Euler brick

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

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

$a^{2}+b^{2}=d^{2}\,$

$b^{2}+c^{2}=e^{2}\,$

$a^{2}+c^{2}=f^{2}.\,$

The smallest Euler brick, discovered by Paul Halcke in 1719, has edges

$(a,b,c)=(240,117,44)\$

and face polyhedron 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).

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

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

Perfect cuboid

Unsolved problem in mathematics:

Does a perfect cuboid exist?

(more unsolved problems in mathematics)

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 equations:

$a^{2}+b^{2}+c^{2}=g^{2}.\,$

Some interesting facts about a perfect cuboid:

• 2 edges must be even and 1 edge must be odd (for a primitive perfect cuboid)
• 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 11.

As of 2010, 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 at least 100 billion.

Solutions have been found where the body 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 a special case of a perfect parallelepiped with all right angles. In 2009, a perfect parallelepiped was shown to exist, answering an open question of Richard Guy. Solutions with only a single oblique angle have been found.

References

• Leech, John (1977). "The Rational Cuboid Revisited". American Mathematical Monthly. 84: 518–533. doi:10.2307/2320014.
• Guy, Richard K. (2004). Unsolved Problems in Number Theory. Springer-Verlag. pp. 275–283. ISBN 0-387-20860-7.