Primitive Euler brick generator
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2024
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866909208096538624 |
|---|---|
| author | Himane, Djamel |
| author_facet | Himane, Djamel |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_13061 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Primitive Euler brick generator Himane, Djamel General Mathematics 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. |
| title | Primitive Euler brick generator |
| topic | General Mathematics |
| url | https://arxiv.org/abs/2405.13061 |