TOPICS
Search

Perfect Cuboid

DOWNLOAD Mathematica NotebookDownload Wolfram Notebook
Brick

A perfect cuboid is a cuboid having integer side lengths, 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)

and an integer space diagonal

d_(abc)=sqrt(a^2+b^2+c^2).
(4)

The problem of finding such a cuboid is also called the brick problem, diagonals problem, perfect box problem, perfect cuboid problem, or rational cuboid problem.

No perfect cuboids are known despite an exhaustive search for all "odd sides" up to 10^(10) (Butler, pers. comm., Dec. 23, 2004).

Solving the perfect cuboid problem is equivalent to solving the Diophantine equations

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

A solution with integer space diagonal and two out of three face diagonals is a=672, b=153, and c=104, giving