Using conics to construct geometric 3-configurations, part I: symmetrically generalizing the Pappus configuration
Abstract The $$(9_{3})$$ Pappus configuration can be realized as a geometric configuration with 3-fold rotational symmetry; using this 3-fold rotation, its Levi graph (the incidence graph for the configuration) can be quotiented by the automorphism corresponding to the 3-fold rotation to form an edg...
Ausführliche Beschreibung
Gespeichert in:
| Autor*in: |
Berman, Leah Wrenn [verfasserIn] |
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| Format: |
Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2016 |
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| Schlagwörter: |
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| Anmerkung: |
© Springer International Publishing 2016 |
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| Übergeordnetes Werk: |
Enthalten in: Journal of geometry - Springer International Publishing, 1971, 108(2016), 2 vom: 08. Dez., Seite 591-609 |
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| Übergeordnetes Werk: |
volume:108 ; year:2016 ; number:2 ; day:08 ; month:12 ; pages:591-609 |
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| DOI / URN: |
10.1007/s00022-016-0361-z |
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OLC2069428672 |
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| 520 | |a Abstract The $$(9_{3})$$ Pappus configuration can be realized as a geometric configuration with 3-fold rotational symmetry; using this 3-fold rotation, its Levi graph (the incidence graph for the configuration) can be quotiented by the automorphism corresponding to the 3-fold rotation to form an edge-labelled $$K_{3,3}$$. This graph quotient, a type of voltage graph, is called the reduced Levi graph of the configuration. In this paper, we generalize this rotational realization of the Pappus configuration by developing a construction technique for a family of geometric configurations with m-fold rotational symmetry for any $$m \ge 3$$ whose reduced Levi graphs are all labelled versions of $$K_{3,3}$$. The constructions use primarily straightedge-and-compass techniques, along with the construction of certain conic sections. | ||
| 650 | 4 | |a Geometric configuration | |
| 650 | 4 | |a Pappus configuration | |
| 650 | 4 | |a Reduced Levi graph | |
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10.1007/s00022-016-0361-z doi (DE-627)OLC2069428672 (DE-He213)s00022-016-0361-z-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Berman, Leah Wrenn verfasserin (orcid)0000-0003-0935-5724 aut Using conics to construct geometric 3-configurations, part I: symmetrically generalizing the Pappus configuration 2016 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer International Publishing 2016 Abstract The $$(9_{3})$$ Pappus configuration can be realized as a geometric configuration with 3-fold rotational symmetry; using this 3-fold rotation, its Levi graph (the incidence graph for the configuration) can be quotiented by the automorphism corresponding to the 3-fold rotation to form an edge-labelled $$K_{3,3}$$. This graph quotient, a type of voltage graph, is called the reduced Levi graph of the configuration. In this paper, we generalize this rotational realization of the Pappus configuration by developing a construction technique for a family of geometric configurations with m-fold rotational symmetry for any $$m \ge 3$$ whose reduced Levi graphs are all labelled versions of $$K_{3,3}$$. The constructions use primarily straightedge-and-compass techniques, along with the construction of certain conic sections. Geometric configuration Pappus configuration Reduced Levi graph Enthalten in Journal of geometry Springer International Publishing, 1971 108(2016), 2 vom: 08. Dez., Seite 591-609 (DE-627)129288993 (DE-600)120140-2 (DE-576)014470527 0047-2468 nnns volume:108 year:2016 number:2 day:08 month:12 pages:591-609 https://doi.org/10.1007/s00022-016-0361-z lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_22 GBV_ILN_24 GBV_ILN_30 GBV_ILN_70 GBV_ILN_2088 GBV_ILN_4277 AR 108 2016 2 08 12 591-609 |
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10.1007/s00022-016-0361-z doi (DE-627)OLC2069428672 (DE-He213)s00022-016-0361-z-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Berman, Leah Wrenn verfasserin (orcid)0000-0003-0935-5724 aut Using conics to construct geometric 3-configurations, part I: symmetrically generalizing the Pappus configuration 2016 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer International Publishing 2016 Abstract The $$(9_{3})$$ Pappus configuration can be realized as a geometric configuration with 3-fold rotational symmetry; using this 3-fold rotation, its Levi graph (the incidence graph for the configuration) can be quotiented by the automorphism corresponding to the 3-fold rotation to form an edge-labelled $$K_{3,3}$$. This graph quotient, a type of voltage graph, is called the reduced Levi graph of the configuration. In this paper, we generalize this rotational