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[{"key": "dc.contributor.advisor", "value": "Rajala, Kai", "language": "", "element": "contributor", "qualifier": "advisor", "schema": "dc"}, {"key": "dc.contributor.author", "value": "Kannisto, Erika", "language": "", "element": "contributor", "qualifier": "author", "schema": "dc"}, {"key": "dc.date.accessioned", "value": "2024-06-26T10:40:14Z", "language": null, "element": "date", "qualifier": "accessioned", "schema": "dc"}, {"key": "dc.date.available", "value": "2024-06-26T10:40:14Z", "language": null, "element": "date", "qualifier": "available", "schema": "dc"}, {"key": "dc.date.issued", "value": "2024", "language": "", "element": "date", "qualifier": "issued", "schema": "dc"}, {"key": "dc.identifier.uri", "value": "https://jyx.jyu.fi/handle/123456789/96167", "language": null, "element": "identifier", "qualifier": "uri", "schema": "dc"}, {"key": "dc.description.abstract", "value": "T\u00e4m\u00e4n tutkielman tarkoituksena on tarkastella kolmiulotteisen avaruuden isometrioita eli kuvauksia, jotka s\u00e4ilytt\u00e4v\u00e4t pisteparien et\u00e4isyydet, sek\u00e4 tetraedrin symmetriaryhm\u00e4\u00e4. Symmetriaryhm\u00e4 tarkoittaa kuvauksia, jotka kuvaavat tetraedrin takaisin itsekseen. Kolmiulotteisen avaruuden isometriat ovat samat kuin tason isometriat: siirto, kierto, peilaus ja siirtopeilaus. N\u00e4m\u00e4 isometriat muodostavat sek\u00e4 tasossa ett\u00e4 kolmiulotteisessa avaruudessa ryhm\u00e4n. Koska tavoitteena on tarkastella tetraedrin symmetriaryhm\u00e4\u00e4, halutaan valita ne isometriat, jotka kuvaavat tetraedrin itsekseen. T\u00e4t\u00e4 varten ainakin origon pit\u00e4\u00e4 pysy\u00e4 paikoillaan, joten siirrot ja siirtopeilaukset ovat poissa laskuista. Kuitenkaan kaikki peilaukset tai kierrotkaan eiv\u00e4t pid\u00e4 origoa paikoillaan; vain ne joiden akselit kulkevat origon l\u00e4pi. Kutsutaan n\u00e4it\u00e4 isometrioita pallogeometrian isometrioiksi. Pallogeometriassa liikutaan yksikk\u00f6pallon eli pallon, jonka s\u00e4de on yksi pinnalla. Peilaukset pallogeometriassa voidaan ajatella siten, ett\u00e4 peilausakselina on isoympyr\u00e4. Isoympyr\u00e4 on tason, jonka kulkee origon l\u00e4pi sek\u00e4 pallokuoren leikkaus. Peilauksiin liittyy kiinte\u00e4sti my\u00f6s kolmen peilauksen lause, jonka mukaan jokainen pallogeometrian isometria voidaan esitt\u00e4\u00e4 yhden, kahden tai kolmen peilauksen yhdisteen\u00e4. Kierrot ovat kahden peilauksen yhdisteit\u00e4. Yleisimpien kiertojen lausekkeet voidaan p\u00e4\u00e4tell\u00e4 tason kiertojen esitysten avulla, mutta se ei ole tehokas tai tarkka tapa. T\u00e4ll\u00f6in otetaan k\u00e4ytt\u00f6\u00f6n kvaterniot eli kompleksikertoimiset kaksi kertaa kaksi -matriisit, joiden avulla kierrot saadaan esitetty\u00e4. Kvaternioiden my\u00f6t\u00e4 p\u00e4\u00e4st\u00e4\u00e4n tutustumaan kompleksilaskentaan, kun kvaterniot m\u00e4\u00e4ritell\u00e4\u00e4n sek\u00e4 neliulotteiseen ett\u00e4 kolmiulotteiseen avaruuteen ja osoitetaan, ett\u00e4 kvaterniot muodostavat jakorenkaan. N\u00e4m\u00e4 tiedot yhdist\u00e4m\u00e4ll\u00e4 l\u00f6ydet\u00e4\u00e4n kuvaus, joka kuvaa pallogeometrian ja yleisemmin kolmiulotteisen avaruuden kierrot. Tutkielman lopuksi tarkastellaan tetraedrin symmetriaryhm\u00e4n kiertoja ja miten ne esitet\u00e4\u00e4n kvaternioiden avulla. Kiertoja on kolmenlaisia ja yhteens\u00e4 12. Ensinn\u00e4kin on niin sanottu nollakierto eli tetraedri, jota ei ole kierretty. T\u00e4t\u00e4 kutsutaan neutraalialkioksi. Toinen kiertotyyppi on niin sanotut puolikierrot, joissa kierret\u00e4\u00e4n 180 asteen verran ja n\u00e4it\u00e4 kiertoja on kolme. Viimeinen kiertotyyppi on niin sanotut kolmasosakierrot, joissa kierret\u00e4\u00e4n 120 asteen verran ja n\u00e4it\u00e4 on yhteens\u00e4 kahdeksan.", "language": "fi", "element": "description", "qualifier": "abstract", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Submitted by Miia Hakanen (mihakane@jyu.fi) on 2024-06-26T10:40:14Z\nNo. of bitstreams: 0", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.description.provenance", "value": "Made available in DSpace on 2024-06-26T10:40:14Z (GMT). No. of bitstreams: 0\n Previous issue date: 2024", "language": "en", "element": "description", "qualifier": "provenance", "schema": "dc"}, {"key": "dc.format.extent", "value": "39", "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.subject.other", "value": "isometriat", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.subject.other", "value": "kiertoryhm\u00e4", "language": "", "element": "subject", "qualifier": "other", "schema": "dc"}, {"key": "dc.title", "value": "Kolmiulotteiset isometriat", "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-202406265011", "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": "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": "Jyv\u00e4skyl\u00e4n yliopisto", "language": "fi", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.contributor.organization", "value": "University of Jyv\u00e4skyl\u00e4", "language": "en", "element": "contributor", "qualifier": "organization", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Matematiikan opettajankoulutus", "language": "fi", "element": "subject", "qualifier": "discipline", "schema": "dc"}, {"key": "dc.subject.discipline", "value": "Teacher education programme in Mathematics", "language": "en", "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": "", "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": "kompleksiluvut", "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": "ryhm\u00e4teoria", "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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