{"id":{"repo_id":"unr","oai_identifier":"oai:scholarwolf.unr.edu:11714/3718"},"canonical_url":"https://search.dev.ndltd.org/etd/unr/oai:scholarwolf.unr.edu:11714/3718","repository":{"repo_id":"unr","name":"University of Nevada - Reno","base_url":"https://scholarwolf.unr.edu/server/oai/request"},"display":{"title":"Geometric, Algebraic, and Topological Connections in the Historical Sphere of the Platonic Solids","abstract":"The Platonic solids have made prominent appearances in the history of pure mathematics: in Euclid's <italic>Elements</italic>, which contains early geometric constructions of the solids and demonstrates the special manner in which each is comprehended by a sphere; in William Hamilton's geometric interpretation of the icosians, a non-abelian group that describes certain \"passages\" between faces of the Platonic solids; and in Henri Poincar&eacute's <italic>Analysis</italic> <italic>situs</italic>, in which 0- and 1-dimensional homology are determined. Incidentally, Poincar&eacute's non-polyhedral construction of a homology 3-sphere with non-trivial fundamental group revealed the same icosahedral relations that Hamilton had invented. Topologists later realized that this important construction could be obtained as a polyhedral manifold using the dodecahedron. On the basis of these historical observations, a salient pattern emerges: The recurring uses of the Platonic solids in key episodes of mathematical development that have connected Euclidean geometry, non-commutative algebra and topology in important ways.","abstract_html":"The Platonic solids have made prominent appearances in the history of pure mathematics: in Euclid&#x27;s &lt;italic&gt;Elements&lt;/italic&gt;, which contains early geometric constructions of the solids and demonstrates the special manner in which each is comprehended by a sphere; in William Hamilton&#x27;s geometric interpretation of the icosians, a non-abelian group that describes certain &quot;passages&quot; between faces of the Platonic solids; and in Henri Poincar&amp;eacute&#x27;s &lt;italic&gt;Analysis&lt;/italic&gt; &lt;italic&gt;situs&lt;/italic&gt;, in which 0- and 1-dimensional homology are determined. Incidentally, Poincar&amp;eacute&#x27;s non-polyhedral construction of a homology 3-sphere with non-trivial fundamental group revealed the same icosahedral relations that Hamilton had invented. Topologists later realized that this important construction could be obtained as a polyhedral manifold using the dodecahedron. On the basis of these historical observations, a salient pattern emerges: The recurring uses of the Platonic solids in key episodes of mathematical development that have connected Euclidean geometry, non-commutative algebra and topology in important ways.","abstract_has_math":false,"creators":["Smith, James Adam"],"institution":null,"degree_name":null,"degree_level":"Master's Degree","degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":["Kumjian, Alexander"],"committee_chairs":[],"committee_members":["Herald, Christopher","Jabuka, Stanislav","Moran, Bruce","Nickles, Tom"],"year":2012,"date_issued":"2012","date_published":"2012","updated_at":"2026-07-27T21:45:45Z","subjects":["History of mathematics","Platonic solids"],"languages":[],"rights":["In Copyright(All Rights Reserved)"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"http://hdl.handle.net/11714/3718","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Kumjian, Alexander"]},{"key":"dc:contributor.committeemember","label":"Committee Member","values":["Herald, Christopher","Jabuka, Stanislav","Moran, Bruce","Nickles, Tom"]},{"key":"dc:creator","label":"Author","values":["Smith, James Adam"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2018-07-26T18:29:57Z"]},{"key":"dc:date.available","label":"Dc Date Available","values":["2018-07-26T18:29:57Z"]},{"key":"dc:date.issued","label":"Date","values":["2012"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"thesis:degree_level","label":"Degree Level","values":["Master's Degree"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["History of mathematics","Platonic solids"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:rights","label":"Dc Rights","values":["In Copyright(All Rights Reserved)"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["http://hdl.handle.net/11714/3718"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["The Platonic solids have made prominent appearances in the history of pure mathematics: in Euclid's <italic>Elements</italic>, which contains early geometric constructions of the solids and demonstrates the special manner in which each is comprehended by a sphere; in William Hamilton's geometric interpretation of the icosians, a non-abelian group that describes certain \"passages\" between faces of the Platonic solids; and in Henri Poincar&eacute's <italic>Analysis</italic> <italic>situs</italic>, in which 0- and 1-dimensional homology are determined. Incidentally, Poincar&eacute's non-polyhedral construction of a homology 3-sphere with non-trivial fundamental group revealed the same icosahedral relations that Hamilton had invented. Topologists later realized that this important construction could be obtained as a polyhedral manifold using the dodecahedron. On the basis of these historical observations, a salient pattern emerges: The recurring uses of the Platonic solids in key episodes of mathematical development that have connected Euclidean geometry, non-commutative algebra and topology in important ways."]},{"key":"dc:format","label":"Dc Format","values":["PDF"]},{"key":"dc:title","label":"Title","values":["Geometric, Algebraic, and Topological Connections in the Historical Sphere of the Platonic Solids"]}]}],"canonical_facts":{"dc:contributor.advisor":["Kumjian, Alexander"],"dc:contributor.committeemember":["Herald, Christopher","Jabuka, Stanislav","Moran, Bruce","Nickles, Tom"],"dc:creator":["Smith, James Adam"],"dc:date.accessioned":["2018-07-26T18:29:57Z"],"dc:date.available":["2018-07-26T18:29:57Z"],"dc:date.issued":["2012"],"dc:description.abstract":["The Platonic solids have made prominent appearances in the history of pure mathematics: in Euclid's <italic>Elements</italic>, which contains early geometric constructions of the solids and demonstrates the special manner in which each is comprehended by a sphere; in William Hamilton's geometric interpretation of the icosians, a non-abelian group that describes certain \"passages\" between faces of the Platonic solids; and in Henri Poincar&eacute's <italic>Analysis</italic> <italic>situs</italic>, in which 0- and 1-dimensional homology are determined. Incidentally, Poincar&eacute's non-polyhedral construction of a homology 3-sphere with non-trivial fundamental group revealed the same icosahedral relations that Hamilton had invented. Topologists later realized that this important construction could be obtained as a polyhedral manifold using the dodecahedron. On the basis of these historical observations, a salient pattern emerges: The recurring uses of the Platonic solids in key episodes of mathematical development that have connected Euclidean geometry, non-commutative algebra and topology in important ways."],"dc:format":["PDF"],"dc:identifier.uri":["http://hdl.handle.net/11714/3718"],"dc:rights":["In Copyright(All Rights Reserved)"],"dc:subject":["History of mathematics","Platonic solids"],"dc:title":["Geometric, Algebraic, and Topological Connections in the Historical Sphere of the Platonic Solids"],"dc:type":["Thesis"],"thesis:degree_level":["Master's Degree"]},"updated_at":"2026-07-27T21:45:45Z"}