Mathematicians have been looking for the 'perfect brick' shape for centuries, but they just proved it might actually be impossible to build.
April 13, 2026
Original Paper
Quartic reductions and elliptic obstructions for perfect Euler bricks
arXiv · 2604.09328
The Takeaway
For hundreds of years, people have tried to find a box where the length, width, height, and all diagonal lines are perfect whole numbers. This paper uses advanced geometry to trap the problem in a corner, showing that if such a 'perfect brick' exists, it must be hidden in a mathematical needle-in-a-haystack.
From the abstract
We show that the perfect Euler brick (perfect cuboid) problem is equivalent to the following elementary question: do there exist coprime integers $a, b, m, n$ such that the two expressions $(2(a^2-b^2)mn)^2 + ((a^2+b^2)(m^2-n^2))^2$ and $(4abmn)^2 + ((a^2+b^2)(m^2-n^2))^2$ are simultaneously perfect squares? Despite their near-identical structure (differing only in the first summand), no solution has ever been found. We reduce this quartic pair to a one-parameter family of genus-3 hyperelliptic