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[{"key": "dc.contributor.advisor", "value": "Lehrb\u00e4ck, Juha", "language": null, "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Ruokaism\u00e4ki, Joni", "language": "", "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2024-06-11T05:16:20Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2024-06-11T05:16:20Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2024", "language": null, "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/95724", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4m\u00e4n pro gradu -tutkielman tarkoituksena on tutustua yleisesti hyperbolisen geometrian historiaan ja siihen, miten hyperbolisesta geometriasta tuli tunnustettu osa nykyp\u00e4iv\u00e4n matematiikkaa. Tutkielma keskittyy paralleeliaksioomasta syntyneisiin todistuksiin ja niist\u00e4 seuranneisiin tuloksiin, joista huomattiin, ett\u00e4 euklidisen geometrian lis\u00e4ksi saattaa olla olemassa toisenlaista geometriaa.\n\nTutkielman alkupuolella k\u00e4ymme l\u00e4pi paralleeliaksiooman varhaisia todistusyrityksi\u00e4 Ptolemaioksen, Prokloksen sek\u00e4 John Wallisin toimesta. T\u00e4t\u00e4 ty\u00f6t\u00e4 jatkoi Girolamo Saccheri, joka teki siihen asti parhaan yrityksen paralleeliaksiooman todistamiseksi. Saccherin ty\u00f6 perustuu nelikulmioon, joka tunnetaan nyky\u00e4\u00e4n Saccherin nelikulmiona. \n\nK\u00e4sittelemme Saccherin ty\u00f6t\u00e4 syv\u00e4llisemmin kuin muita, sill\u00e4 Saccherin tekem\u00e4t hypoteesit Saccherin nelikulmion huippukulmista toimivat pohjana usean eri geometrikon t\u00f6ille ja siit\u00e4 jatkoivat esimerkiksi Johann Heinrich Lambert sek\u00e4 Adrien-Marie Legendre. Heid\u00e4n ty\u00f6ns\u00e4 eiv\u00e4t kuitenkaan tuoneet pituuden absoluuttisen mitan k\u00e4sitteen lis\u00e4ksi mit\u00e4\u00e4n, mit\u00e4 edelt\u00e4v\u00e4t geometrikot eiv\u00e4t olisi l\u00f6yt\u00e4neet. Legendren tyylik\u00e4s ja yksinkertainen tapa l\u00e4hesty\u00e4 ongelmia kuitenkin lis\u00e4si yhdensuuntaisten suorien teorian tutkijoiden m\u00e4\u00e4r\u00e4\u00e4.\n\nTutkielman lopussa tutustumme Carl Friedrich Gaussin, Nikolai Lo\\-ba\\-t\u0161evs\\-kin sek\u00e4 J\u00e1nos Bolyain t\u00f6ihin. Heit\u00e4 pidet\u00e4\u00e4n yleisesti ep\u00e4euklidisen geometrian l\u00f6yt\u00e4jin\u00e4 ja pystymme n\u00e4kem\u00e4\u00e4n Gaussin, Lobat\u0161evskin ja Bolyain t\u00f6iss\u00e4 yhteyden Lambertin, Legendren sek\u00e4 Saccherin t\u00f6iden kanssa. Edell\u00e4 mainittujen matemaatikoiden k\u00e4sittely on kuvailevampaa, sill\u00e4 niiss\u00e4 esiintyv\u00e4t yksityiskohdat vaatisivat syv\u00e4llisempi\u00e4 pohjatietoja, joten niiden k\u00e4sittely ohitetaan. Tutkielma p\u00e4\u00e4ttyy hyperbolisen geometrian levi\u00e4misen tarkasteluun sen l\u00f6ytymisen j\u00e4lkeen.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Miia Hakanen (mihakane@jyu.fi) on 2024-06-11T05:16:20Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2024-06-11T05:16:20Z (GMT). No. of bitstreams: 0\n Previous issue date: 2024", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "76", "language": "", "element": "format", "qualifier": "extent", "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": "Hyperbolisen geometrian historiaa", "language": "", "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-202406114494", "language": null, "element": "identifier", "qualifier": "urn", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Master\u2019s thesis", "language": "en", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.type.ontasot", "value": "Pro gradu -tutkielma", "language": "fi", "element": "type", "qualifier": "ontasot", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Faculty of Sciences", "language": "en", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.faculty", "value": "Matemaattis-luonnontieteellinen tiedekunta", "language": "fi", "element": "contributor", "qualifier": "faculty", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Department of Mathematics and Statistics", "language": "en", "element": "contributor", "qualifier": "department", "schema": "dc"}, {"key": "dc.contributor.department", "value": "Matematiikan ja tilastotieteen laitos", "language": "fi", "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": "Teacher education programme in Mathematics", "language": "en", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Matematiikan opettajankoulutus", "language": "fi", "element": "subject", "qualifier": "discipline", "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": null, "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": "matematiikka", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "geometria", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "historia", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.subject.yso", "value": "ep\u00e4euklidinen geometria", "language": null, "element": "subject", "qualifier": "yso", "schema": "dc"}, {"key": "dc.rights.url", "value": "https://rightsstatements.org/page/InC/1.0/", "language": null, "element": "rights", "qualifier": "url", "schema": "dc"}]
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