realization of the Pappus configuration by developing a construction technique for a family of geometric configurations with m-fold rotational symmetry for any $$m \ge 3$$ whose reduced Levi graphs are all labelled versions of $$K_{3,3}$$. The constructions use primarily straightedge-and-compass techniques, along with the construction of certain conic sections. Geometric configuration Pappus configuration Reduced Levi graph Enthalten in Journal of geometry Springer International Publishing, 1971 108(2016), 2 vom: 08. Dez., Seite 591-609 (DE-627)129288993 (DE-600)120140-2 (DE-576)014470527 0047-2468 nnns volume:108 year:2016 number:2 day:08 month:12 pages:591-609 https://doi.org/10.1007/s00022-016-0361-z lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_22 GBV_ILN_24 GBV_ILN_30 GBV_ILN_70 GBV_ILN_2088 GBV_ILN_4277 AR 108 2016 2 08 12 591-609 |
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10.1007/s00022-016-0361-z doi (DE-627)OLC2069428672 (DE-He213)s00022-016-0361-z-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Berman, Leah Wrenn verfasserin (orcid)0000-0003-0935-5724 aut Using conics to construct geometric 3-configurations, part I: symmetrically generalizing the Pappus configuration 2016 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer International Publishing 2016 Abstract The $$(9_{3})$$ Pappus configuration can be realized as a geometric configuration with 3-fold rotational symmetry; using this 3-fold rotation, its Levi graph (the incidence graph for the configuration) can be quotiented by the automorphism corresponding to the 3-fold rotation to form an edge-labelled $$K_{3,3}$$. This graph quotient, a type of voltage graph, is called the reduced Levi graph of the configuration. In this paper, we generalize this rotational realization of the Pappus configuration by developing a construction technique for a family of geometric configurations with m-fold rotational symmetry for any $$m \ge 3$$ whose reduced Levi graphs are all labelled versions of $$K_{3,3}$$. The constructions use primarily straightedge-and-compass techniques, along with the construction of certain conic sections. Geometric configuration Pappus configuration Reduced Levi graph Enthalten in Journal of geometry Springer International Publishing, 1971 108(2016), 2 vom: 08. Dez., Seite 591-609 (DE-627)129288993 (DE-600)120140-2 (DE-576)014470527 0047-2468 nnns volume:108 year:2016 number:2 day:08 month:12 pages:591-609 https://doi.org/10.1007/s00022-016-0361-z lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_22 GBV_ILN_24 GBV_ILN_30 GBV_ILN_70 GBV_ILN_2088 GBV_ILN_4277 AR 108 2016 2 08 12 591-609 |
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10.1007/s00022-016-0361-z doi (DE-627)OLC2069428672 (DE-He213)s00022-016-0361-z-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Berman, Leah Wrenn verfasserin (orcid)0000-0003-0935-5724 aut Using conics to construct geometric 3-configurations, part I: symmetrically generalizing the Pappus configuration 2016 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer International Publishing 2016 Abstract The $$(9_{3})$$ Pappus configuration can be realized as a geometric configuration with 3-fold rotational symmetry; using this 3-fold rotation, its Levi graph (the incidence graph for the configuration) can be quotiented by the automorphism corresponding to the 3-fold rotation to form an edge-labelled $$K_{3,3}$$. This graph quotient, a type of voltage graph, is called the reduced Levi graph of the configuration. In this paper, we generalize this rotational realization of the Pappus configuration by developing a construction technique for a family of geometric configurations with m-fold rotational symmetry for any $$m \ge 3$$ whose reduced Levi graphs are all labelled versions of $$K_{3,3}$$. The constructions use primarily straightedge-and-compass techniques, along with the construction of certain conic sections. Geometric configuration Pappus configuration Reduced Levi graph Enthalten in Journal of geometry Springer International Publishing, 1971 108(2016), 2 vom: 08. Dez., Seite 591-609 (DE-627)129288993 (DE-600)120140-2 (DE-576)014470527 0047-2468 nnns volume:108 year:2016 number:2 day:08 month:12 pages:591-609 https://doi.org/10.1007/s00022-016-0361-z lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_22 GBV_ILN_24 GBV_ILN_30 GBV_ILN_70 GBV_ILN_2088 GBV_ILN_4277 AR 108 2016 2 08 12 591-609 |
