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 where the longest diagonal is also a whole number but such a brick has not yet been found.
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 primitive solutions, given as edges — face diagonals, are below:
Generating formula
Euler found at least two parametric solutions to the problem, but neither gives all solutions. An infinitude of Euler bricks can be generated with Sounderson's parametric formula. Let be a Pythagorean triple Then the edges give face diagonals There are many Euler bricks which are not parametrized as above, for instance the Euler brick with edges and face diagonals.
A perfect cuboid is an Euler brick whose space diagonal also has integer length. In other words, the following equation is added to the system of Diophantine equations defining an Euler brick: where is the space diagonal. , no example of a perfect cuboid had been found and no one has proven that none exist. Exhaustive computer searches show that, if a perfect cuboid exists,
Some facts are known about properties that must be satisfied by a primitive perfect cuboid, if one exists, based on modular arithmetic:
One edge, two face diagonals and the body diagonal must be odd, one edge and the remaining face diagonal must be divisible by 4, and the remaining edge must be divisible by 16.
Two edges must have length divisible by 3 and at least one of those edges must have length divisible by 9.
One edge must have length divisible by 5.
One edge must have length divisible by 7.
One edge must have length divisible by 11.
One edge must have length divisible by 19.
One edge or space diagonal must be divisible by 13.
One edge, face diagonal or space diagonal must be divisible by 17.
One edge, face diagonal or space diagonal must be divisible by 29.
One edge, face diagonal or space diagonal must be divisible by 37.
The space diagonal can only contain prime divisors ≡ 1.
Almost-perfect cuboids
An almost-perfect cuboid has 6 out of the 7 lengths as rational. Such cuboids can be sorted into three types, called Body, Edge, and Face cuboids. In the case of the Body cuboid, the body diagonal is irrational. For the Edge cuboid, one of the edges is irrational. The Face cuboid has just one of the face diagonals irrational. The Body cuboid is commonly referred to as the Euler cuboid in honor of Leonard Euler, who discussed this type of cuboid. He was also aware of Face cuboids, and provided the example. The three integer cuboid edge lengths and three integer diagonal lengths of a face cuboid can also be interpreted as the edge lengths of a Heronian tetrahedron that is also a Schläfli orthoscheme. There are infinitely many face cuboids, and infinitely many Heronian orthoschemes. Only recently have cuboids in complex numbers become known. Randall L. Rathbun published 155,151 found cuboids with the smallest integer edge less than 157,000,000,000: 56,575 were Euler cuboids, 15,449 were Edge cuboids with a complex number edge length, 30,081 were Edge cuboids, and 53,046 were Face cuboids. The smallest solutions for each type of almost-perfect cuboids, given as edges, face diagonals and the space diagonal :
A perfect parallelepiped is a parallelepiped with integer-length edges, face diagonals, and body diagonals, but not necessarily with all right angles; a perfect cuboid is a special case of a perfect parallelepiped. In 2009, dozens of perfect parallelepipeds were shown to exist, answering an open question of Richard Guy. A small example has edges 271, 106, and 103, face diagonals 101, 266, 255, 183, 312, and 323, and body diagonals 374, 300, 278, and 272. Some of these perfect parallelepipeds have two rectangular faces.
