Primitive Euler brick generator

TL;DR

This paper explores formulas for generating primitive Euler bricks, aiming to discover new parameterizations or initial guesses that can lead to understanding their generation better.

Contribution

It investigates alternative formulas and initial guesses for primitive Euler bricks beyond existing parameterizations, enhancing understanding of their generation.

Findings

Identified potential new formulas for primitive Euler brick generation

Analyzed the limitations of current parameterizations

Proposed directions for future research in Euler brick generation

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.