{"id":2983,"date":"2019-04-12T16:42:28","date_gmt":"2019-04-12T16:42:28","guid":{"rendered":"http:\/\/www.goodsoul.us\/?p=2983"},"modified":"2019-04-12T16:42:32","modified_gmt":"2019-04-12T16:42:32","slug":"new-geometrical-proof-of-fermats-theorem","status":"publish","type":"post","link":"https:\/\/www.goodsoul.us\/?p=2983","title":{"rendered":"New Geometrical Proof of Fermat\u2019s Theorem"},"content":{"rendered":"\r\n
\"\"<\/figure>\r\n\r\n\r\n\r\n

By  Interesting\r\nEngineering<\/a><\/strong><\/p>\r\n\r\n\r\n\r\n

January, 26th 2017<\/strong><\/p>\r\n\r\n\r\n\r\n

\"\"<\/figure>\r\n\r\n\r\n\r\n

Last year (2016), in the Interesting Engineering article<\/a> entitled \u201cRevolution in\r\nthe Pythagoras\u2019 Theorem?\u201d, Dr. Luis Teia presented the proof of the Pythagoras\u2019\r\ntheorem in 3D. This year, Teia explains in his recent (Feb 2017) peer reviewed\r\npaper, entitled Fermat\u2019s Theorem \u2013 a Geometrical View<\/em><\/a> published in\r\nthe Journal of Mathematics Research, how this 3D understanding of the\r\nPythagoras\u2019 Theorem provided the geometrical foundation to prove Fermat\u2019s Last\r\nTheorem. Fermat\u2019s Last Theorem, also known as Fermat\u2019s conjecture, is more than\r\njust about triples, it is about the fundamental nature of an integer number,\r\nand it\u2019s mathematical and geometrical meaning. It raises the philosophical\r\nquestion: What is a unit?<\/em> In the language of mathematics, a\r\nunit is defined by the number 1. In the language of geometry, a unit is defined\r\nby an element of side length one. A perspective of a problem depends on the\r\nlanguage we use to observe it, and a change in perspective is often all it\r\ntakes to see the solution.<\/p>\r\n\r\n\r\n\r\n

What is Fermat\u2019s Theorem?<\/strong><\/p>\r\n\r\n\r\n\r\n

Fermat\u2019s Last Theorem questions not only what is a triple, but more importantly, what is an integer in the context of equations of the type Xn<\/sup>\u00a0+Yn<\/sup>\u00a0= Zn<\/sup>. The picture below shows in a pictorial way the difference between the Pythagoras\u2019 theorem and Fermat\u2019s Last theorem<\/g>. These two are sometimes confused. Fermat\u2019s Last Theorem is a mathematical conjecture about integer numbers, while the 3D Pythagoras\u2019 theorem is a mathematical and geometrical proof about real numbers. The Pythagoras\u2019 theorem in 1D is the principle of summation (i.e., X+Y=Z). In it, all integers form triples [e.g., 1+2=3 forms the 1D triple (1,2,3) while 3+4=7 forms (3,4,7)]. In the middle is the well-known Pythagoras\u2019 theorem in 2D, where only some integers form triples. Fermat\u2019s Last Theorem states that no triples can be found for Pythagoras\u2019 theorem in 3D, or for any higher dimension. <\/p>\r\n\r\n\r\n\r\n

\"\"<\/figure>\r\n\r\n\r\n\r\n

<\/strong>  The Pythagoras\u2019 Theorem in 1D, 2D and 3D, and\r\nFermat\u2019s Last Theorem [Image Source: Teia<\/a><\/em>]<\/strong><\/p>\r\n\r\n\r\n\r\n

