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Euler Brick

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EulerBrick

An Euler brick is a cuboid with integer edge lengths a>b>c>0 and integer face diagonals

d_(ab)=sqrt(a^2+b^2)
(1)
d_(ac)=sqrt(a^2+c^2)
(2)
d_(bc)=sqrt(b^2+c^2).
(3)

If the space diagonal is also an integer, the Euler brick is called a perfect cuboid, although no examples of perfect cuboids are currently known.

The smallest Euler brick has sides (a,b,c)=(240,117,44) and face diagonals d_(ab)=267, d_(ac)=244, and d_(bc)=125, and was discovered by Halcke (1719; Dickson 2005, pp. 497-500). 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 primitive Euler bricks, with greatest common divisor GCD(a,b,c)=1, are (240, 117, 44), (275, 252, 240), (693, 480, 140), (720, 132, 85), (792, 231, 160), ... (OEIS A031173, A031174, and A031175).

Interest in this problem was high during the 18th century, and Saunderson (1740) found a parametric solution always giving Euler bricks (but not giving all possible Euler bricks), while in 1770 and 1772, Euler found at least two parametric solutions. Saunderson's solution takes a positive Pythagorean triple (a^',b^',c^'), with c^' the hypotenuse, and gives edge lengths

(a,b,c)=(|a^'(4b^('2)-c^('2))|,|b^'(4a^('2)-c^('2))|,4a^'b^'c^'),
(4)

which can be reordered to satisfy a>b>c. Before reordering, the corresponding face diagonals are

d_(ab)=c^('3)
(5)
d_(ac)=a^'(4b^('2)+c^('2))
(6)
d_(bc)=b^'(4a^('2)+c^('2)).
(7)

(Saunderson 1740; Dickson 2005, p. 497).

See also

Cuboid, Cyclic Quadrilateral, Face Diagonal, Heronian Tetrahedron, Heronian Triangle, Parallelepiped, Perfect Cuboid, Polyhedron Diagonal, Pythagorean Quadruple, Pythagorean Triple, Rational Distance Problem, Space Diagonal

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References

Dickson, L. E. History of the Theory of Numbers, Vol. 2: Diophantine Analysis. New York: Dover, 2005.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.Halcke, P. Deliciae Mathematicae; oder, Mathematisches sinnen-confect. Hamburg, Germany: N. Sauer, p. 265, 1719.Leech, J. "The Rational Cuboid Revisited." Amer. Math. Monthly 84, 518-533, 1977. Erratum in Amer. Math. Monthly 85, 472, 1978.Peterson, I. "MathTrek: Euler Bricks and Perfect Polyhedra." Oct. 23, 1999. https://web.archive.org/web/20121113000137/http://www.sciencenews.org/sn_arc99/10_23_99/mathland.htm.Rathbun, R. L. "Integer Cuboid Search Update." 8 Jan 2001. https://listserv.nodak.edu/cgi-bin/wa.exe?A2=ind0101&L=NMBRTHRY&P=1272.Saunderson, N. The Elements of Algebra in 10 Books, Vol. 2. Cambridge, England: University Press, pp. 429-431, 1740.Sloane, N. J. A. Sequences A031173, A031174, and A031175 in "The On-Line Encyclopedia of Integer Sequences."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, England: Penguin, p. 127, 1986.

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Euler Brick

Cite this as:

Weisstein, Eric W. "Euler Brick." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/EulerBrick.html

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