{"title":"原始欧拉砖生成器","authors":"Djamel Himane","doi":"arxiv-2405.13061","DOIUrl":null,"url":null,"abstract":"The smallest Euler brick, discovered by Paul Halcke, has edges $(177, 44,\n240) $ and face diagonals $(125, 267, 244 ) $, generated by the primitive\nPythagorean triple $ (3, 4, 5) $. Let $ (u,v,w) $ primitive Pythagorean triple,\nSounderson made a generalization parameterization of the edges\n\\begin{equation*} a = \\vert u(4v^2 - w^2) \\vert, \\quad b = \\vert v(4u^2 -\nw^2)\\vert, \\quad c = \\vert 4uvw \\vert \\end{equation*} give face diagonals\n\\begin{equation*} {\\displaystyle d=w^{3},\\quad e=u(4v^{2}+w^{2}),\\quad\nf=v(4u^{2}+w^{2})} \\end{equation*} leads to an Euler brick. Finding other\nformulas that generate these primitive bricks, other than formula above, or\nmaking initial guesses that can be improved later, is the key to understanding\nhow they are generated.","PeriodicalId":501502,"journal":{"name":"arXiv - MATH - General Mathematics","volume":"59 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-05-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Primitive Euler brick generator\",\"authors\":\"Djamel Himane\",\"doi\":\"arxiv-2405.13061\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"The smallest Euler brick, discovered by Paul Halcke, has edges $(177, 44,\\n240) $ and face diagonals $(125, 267, 244 ) $, generated by the primitive\\nPythagorean triple $ (3, 4, 5) $. Let $ (u,v,w) $ primitive Pythagorean triple,\\nSounderson made a generalization parameterization of the edges\\n\\\\begin{equation*} a = \\\\vert u(4v^2 - w^2) \\\\vert, \\\\quad b = \\\\vert v(4u^2 -\\nw^2)\\\\vert, \\\\quad c = \\\\vert 4uvw \\\\vert \\\\end{equation*} give face diagonals\\n\\\\begin{equation*} {\\\\displaystyle d=w^{3},\\\\quad e=u(4v^{2}+w^{2}),\\\\quad\\nf=v(4u^{2}+w^{2})} \\\\end{equation*} leads to an Euler brick. Finding other\\nformulas that generate these primitive bricks, other than formula above, or\\nmaking initial guesses that can be improved later, is the key to understanding\\nhow they are generated.\",\"PeriodicalId\":501502,\"journal\":{\"name\":\"arXiv - MATH - General Mathematics\",\"volume\":\"59 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-05-20\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"arXiv - MATH - General Mathematics\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/arxiv-2405.13061\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - General Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2405.13061","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
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.