euler brick, discovered by Paul Halcke in 1719, has edges $(a, b, c) = (240, 117, 44)$ and face diagonals 267, 244, and 125.

Other solutions are: Given as: length (a, b, c)

(275, 252, 240),
(693, 480, 140),
(720, 132, 85), and
(792, 231, 160).

Perfect cuboid

Unsolved problems in mathematics
Does a perfect cuboid exist?

A perfect cuboid (also called a perfect box) is an Euler brick whose space diagonal is also an integer.

In other words the following equation is added to the above Diophantine equations:

$a^2 %2B b^2 %2B c^2 = g^2.\,$

Some interesting facts about a primitive perfect cuboid:

2 of the edges {a,b,c} must be even and 1 edge must be odd
1 edge must be divisible by 4 and 1 edge must be divisible by 16
1 edge must be divisible by 3 and 1 edge must be divisible by 9
1 edge must be divisible by 5
1 edge must be divisible by 7
1 edge must be divisible by 11
1 edge must be divisible by 19.

As of January 2011, no example of a perfect cuboid had been found and no one had proven that it cannot exist. Exhaustive computer searches show that, if a perfect cuboid exists, one of its sides must be greater than 1 trillion (10¹²).^[2]^[3]

Solutions have been found where the space diagonal and two of the three face diagonals are integers, such as:

$(a, b, c) = (672, 153, 104).\,$

Solutions are also known where all four diagonals but only two of the three edges are integers, such as:

$(a, b, c) = (18720, \sqrt{211773121}, 7800)$

and

$(a, b, c) = (520, 576, \sqrt{618849}).$

Perfect parallelepiped

A perfect cuboid is the special case of a perfect parallelepiped with all right angles. In 2009, a perfect parallelepiped was shown to exist,^[4] answering an open question of Richard Guy. Solutions with only a single oblique angle have been found.

Notes

^ Weisstein, Eric W., "Euler Brick" from MathWorld.
^ Durango Bill. The “Integer Brick” Problem
^ Weisstein, Eric W., "Perfect Cuboid" from MathWorld.
^ Sawyer, Jorge F.; Reiter, Clifford A. (2009). Perfect parallelepipeds exist. arXiv:0907.0220 .

References

Leech, John (1977). "The Rational Cuboid Revisited". American Mathematical Monthly 84 (7): 518–533. doi:10.2307/2320014. JSTOR 2320014.
Guy, Richard K. (2004). Unsolved Problems in Number Theory. Springer-Verlag. pp. 275–283. ISBN 0-387-20860-7.
Roberts, Tim (2010). "Some constraints on the existence of a perfect cuboid". Australian Mathematical Society Gazette 37: 29–31. ISSN 1326-2297.