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10.1007/s00022-016-0361-z doi (DE-627)OLC2069428672 (DE-He213)s00022-016-0361-z-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Berman, Leah Wrenn verfasserin (orcid)0000-0003-0935-5724 aut Using conics to construct geometric 3-configurations, part I: symmetrically generalizing the Pappus configuration 2016 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer International Publishing 2016 Abstract The $$(9_{3})$$ Pappus configuration can be realized as a geometric configuration with 3-fold rotational symmetry; using this 3-fold rotation, its Levi graph (the incidence graph for the configuration) can be quotiented by the automorphism corresponding to the 3-fold rotation to form an edge-labelled $$K_{3,3}$$. This graph quotient, a type of voltage graph, is called the reduced Levi graph of the configuration. In this paper, we generalize this rotational realization of the Pappus configuration by developing a construction technique for a family of geometric configurations with m-fold rotational symmetry for any $$m \ge 3$$ whose reduced Levi graphs are all labelled versions of $$K_{3,3}$$. The constructions use primarily straightedge-and-compass techniques, along with the construction of certain conic sections. Geometric configuration Pappus configuration Reduced Levi graph Enthalten in Journal of geometry Springer International Publishing, 1971 108(2016), 2 vom: 08. Dez., Seite 591-609 (DE-627)129288993 (DE-600)120140-2 (DE-576)014470527 0047-2468 nnns volume:108 year:2016 number:2 day:08 month:12 pages:591-609 https://doi.org/10.1007/s00022-016-0361-z lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_22 GBV_ILN_24 GBV_ILN_30 GBV_ILN_70 GBV_ILN_2088 GBV_ILN_4277 AR 108 2016 2 08 12 591-609 |
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Using conics to construct geometric 3-configurations, part I: symmetrically generalizing the Pappus configuration |
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Abstract The $$(9_{3})$$ Pappus configuration can be realized as a geometric configuration with 3-fold rotational symmetry; using this 3-fold rotation, its Levi graph (the incidence graph for the configuration) can be quotiented by the automorphism corresponding to the 3-fold rotation to form an edge-labelled $$K_{3,3}$$. This graph quotient, a type of voltage graph, is called the reduced Levi graph of the configuration. In this paper, we generalize this rotational realization of the Pappus configuration by developing a construction technique for a family of geometric configurations with m-fold rotational symmetry for any $$m \ge 3$$ whose reduced Levi graphs are all labelled versions of $$K_{3,3}$$. The constructions use primarily straightedge-and-compass techniques, along with the construction of certain conic sections. © Springer International Publishing 2016 |
| abstractGer |
Abstract The $$(9_{3})$$ Pappus configuration can be realized as a geometric configuration with 3-fold rotational symmetry; using this 3-fold rotation, its Levi graph (the incidence graph for the configuration) can be quotiented by the automorphism corresponding to the 3-fold rotation to form an edge-labelled $$K_{3,3}$$. This graph quotient, a type of voltage graph, is called the reduced Levi graph of the configuration. In this paper, we generalize this rotational realization of the Pappus configuration by developing a construction technique for a family of geometric configurations with m-fold rotational symmetry for any $$m \ge 3$$ whose reduced Levi graphs are all labelled versions of $$K_{3,3}$$. The constructions use primarily straightedge-and-compass techniques, along with the construction of certain conic sections. © Springer International Publishing 2016 |
| abstract_unstemmed |
Abstract The $$(9_{3})$$ Pappus configuration can be realized as a geometric configuration with 3-fold rotational symmetry; using this 3-fold rotation, its Levi graph (the incidence graph for the configuration) can be quotiented by the automorphism corresponding to the 3-fold rotation to form an edge-labelled $$K_{3,3}$$. This graph quotient, a type of voltage graph, is called the reduced Levi graph of the configuration. In this paper, we generalize this rotational realization of the Pappus configuration by developing a construction technique for a family of geometric configurations with m-fold rotational symmetry for any $$m \ge 3$$ whose reduced Levi graphs are all labelled versions of $$K_{3,3}$$. The constructions use primarily straightedge-and-compass techniques, along with the construction of certain conic sections. © Springer International Publishing 2016 |
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Using conics to construct geometric 3-configurations, part I: symmetrically generalizing the Pappus configuration |
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https://doi.org/10.1007/s00022-016-0361-z |
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| doi_str |
10.1007/s00022-016-0361-z |
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2024-07-03T22:09:40.915Z |
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1803597466479624192 |
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This graph quotient, a type of voltage graph, is called the reduced Levi graph of the configuration. In this paper, we generalize this rotational realization of the Pappus configuration by developing a construction technique for a family of geometric configurations with m-fold rotational symmetry for any $$m \ge 3$$ whose reduced Levi graphs are all labelled versions of $$K_{3,3}$$. 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