The 3D Pythagoras\u2019 Theorem<\/strong><\/p>\r\n\r\n\r\n\r\n

The Pythagoras\u2019 theorem in 1D is governed by\r\nlines, while in 2D by squares (see picture below). Just like squares appear\r\nnaturally when transforming the Pythagoras\u2019 theorem from 1D to 2D, octahedrons\r\nalso appear naturally when transforming the Pythagoras\u2019 theorem from 2D to 3D.\r\nAs shown by Dr. Teia (in his book<\/a>published in 2015), the 3D Pythagoras\u2019 theorem is governed\r\nby octahedrons. Therefore, any number (real or integer) within the Pythagoras\u2019\r\ntheorem is expressible geometrically by a line in 1D, a square in 2D and an\r\noctahedron in 3D. How does this geometrical notion affect our understanding\r\nof integers, and more importantly of triples?<\/em><\/p>\r\n\r\n\r\n\r\n

\"\"<\/figure>\r\n\r\n\r\n\r\n

<\/strong>The 1D, 2D and 3D Pythagoras\u2019 Theorem [Image Source: ]<\/strong><\/p>\r\n\r\n\r\n\r\n

Hypothesis<\/strong><\/p>\r\n\r\n\r\n\r\n

 <\/strong>The hypothesis of this new proof is that a triple only exists,\r\nif all integer elements within that triple also exist [e.g., 1, 2, 3 for the 1D\r\ntriple (1,2,3), and 3, 4, 5 for the 2D triple (3,4,5)]. In turn, an integer\r\nelement only exits if it obeys two conditions: it satisfies the Pythagoras\u2019\r\ntheorem of the respective dimension (Condition 1), and it can be completely\r\nsuccessfully split into multiple unit scalars (Condition 2). One can therefore\r\nhypothesize that integer elements do not exist if either Condition 1 or 2 is\r\nnot satisfied. By consequence, if the integer does not exist, then the\r\nassociated triples also do not exist.<\/p>\r\n\r\n\r\n\r\n

The Geometrical Integer<\/strong><\/p>\r\n\r\n\r\n\r\n

 <\/strong>Integers are clear multiples of a unit. The unit line, or line\r\nof length 1, is the fundamental geometric scalar that composes all integer\r\nelements in the 1D Pythagoras\u2019 universe. Likewise, the unit square, or square\r\nof side 1, is the fundamental geometric scalar that composes all integer\r\nelements in the 2D Pythagoras\u2019 universe. Generally, one can conclude that in\r\norder for an integer element to exist, it needs to be completely split into\r\nmultiples of the fundamental unit scalar particular to that dimension (i.e.,\r\nunit line in 1D or unit square in 2D). In 3D, despite octahedrons validating\r\nthe 3D Pythagoras\u2019 Theorem (satisfying Condition 1), an octahedron with side\r\ninteger N is not a multiple of unit octahedrons, as tetrahedrons appear in the\r\nmiddle (refer to figure below right) [not satisfying Condition 2]. Therefore,\r\ngeometrical integers do not exist in the 3D domain of the Pythagoras\u2019 theorem,\r\nand neither do their triples. This satisfies Fermat\u2019s theorem for three\r\ndimensions.<\/p>\r\n\r\n\r\n\r\n

\"\"<\/figure>\r\n\r\n\r\n\r\n

<\/strong>The geometrical definition of integers in 1D, 2D and not in\r\n3D [Image Source: ]<\/strong><\/p>\r\n\r\n\r\n\r\n

Higher Dimensions<\/strong><\/p>\r\n\r\n\r\n\r\n

The geometrical interdependency between\r\nintegers in 1D and 2D suggests that all integers of higher dimensions are\r\nbuilt, and hence are dependent, on the integers of lower dimensions (e.g.\r\nsquares are built with lines). This interdependency coupled with the absence of\r\nintegers in 3D suggests that there is no integer above n > 2, and therefore\r\nthere is also no triples that satisfy Xn<\/sup> + Yn<\/sup> = Zn<\/sup> for n > 2.<\/p>\r\n\r\n\r\n\r\n

