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[{"key": "dc.contributor.author", "value": "Joutsen, Elina", "language": null, "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2018-03-16T18:48:49Z", "language": "", "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2018-03-16T18:48:49Z", "language": "", "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2018", "language": null, "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.other", "value": "oai:jykdok.linneanet.fi:1862217", "language": null, "element": "identifier", "qualifier": "other", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/57340", "language": "", "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "Eukleides Aleksandrialainen oli kreikkalainen matemaatikko, joka loi noin 300 eaa.\r\neuklidisen geometrian. H\u00e4n julkaisi euklidisen geometrian perustana olevat aksioomat\r\nja perusolettamukset teoksessaan Alkeet. Eukleideen teos on s\u00e4ilynyt koulujen geometrian\r\nopetuksen pohjana jopa 1800\u2013luvulle asti. P\u00e4\u00e4l\u00e4hteen\u00e4 tutkielmassa on k\u00e4ytetty\r\nEukleideen teoksen Pekka Aschanin suomennosta ja sen nykysuomennosta kommentteineen,\r\njonka on toimittanut Lauri Kahanp\u00e4\u00e4 teoksessa Alkeet, Kuusi ensimm\u00e4ist\u00e4\r\nkirjaa eli tasogeometria.\r\nTutkielma tarkastelee Eukleideen muodostamaa teoriaa. Tavoitteena on ratkaista\r\nnelj\u00e4 vaativaa ympyr\u00e4n ja kolmion v\u00e4list\u00e4 ongelmaa Eukleideen teorian pohjalta.\r\nEukleideen aksioomaj\u00e4rjestelm\u00e4 perustuu viiteen aksioomaan, joiden perusteella geometria\r\npyrit\u00e4\u00e4n m\u00e4\u00e4rittelem\u00e4\u00e4n t\u00e4ydellisesti. Alussa esitell\u00e4\u00e4n aksioomaj\u00e4rjestelm\u00e4n\r\nkannalta t\u00e4rke\u00e4t yleiset k\u00e4sitteet, mink\u00e4 j\u00e4lkeen kerrotaan lyhyesti aksioomaj\u00e4rjestelm\u00e4st\u00e4\r\nja sen vaatimuksista sek\u00e4 esitell\u00e4\u00e4n Eukleideen viisi aksioomaa.\r\nTutkielman t\u00e4rkein teema on tarvittavan euklidisen teorian kokoaminen geometristen\r\nongelmien ratkaisemiseksi. Tutkielman seuraavassa vaiheessa tarkastellaan kolmioiden\r\nja ympyr\u00f6iden geometrisia ominaisuuksia. Lis\u00e4ksi esitell\u00e4\u00e4n kyseisten ongelmien\r\nratkaisemisen kannalta tarpeellisia k\u00e4yt\u00e4nn\u00f6n esimerkkej\u00e4, jotka perustuvat\r\nharppi\u2013viivain konstruktioihin. Teorian pohjalta ratkaistaan n\u00e4am\u00e4 nelj\u00e4 ongelmaa:\r\nannetun ympyr\u00e4n sis\u00e4\u00e4n on piirrett\u00e4v\u00e4 kolmio, annetun ympyr\u00e4n ymp\u00e4ri on piirrett\u00e4v\u00e4\r\nkolmio, annetun kolmion sis\u00e4\u00e4n on piirrett\u00e4v\u00e4 ympyr\u00e4 sek\u00e4 annetun kolmion\r\nymp\u00e4ri on piirrett\u00e4v\u00e4 ympyr\u00e4.\r\nTarkastelun lopuksi esitell\u00e4\u00e4n Eukleideen aksioomaj\u00e4rjestelm\u00e4\u00e4 nykyaikaisempi\r\nHilbertin aksioomaj\u00e4rjestelm\u00e4 euklidiselle geometrialle. David Hilbertin aksioomaj\u00a8arjestelm\u00a8a\r\njulkaistiin vuonna 1899 ja se on huomattavasti laajempi ja tarkempi kuin\r\nEukleideen aksioomaj\u00e4rjestelm\u00e4. Lopuksi verrataan Eukleideen aksioomaj\u00e4rjestelm\u00e4\u00e4\r\nHilbertin aksioomaj\u00e4rjestelm\u00e4\u00e4n ja erityisesti tarkastellaan Eukleideen viidett\u00e4 aksioomaa.\r\nVertailun tarkoituksena on havainnollistaa Eukleideen teorian mahdollisia\r\nongelmakohtia.", "language": "", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted using Plone Publishing form by Elina Joutsen (elsojout) on 2018-03-16 18:48:48.682556. Form: Pro gradu -lomake (https://kirjasto.jyu.fi/julkaisut/julkaisulomakkeet/pro-gradu-lomake). JyX data: [jyx_publishing-allowed (fi) =True]", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by jyx lomake-julkaisija (jyx-julkaisija.group@korppi.jyu.fi) on 2018-03-16T18:48:49Z\r\nNo. of bitstreams: 2\r\nURN:NBN:fi:jyu-201803161748.pdf: 476954 bytes, checksum: d91b3e4d13c31c29016bc8e0c3ec7424 (MD5)\r\nlicense.html: 4777 bytes, checksum: ebe5c23c29b64b4db46935711c18c7a9 (MD5)", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2018-03-16T18:48:49Z (GMT). No. of bitstreams: 2\r\nURN:NBN:fi:jyu-201803161748.pdf: 476954 bytes, checksum: d91b3e4d13c31c29016bc8e0c3ec7424 (MD5)\r\nlicense.html: 4777 bytes, checksum: ebe5c23c29b64b4db46935711c18c7a9 (MD5)\r\n Previous issue date: 2018", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "1 verkkoaineisto (51 sivua)", "language": null, "element": "format", "qualifier": "extent", "schema": "dc"}, {"key": "dc.format.mimetype", "value": "application/pdf", "language": null, "element": "format", "qualifier": "mimetype", "schema": "dc"}, {"key": "dc.language.iso", "value": "fin", "language": null, "element": "language", "qualifier": "iso", "schema": "dc"}, {"key": "dc.rights", "value": "In Copyright", "language": "en", "element": "rights", "qualifier": null, "schema": "dc"}, {"key": "dc.title", "value": "Eukleideen geometriaa", "language": null, "element": "title", "qualifier": null, "schema": "dc"}, {"key": "dc.type", "value": "master thesis", "language": null, "element": "type", "qualifier": null, "schema": "dc"}, {"key": "dc.identifier.urn", "value": "URN:NBN:fi:jyu-201803161748", "language": null, "element": "identifier", "qualifier": "urn", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Pro gradu -tutkielma", "language": "fi", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Master\u2019s thesis", "language": "en", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Matemaattis-luonnontieteellinen tiedekunta", "language": "fi", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Faculty of Sciences", "language": "en", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Matematiikan ja tilastotieteen laitos", "language": "fi", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Department of Mathematics and Statistics", "language": "en", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "University of Jyv\u00e4skyl\u00e4", "language": "en", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "Jyv\u00e4skyl\u00e4n yliopisto", "language": "fi", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Matematiikka", "language": "fi", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Mathematics", "language": "en", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.date.updated", "value": "2018-03-16T18:48:49Z", "language": "", "element": "date", "qualifier": "updated", "schema": "dc"}, {"key": "yvv.contractresearch.funding", "value": "0", "language": "", "element": "contractresearch", "qualifier": "funding", "schema": "yvv"}, {"key": "dc.type.coar", "value": "http://purl.org/coar/resource_type/c_bdcc", "language": null, "element": "type", "qualifier": "coar", "schema": "dc"}, {"key": "dc.rights.accesslevel", "value": "openAccess", "language": "fi", "element": "rights", "qualifier": "accesslevel", "schema": "dc"}, {"key": "dc.type.publication", "value": "masterThesis", "language": null, "element": "type", "qualifier": "publication", "schema": "dc"}, {"key": "dc.subject.oppiainekoodi", "value": "4041", "language": null, "element": "subject", "qualifier": "oppiainekoodi", "schema": "dc"}, {"key": "dc.subject.yso", "value": "geometria", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "euklidinen geometria", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "aksioomat", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.format.content", "value": "fulltext", "language": null, "element": "format", "qualifier": "content", "schema": "dc"}, {"key": "dc.rights.url", "value": "https://rightsstatements.org/page/InC/1.0/", "language": null, "element": "rights", "qualifier": "url", "schema": "dc"}, {"key": "dc.type.okm", "value": "G2", "language": null, "element": "type", "qualifier": "okm", "schema": "dc"}]
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