Purposing an Algebraic Solution to the Four-Color Problem
Autor(a) principal: | |
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Data de Publicação: | 2021 |
Tipo de documento: | preprint |
Idioma: | eng |
Título da fonte: | SciELO Preprints |
Texto Completo: | https://preprints.scielo.org/index.php/scielo/preprint/view/3156 |
Resumo: | The Four-Color Theorem was originated with the coloring of Countries in a MAP and it was a challenging problem that remained open since 1853 for more than 170 years. By the end of Sec XX, this problem was solved using computational tools but until today there is no algebraic proof of it. In this article, the original problem of coloring MAPS over a Spherical Surface is briefly reviewed. A Spherical MAP is converted into a Planar MAP using polar coordinates and the frontiers of the Countries are described as real implicit equations and then deployed from the real space into the complex space. In the complex space the rules to color MAPs are described as system of algebraic equations and inequations. One example of MAP is solved (colored) and the explanation about why these systems are solvable is done. Beginning from the example, a general theory to coloring MAPs is derived. As all the transformations used admits inverse, the obtained planar MAP solution can be reversed as a solution to the Spherical MAP. All operations involve simple algebraic transformations and some Calculus concepts |
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Purposing an Algebraic Solution to the Four-Color ProblemProponiendo una Solución Algebrica para el Teorema de las 4 ColoresProposta de Solução Algébrica para o Teorema das 4 CoresTeorema das Quatro CoresDesbobramento complexoFour-Color TheoremComplex deploymentTeorema de los Quatro ColoresDes´pliegue complejoThe Four-Color Theorem was originated with the coloring of Countries in a MAP and it was a challenging problem that remained open since 1853 for more than 170 years. By the end of Sec XX, this problem was solved using computational tools but until today there is no algebraic proof of it. In this article, the original problem of coloring MAPS over a Spherical Surface is briefly reviewed. A Spherical MAP is converted into a Planar MAP using polar coordinates and the frontiers of the Countries are described as real implicit equations and then deployed from the real space into the complex space. In the complex space the rules to color MAPs are described as system of algebraic equations and inequations. One example of MAP is solved (colored) and the explanation about why these systems are solvable is done. Beginning from the example, a general theory to coloring MAPs is derived. As all the transformations used admits inverse, the obtained planar MAP solution can be reversed as a solution to the Spherical MAP. All operations involve simple algebraic transformations and some Calculus conceptsEl teorema de los cuatro colores se originó de la coloración de países en un mapa y fue un problema desafiante que ha permanecido abierto desde 1853 durante más de 170 años. Al final de la Siglo XX, se resolvió este problema utilizando herramientas computacionales, pero hasta hoy no hay una prueba algébrica de ello. En este artículo, se repasa brevemente el problema original de pintar países sobre una superficie esférica. Un mapa esférico se convierte en un mapa plano utilizando coordenadas polares, y las fronteras de los países se describen como ecuaciones implícitas reales y luego se despliegan desde el espacio real al espacio complejo. En un espacio complejo, las reglas para colorear MAPAS se describen mediante un sistema de ecuaciones y desigualdades algébricas. Se resuelve un ejemplo de MAPA (se lo colorea) y se da una explicación de por qué se pueden resolver estos sistemas. Del ejemplo, se deriva una teoría general para resolver MAPAS. Como todas las transformaciones utilizadas admiten inversa, la solución plana obtenida se puede revertir al MAP esférico. Todas las operaciones involucran transformaciones algébricas sencillas y algunos conceptos de cálculo.O Teorema das Quatro Cores originou-se da coloração de Países em um MAPA e foi um problema desafiador que permaneceu em aberto desde 1853 por mais de 170 anos. No final da Seção XX, esse problema foi resolvido usando ferramentas computacionais, mas até hoje não há prova algébrica do mesmo. Neste artigo, o problema original de pintar Países sobre uma Superfície Esférica é brevemente revisado. Um mapa esférico é convertido em um mapa plano usando coordenadas polares e as fronteiras dos países são descritas como equações implícitas reais e, em seguida, desdobradas do espaço real para o espaço complexo. No espaço complexo, as regras para colorir MAPAS são descritas por um sistema de equações e inequações algébricas. Um exemplo de MAPA é resolvido (colorido) e a explicação sobre por que esses sistemas podem ser resolvidos é dada. A partir do exemplo, uma teoria geral para a resolução de MAPAS é derivada. Como