Primitive Euler brick generator

Abstract

The smallest Euler brick, discovered by Paul Halcke, has edges $(177,44,240)$ and face diagonals $(125,267,244)$, generated by the primitive Pythagorean triple $(3,4,5)$. Let $(u,v,w)$ primitive Pythagorean triple, Sounderson made a generalization parameterization of the edges \begin{equation*} a = \vert u(4v^2 - w^2) \vert, \quad b = \vert v(4u^2 - w^2)\vert, \quad c = \vert 4uvw \vert \end{equation*} give face diagonals \begin{equation*} {\displaystyle d=w^{3},\quad e=u(4v^{2}+w^{2}),\quad f=v(4u^{2}+w^{2})} \end{equation*} leads to an Euler brick. Finding other formulas that generate these primitive bricks, other than formula above, or making initial guesses that can be improved later, is the key to understanding how they are generated.