# Cuboid nearest to a cube

Cuboid nearest to a cube.

While answering this question, euler bricks: way to calculate them? I noticed one result was not too far from cube shaped, and wondered if there was a more cubic cuboid.

$$x^2+y^2=u^2$$
$$y^2+z^2=v^2$$
$$x^2+z^2=w^2$$

$x,y,z,u,v,w$ positive integers, and $x<y<z$

The result I noticed was $(240,252,275)$, and decided to use $\alpha=\large \frac{z^2}{xy}$ as a measure of nearness to a cube. For $(240,252,275)$ we have $\alpha=1.25041336$

Diagram: https://en.wikipedia.org/wiki/Euler_brick#/media/File:Euler_brick_examples.svg

Despite a fair bit of calculation, I can only find one more cubic cuboid:
$$(1008,1100,1155)$$

This has $\alpha=1.203125$ and is produced from the following solution generator using $(240,252,275)$,
“If $(x,y,z)$ is a solution, then $(xy,xz,yz)$ is also a solution”.

My questions.
Is there a better measure of nearness to a cube than $\alpha= \large\frac{z^2}{xy}$ ?

Is there a better solution than $(1008,1100,1155)$ ?

Thank you.

#### Solutions Collecting From Web of "Cuboid nearest to a cube"

Answer to the $2$nd question:
yes, there are cubic cuboids with rather less measure.

If use measure $\alpha = \dfrac{z^2}{xy}$, then the best one currently known for me (see the table below) has $\alpha \approx \color{red}{1.0352}$.

Here are few noteworthy examples:

\begin{array}{|r|c|}
\hline
(x,y,z) & \alpha \\
\hline
(2\:278\:100,\; 2\:423\:952,\; 2\:564\:661) & \approx 1.191 \\
(4\:160\:772,\; 4\:540\:525,\; 4\:717\:440) & \approx 1.178 \\
(14\:358\:336,\; 15\:041\:873,\; 15\:526\:440) & \approx 1.116 \\
(43\:875\:188,\; 44\:127\:291,\; 46\:181\:520) & \approx 1.102 \\
(5\:122\:780,\; 5\:245\:200,\; 5\:288\:547) & \approx 1.0409 \\
(15\:301\:440,\; 15\:748\:920,\; 15\:798\:809) & \approx 1.0358 \\
(108\:192\:528,\; 109\:141\:700,\; 110\:562\:771) & \approx 1.0352 \\
\hline
\end{array}