todas as transformações utilizadas admitem inversa, a solução planar obtida pode ser revertida para o MAPA esférico. Todas as operações envolvem transformações algébricas simples e alguns conceitos de cálculo.SciELO PreprintsSciELO PreprintsSciELO Preprints2021-11-29info:eu-repo/semantics/preprintinfo:eu-repo/semantics/publishedVersionapplication/pdfhttps://preprints.scielo.org/index.php/scielo/preprint/view/315610.1590/SciELOPreprints.3156enghttps://preprints.scielo.org/index.php/scielo/article/view/3156/5696Copyright (c) 2021 José Jansenhttps://creativecommons.org/licenses/by/4.0info:eu-repo/semantics/openAccessJansen, Joséreponame:SciELO Preprintsinstname:SciELOinstacron:SCI2021-11-07T14:07:42Zoai:ops.preprints.scielo.org:preprint/3156Servidor de preprintshttps://preprints.scielo.org/index.php/scieloONGhttps://preprints.scielo.org/index.php/scielo/oaiscielo.submission@scielo.orgopendoar:2021-11-07T14:07:42SciELO Preprints - SciELOfalse |
dc.title.none.fl_str_mv |
Purposing an Algebraic Solution to the Four-Color Problem Proponiendo una Solución Algebrica para el Teorema de las 4 Colores Proposta de Solução Algébrica para o Teorema das 4 Cores |
title |
Purposing an Algebraic Solution to the Four-Color Problem |
spellingShingle |
Purposing an Algebraic Solution to the Four-Color Problem Jansen, José Teorema das Quatro Cores Desbobramento complexo Four-Color Theorem Complex deployment Teorema de los Quatro Colores Des´pliegue complejo |
title_short |
Purposing an Algebraic Solution to the Four-Color Problem |
title_full |
Purposing an Algebraic Solution to the Four-Color Problem |
title_fullStr |
Purposing an Algebraic Solution to the Four-Color Problem |
title_full_unstemmed |
Purposing an Algebraic Solution to the Four-Color Problem |
title_sort |
Purposing an Algebraic Solution to the Four-Color Problem |
author |
Jansen, José |
author_facet |
Jansen, José |
author_role |
author |
dc.contributor.author.fl_str_mv |
Jansen, José |
dc.subject.por.fl_str_mv |
Teorema das Quatro Cores Desbobramento complexo Four-Color Theorem Complex deployment Teorema de los Quatro Colores Des´pliegue complejo |
topic |
Teorema das Quatro Cores Desbobramento complexo Four-Color Theorem Complex deployment Teorema de los Quatro Colores Des´pliegue complejo |
description |
The Four-Color Theorem was originated with the coloring of Countries in a MAP and it was a challenging problem that remained open since 1853 for more than 170 years. By the end of Sec XX, this problem was solved using computational tools but until today there is no algebraic proof of it. In this article, the original problem of coloring MAPS over a Spherical Surface is briefly reviewed. A Spherical MAP is converted into a Planar MAP using polar coordinates and the frontiers of the Countries are described as real implicit equations and then deployed from the real space into the complex space. In the complex space the rules to color MAPs are described as system of algebraic equations and inequations. One example of MAP is solved (colored) and the explanation about why these systems are solvable is done. Beginning from the example, a general theory to coloring MAPs is derived. As all the transformations used admits inverse, the obtained planar MAP solution can be reversed as a solution to the Spherical MAP. All operations involve simple algebraic transformations and some Calculus concepts |
publishDate |
2021 |
dc.date.none.fl_str_mv |
2021-11-29 |
dc.type.driver.fl_str_mv |
info:eu-repo/semantics/preprint info:eu-repo/semantics/publishedVersion |
format |
preprint |
status_str |
publishedVersion |
dc.identifier.uri.fl_str_mv |
https://preprints.scielo.org/index.php/scielo/preprint/view/3156 10.1590/SciELOPreprints.3156 |
url |
https://preprints.scielo.org/index.php/scielo/preprint/view/3156 |
identifier_str_mv |
10.1590/SciELOPreprints.3156 |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
https://preprints.scielo.org/index.php/scielo/article/view/3156/5696 |
dc.rights.driver.fl_str_mv |
Copyright (c) 2021 José Jansen https://creativecommons.org/licenses/by/4.0 info:eu-repo/semantics/openAccess |
rights_invalid_str_mv |
Copyright (c) 2021 José Jansen https://creativecommons.org/licenses/by/4.0 |
eu_rights_str_mv |
openAccess |
dc.format.none.fl_str_mv |
application/pdf |
dc.publisher.none.fl_str_mv |
SciELO Preprints SciELO Preprints SciELO Preprints |
publisher.none.fl_str_mv |
SciELO Preprints SciELO Preprints SciELO Preprints |
dc.source.none.fl_str_mv |
reponame:SciELO Preprints instname:SciELO instacron:SCI |
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SciELO |
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SCI |
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SCI |
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SciELO Preprints |
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SciELO Preprints |
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SciELO Preprints - SciELO |
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