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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. A perfect Euler brick is one whose space diagonal is also an integer, but such a brick has not yet been found.
The definition of an Euler brick in geometric terms is equivalent to a solution to the following system of Diophantine equations:
where a, b, c are the edges and d, e, f are the diagonals.
The smallest Euler brick, discovered by Paul Halcke in 1719, has edges (a, b, c) = (44, 117, 240) and face diagonals (d, e, f ) = (125, 244, 267). Some other small primitive solutions, given as edges (a, b, c) — face diagonals (d, e, f), are below:
Euler found at least two parametric solutions to the problem, but neither gives all solutions.
An infinitude of Euler bricks can be generated with Saunderson's parametric formula. Let (u, v, w) be a Pythagorean triple (that is, u2 + v2 = w2.) Then the edges
give face diagonals
There are many Euler bricks which are not parametrized as above, for instance the Euler brick with edges (a, b, c) = (240, 252, 275) and face diagonals (d, e, f ) = (348, 365, 373).
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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. A perfect Euler brick is one whose space diagonal is also an integer, but such a brick has not yet been found.
The definition of an Euler brick in geometric terms is equivalent to a solution to the following system of Diophantine equations:
where a, b, c are the edges and d, e, f are the diagonals.
The smallest Euler brick, discovered by Paul Halcke in 1719, has edges (a, b, c) = (44, 117, 240) and face diagonals (d, e, f ) = (125, 244, 267). Some other small primitive solutions, given as edges (a, b, c) — face diagonals (d, e, f), are below:
Euler found at least two parametric solutions to the problem, but neither gives all solutions.
An infinitude of Euler bricks can be generated with Saunderson's parametric formula. Let (u, v, w) be a Pythagorean triple (that is, u2 + v2 = w2.) Then the edges
give face diagonals
There are many Euler bricks which are not parametrized as above, for instance the Euler brick with edges (a, b, c) = (240, 252, 275) and face diagonals (d, e, f ) = (348, 365, 373).