Euler Brick

$\begin{figure}\begin{center}\BoxedEPSF{Brick.epsf}\end{center}\end{figure}$

A Rectangular Parallelepiped (``Brick'') with integral edges and face diagonals $d_{ij}$ given by

$\displaystyle d_{ab}$	$\textstyle =$	$\displaystyle \sqrt{a^2+b^2}$	(1)
$\displaystyle d_{ac}$	$\textstyle =$	$\displaystyle \sqrt{a^2+c^2}$	(2)
$\displaystyle d_{bc}$	$\textstyle =$	$\displaystyle \sqrt{b^2+c^2}.$	(3)

The problem is also called the Brick, Diagonals Problem, Perfect Box, Perfect Cuboid, or Rational Cuboid problem.

Euler found the smallest solution, which has sides , , and and face Diagonals $d_{ab}=267$ , $d_{ac}=244$ , and $d_{bc}=125$ . Kraitchik gave 257 cuboids with the Odd edge less than 1 million (Guy 1994, p. 174). F. Helenius has compiled a list of the 5003 smallest (measured by the longest edge) Euler bricks. The first few are (240, 117, 44), (275, 252, 240), (693, 480, 140), (720, 132, 85), (792, 231, 160), ... (Sloane's A031173, A031174, and A031175). Parametric solutions for Euler bricks are also known.

No solution is known in which the oblique Space Diagonal

$\begin{displaymath} d_{abc}=\sqrt{a^2+b^2+c^2} \end{displaymath}$

(4)

is also an Integer. If such a brick exists, the smallest side must be at least 1,281,000,000 (R. Rathbun 1996). Such a solution is equivalent to solving the Diophantine Equations

$\displaystyle A^2+B^2$	$\textstyle =$	$\displaystyle C^2$	(5)
$\displaystyle A^2+D^2$	$\textstyle =$	$\displaystyle E^2$	(6)
$\displaystyle B^2+D^2$	$\textstyle =$	$\displaystyle F^2$	(7)
$\displaystyle B^2+E^2$	$\textstyle =$	$\displaystyle G^2.$	(8)

A solution with integral Space Diagonal and two out of three face diagonals is

, and

, giving $d_{ab}=3\sqrt{52777}$ , $d_{ac}=680$ , $d_{bc}=185$ , and $d_{abc}=697$ . A solution giving integral space and face diagonals with only a single nonintegral Edge is

, $b=\sqrt{211773121}$ , and

, giving $d_{ab}=23711$ , $d_{ac}=20280$ , $d_{bc}=16511$ , and $d_{abc}=24961$ .

References

Guy, R. K. ``Is There a Perfect Cuboid? Four Squares whose Sums in Pairs are Square. Four Squares whose Differences are Square.'' §D18 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 173-181, 1994.

Helenius, F. First 1000 Primitive Euler Bricks. notebooks/EulerBricks.dat.

Leech, J. ``The Rational Cuboid Revisited.'' Amer. Math. Monthly 84, 518-533, 1977. Erratum in Amer. Math. Monthly 85, 472, 1978.

Sloane, N. J. A. Sequences A031173, A031174, and A031175 in ``An On-Line Version of the Encyclopedia of Integer Sequences.'' http://www.research.att.com/~njas/sequences/eisonline.html.

Rathbun, R. L. Personal communication, 1996.

Spohn, W. G. ``On the Integral Cuboid.'' Amer. Math. Monthly 79, 57-59, 1972.

Spohn, W. G. ``On the Derived Cuboid.'' Canad. Math. Bull. 17, 575-577, 1974.

Wells, D. G. The Penguin Dictionary of Curious and Interesting Numbers. London: Penguin, p. 127, 1986.