
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 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.
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Taxonomy
TopicsAdvanced Numerical Analysis Techniques
