{"id":{"repo_id":"unt","oai_identifier":"info:ark/67531/metadc3269"},"canonical_url":"https://search.dev.ndltd.org/etd/unt/info:ark/67531/metadc3269","repository":{"repo_id":"unt","name":"University of North Texas","base_url":"https://digital.library.unt.edu/oai/"},"display":{"title":"Understanding Ancient Math Through Kepler: A Few Geometric Ideas from The Harmony of the World","abstract":"Euclid's geometry is well-known for its theorems concerning triangles and circles. Less popular are the contents of the tenth book, in which geometry is a means to study quantity in general. Commensurability and rational quantities are first principles, and from them are derived at least eight species of irrationals. A recently republished work by Johannes Kepler contains examples using polygons to illustrate these species. In addition, figures having these quantities in their construction form solid shapes (polyhedra) having origins though Platonic philosophy and Archimedean works. Kepler gives two additional polyhedra, and a simple means for constructing the “divine” proportion is given.","abstract_html":"Euclid's geometry is well-known for its theorems concerning triangles and circles. Less popular are the contents of the tenth book, in which geometry is a means to study quantity in general. Commensurability and rational quantities are first principles, and from them are derived at least eight species of irrationals. A recently republished work by Johannes Kepler contains examples using polygons to illustrate these species. In addition, figures having these quantities in their construction form solid shapes (polyhedra) having origins though Platonic philosophy and Archimedean works. Kepler gives two additional polyhedra, and a simple means for constructing the “divine” proportion is given.","abstract_has_math":false,"creators":["Arthur, Christopher"],"institution":"University of North Texas","degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":["Anghel, Nicolae","Brozovic, Douglas","Allen, John Ed, 1937-"],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2002,"date_issued":"2002-08","date_published":"2002-08","updated_at":"2026-07-24T05:34:52Z","subjects":["Kepler, Johannes, 1571-1630. Harmonices mundi.","Geometry.","Book X","commensurability","La Grange","manifold","binary operations"],"languages":["English"],"rights":["Public","Copyright","Arthur, Christopher","Copyright is held by the author, unless otherwise noted. 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Commensurability and rational quantities are first principles, and from them are derived at least eight species of irrationals. A recently republished work by Johannes Kepler contains examples using polygons to illustrate these species. In addition, figures having these quantities in their construction form solid shapes (polyhedra) having origins though Platonic philosophy and Archimedean works. Kepler gives two additional polyhedra, and a simple means for constructing the “divine” proportion is given."]},{"key":"dc:format","label":"Dc Format","values":["Text"]},{"key":"dc:title","label":"Title","values":["Understanding Ancient Math Through Kepler: A Few Geometric Ideas from The Harmony of the World"]}]}],"canonical_facts":{"dc:contributor":["Anghel, Nicolae","Brozovic, Douglas","Allen, John Ed, 1937-"],"dc:creator":["Arthur, Christopher"],"dc:date":["2002-08"],"dc:description":["Euclid's geometry is well-known for its theorems concerning triangles and circles. Less popular are the contents of the tenth book, in which geometry is a means to study quantity in general. Commensurability and rational quantities are first principles, and from them are derived at least eight species of irrationals. A recently republished work by Johannes Kepler contains examples using polygons to illustrate these species. In addition, figures having these quantities in their construction form solid shapes (polyhedra) having origins though Platonic philosophy and Archimedean works. Kepler gives two additional polyhedra, and a simple means for constructing the “divine” proportion is given."],"dc:format":["Text"],"dc:identifier":["oclc: 50444717","doi: 10.12794/metadc3269","https://digital.library.unt.edu/ark:/67531/metadc3269/","ark: ark:/67531/metadc3269"],"dc:language":["English"],"dc:publisher":["University of North Texas"],"dc:rights":["Public","Copyright","Arthur, Christopher","Copyright is held by the author, unless otherwise noted. All rights reserved."],"dc:subject":["Kepler, Johannes, 1571-1630. Harmonices mundi.","Geometry.","Book X","commensurability","La Grange","manifold","binary operations"],"dc:title":["Understanding Ancient Math Through Kepler: A Few Geometric Ideas from The Harmony of the World"],"dc:type":["Thesis or Dissertation"]},"updated_at":"2026-07-24T05:34:52Z"}