Conclusion<\/strong><\/p>\r\n\r\n\r\n\r\n

The geometrical solution to Fermat\u2019s riddle\r\ndoesn\u2019t come from the notion of triples, but rather from the notion of\r\nintegers. If integers don\u2019t exist, then neither can triples. Alas, the\r\ncentennial elusiveness of the proof results from the repetitive usage of\r\navailable \u201ctools\u201d, rather than inventing new tools (the 3D Pythagoras\u2019 theorem)\r\nto find the solution. The simplicity of this geometrical proof (founded on the\r\nabsence of integers within the domain of the Pythagoras\u2019 theorem for dimensions\r\nabove 2D) makes us wonder whether this is not the famous \u201celegant solution\u201d\r\nthat Fermat spoke of, of which he left no other records except a written note\r\nsaying:<\/p>\r\n\r\n\r\n\r\n

\u201cI have discovered a truly remarkable proof of\r\nthis theorem, which this margin is too small to contain.\u201d<\/p>\r\n\r\n\r\n\r\n

–Pierre de Fermat (1665)<\/p>\r\n\r\n\r\n\r\n

As for Dr. Luis Teia, his next challenge will\r\nbe to explain the geometrical meaning of the formula on partitions from the\r\nmathematician Srinivasa Ramanujan.<\/p>\r\n","protected":false},"excerpt":{"rendered":"

By  Interesting Engineering January, 26th 2017 Last year (2016), in the Interesting Engineering article entitled \u201cRevolution in the Pythagoras\u2019 Theorem?\u201d, Dr. Luis Teia […]<\/p>\n","protected":false},"author":6,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"site-sidebar-layout":"default","site-content-layout":"","ast-site-content-layout":"default","site-content-style":"default","site-sidebar-style":"default","ast-global-header-display":"","ast-banner-title-visibility":"","ast-main-header-display":"","ast-hfb-above-header-display":"","ast-hfb-below-header-display":"","ast-hfb-mobile-header-display":"","site-post-title":"","ast-breadcrumbs-content":"","ast-featured-img":"","footer-sml-layout":"","ast-disable-related-posts":"","theme-transparent-header-meta":"","adv-header-id-meta":"","stick-header-meta":"","header-above-stick-meta":"","header-main-stick-meta":"","header-below-stick-meta":"","astra-migrate-meta-layouts":"default","ast-page-background-enabled":"default","ast-page-background-meta":{"desktop":{"background-color":"var(--ast-global-color-5)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"ast-content-background-meta":{"desktop":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"tablet":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""},"mobile":{"background-color":"var(--ast-global-color-4)","background-image":"","background-repeat":"repeat","background-position":"center center","background-size":"auto","background-attachment":"scroll","background-type":"","background-media":"","overlay-type":"","overlay-color":"","overlay-opacity":"","overlay-gradient":""}},"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[1],"tags":[],"class_list":["post-2983","post","type-post","status-publish","format-standard","hentry","category-uncategorized"],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/p2R4NI-M7","_links":{"self":[{"href":"https:\/\/www.goodsoul.us\/index.php?rest_route=\/wp\/v2\/posts\/2983","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.goodsoul.us\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.goodsoul.us\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.goodsoul.us\/index.php?rest_route=\/wp\/v2\/users\/6"}],"replies":[{"embeddable":true,"href":"https:\/\/www.goodsoul.us\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=2983"}],"version-history":[{"count":1,"href":"https:\/\/www.goodsoul.us\/index.php?rest_route=\/wp\/v2\/posts\/2983\/revisions"}],"predecessor-version":[{"id":2989,"href":"https:\/\/www.goodsoul.us\/index.php?rest_route=\/wp\/v2\/posts\/2983\/revisions\/2989"}],"wp:attachment":[{"href":"https:\/\/www.goodsoul.us\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=2983"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.goodsoul.us\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=2983"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.goodsoul.us\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=2983"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}