{"id":{"repo_id":"uic","oai_identifier":"oai:figshare.com:article/31451416"},"canonical_url":"https://search.dev.ndltd.org/etd/uic/oai:figshare.com:article/31451416","repository":{"repo_id":"uic","name":"University of Illinois - Chicago","base_url":"https://api.figshare.com/v2/oai"},"display":{"title":"Higher Scissors Congruence Groups of the Euclidean Plane","abstract":"Given a polygon in the Euclidean plane, consider the class of all polygons that are scissors congruent to the given polygon – that is, the class of all polygons which may be cut up finitely many times and then have their pieces rearranged into the given polygon. The problem of classical scissors congruence is to understand the different types of polygons (or polytopes, as they are called in higher dimensions) that exist up to this equivalence relation, in any dimension. Complementarily, higher scissors congruence groups capture information about the ways in which different polytopes are indeed scissors congruent – how to count and describe cutting and pasting equivalent polytopes together. No one has been able to compute these groups before: the only existing calculations were in one-dimensional geometry. Moreover, computations of higher scissors congruence groups are examples of computations in algebraic K-theory, which are known to be exceedingly difficult. In the Euclidean plane, I obtain a complete answer for the computation of an approximation of all higher scissors congruence groups, where that approximation is defined by permitting rational rotations exclusively. In the general case, where I permit all rotations, I provide infinite families of classes in the higher scissors congruence groups for degree 2 and above. In fact, I prove that these higher scissors congruence groups are uncountable. Applying homological methods here will shed light on a conjecture that says a homological tool called a trace map detects all higher scissors congruence groups. To prove or disprove this conjecture would be a great achievement for understanding the interface between homological algebra and scissors congruence computations.","abstract_html":"Given a polygon in the Euclidean plane, consider the class of all polygons that are scissors congruent to the given polygon – that is, the class of all polygons which may be cut up finitely many times and then have their pieces rearranged into the given polygon. The problem of classical scissors congruence is to understand the different types of polygons (or polytopes, as they are called in higher dimensions) that exist up to this equivalence relation, in any dimension. Complementarily, higher scissors congruence groups capture information about the ways in which different polytopes are indeed scissors congruent – how to count and describe cutting and pasting equivalent polytopes together. No one has been able to compute these groups before: the only existing calculations were in one-dimensional geometry. Moreover, computations of higher scissors congruence groups are examples of computations in algebraic K-theory, which are known to be exceedingly difficult. In the Euclidean plane, I obtain a complete answer for the computation of an approximation of all higher scissors congruence groups, where that approximation is defined by permitting rational rotations exclusively. In the general case, where I permit all rotations, I provide infinite families of classes in the higher scissors congruence groups for degree 2 and above. In fact, I prove that these higher scissors congruence groups are uncountable. Applying homological methods here will shed light on a conjecture that says a homological tool called a trace map detects all higher scissors congruence groups. To prove or disprove this conjecture would be a great achievement for understanding the interface between homological algebra and scissors congruence computations.","abstract_has_math":false,"creators":["Lydia Holley (23291653)"],"institution":null,"degree_name":null,"degree_level":null,"degree_discipline":null,"degree_department":null,"school":null,"contributors":[],"advisors":[],"committee_chairs":[],"committee_members":[],"year":2025,"date_issued":"2025-12-01T00:00:00Z","date_published":"2025-12-01T00:00:00Z","updated_at":"2026-07-27T21:34:25Z","subjects":["Algebraic Topology","K-Theory"],"languages":[],"rights":["In Copyright"],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://doi.org/10.25417/uic.31451416.v1","outbound_label":"DOI","outbound_source":"dc:identifier"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:creator","label":"Author","values":["Lydia Holley (23291653)"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date","label":"Dc Date","values":["2025-12-01T00:00:00Z"]},{"key":"dc:relation","label":"Dc Relation","values":["https://figshare.com/articles/thesis/Higher_Scissors_Congruence_Groups_of_the_Euclidean_Plane/31451416"]},{"key":"dc:type","label":"Dc Type","values":["Text","Thesis"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["Algebraic Topology","K-Theory"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:rights","label":"Dc Rights","values":["In Copyright"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier","label":"Identifier","values":["10.25417/uic.31451416.v1"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description","label":"Description","values":["Given a polygon in the Euclidean plane, consider the class of all polygons that are scissors congruent to the given polygon – that is, the class of all polygons which may be cut up finitely many times and then have their pieces rearranged into the given polygon. The problem of classical scissors congruence is to understand the different types of polygons (or polytopes, as they are called in higher dimensions) that exist up to this equivalence relation, in any dimension. Complementarily, higher scissors congruence groups capture information about the ways in which different polytopes are indeed scissors congruent – how to count and describe cutting and pasting equivalent polytopes together. No one has been able to compute these groups before: the only existing calculations were in one-dimensional geometry. Moreover, computations of higher scissors congruence groups are examples of computations in algebraic K-theory, which are known to be exceedingly difficult. In the Euclidean plane, I obtain a complete answer for the computation of an approximation of all higher scissors congruence groups, where that approximation is defined by permitting rational rotations exclusively. In the general case, where I permit all rotations, I provide infinite families of classes in the higher scissors congruence groups for degree 2 and above. In fact, I prove that these higher scissors congruence groups are uncountable. Applying homological methods here will shed light on a conjecture that says a homological tool called a trace map detects all higher scissors congruence groups. To prove or disprove this conjecture would be a great achievement for understanding the interface between homological algebra and scissors congruence computations."]},{"key":"dc:title","label":"Title","values":["Higher Scissors Congruence Groups of the Euclidean Plane"]}]}],"canonical_facts":{"dc:creator":["Lydia Holley (23291653)"],"dc:date":["2025-12-01T00:00:00Z"],"dc:description":["Given a polygon in the Euclidean plane, consider the class of all polygons that are scissors congruent to the given polygon – that is, the class of all polygons which may be cut up finitely many times and then have their pieces rearranged into the given polygon. The problem of classical scissors congruence is to understand the different types of polygons (or polytopes, as they are called in higher dimensions) that exist up to this equivalence relation, in any dimension. Complementarily, higher scissors congruence groups capture information about the ways in which different polytopes are indeed scissors congruent – how to count and describe cutting and pasting equivalent polytopes together. No one has been able to compute these groups before: the only existing calculations were in one-dimensional geometry. Moreover, computations of higher scissors congruence groups are examples of computations in algebraic K-theory, which are known to be exceedingly difficult. In the Euclidean plane, I obtain a complete answer for the computation of an approximation of all higher scissors congruence groups, where that approximation is defined by permitting rational rotations exclusively. In the general case, where I permit all rotations, I provide infinite families of classes in the higher scissors congruence groups for degree 2 and above. In fact, I prove that these higher scissors congruence groups are uncountable. Applying homological methods here will shed light on a conjecture that says a homological tool called a trace map detects all higher scissors congruence groups. To prove or disprove this conjecture would be a great achievement for understanding the interface between homological algebra and scissors congruence computations."],"dc:identifier":["10.25417/uic.31451416.v1"],"dc:relation":["https://figshare.com/articles/thesis/Higher_Scissors_Congruence_Groups_of_the_Euclidean_Plane/31451416"],"dc:rights":["In Copyright"],"dc:subject":["Algebraic Topology","K-Theory"],"dc:title":["Higher Scissors Congruence Groups of the Euclidean Plane"],"dc:type":["Text","Thesis"]},"updated_at":"2026-07-27T21:34:25Z"}