Primitive Euler brick generator
General Mathematics
2024-05-24 v1
Abstract
The smallest Euler brick, discovered by Paul Halcke, has edges and face diagonals , generated by the primitive Pythagorean triple . Let 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.
Cite
@article{arxiv.2405.13061,
title = {Primitive Euler brick generator},
author = {Djamel Himane},
journal= {arXiv preprint arXiv:2405.13061},
year = {2024}
}