Chiral Astral Realizations of Cyclic 3-Configurations
Abstract A cyclic $$(n_{3})$$ configuration is a combinatorial configuration whose automorphism group contains a cyclic permutation of the points of the configuration; that is, the points of the configuration may be considered to be elements of $${\mathbb {Z}}_{n}$$, and the lines of the configurati...
Ausführliche Beschreibung
Gespeichert in:
| Autor*in: |
Berman, Leah Wrenn [verfasserIn] |
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| Format: |
Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2020 |
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| Schlagwörter: |
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| Anmerkung: |
© Springer Science+Business Media, LLC, part of Springer Nature 2020 |
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| Übergeordnetes Werk: |
Enthalten in: Discrete & computational geometry - Springer US, 1986, 64(2020), 2 vom: 27. Apr., Seite 542-565 |
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| Übergeordnetes Werk: |
volume:64 ; year:2020 ; number:2 ; day:27 ; month:04 ; pages:542-565 |
| Links: |
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| DOI / URN: |
10.1007/s00454-020-00203-1 |
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OLC2118740980 |
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| 520 | |a Abstract A cyclic $$(n_{3})$$ configuration is a combinatorial configuration whose automorphism group contains a cyclic permutation of the points of the configuration; that is, the points of the configuration may be considered to be elements of $${\mathbb {Z}}_{n}$$, and the lines of the configuration as cyclic shifts of a single fixed starting block [0, a, b], where $$a, b \in {\mathbb {Z}}_{n}$$. We denote such configurations as $$\mathrm{Cyc}_{n}(0,a,b)$$. One of the fundamental questions in the study of configurations is that of geometric realizability. In the case where $$n = 2m$$, it is combinatorially possible to divide the points and lines of the configuration into two classes according to parity, so it is natural to ask whether the configuration can be realized using those classes. We provide methods for producing geometric realizations of configurations $$\mathrm{Cyc}_{2m}(0,a,b)$$ that have two symmetry classes under the maximal rotational subgroup of the geometric symmetry group (that is, chiral astral realizations), and we provide a number of constraints on a and b that guarantee such a realization exists. Experiments on up to 500 points suggest that, with the exception of some small sporadic examples and a single infinite family $$\mathrm{Cyc}_{2(k+1)}(0,1,k)$$, $$k \ge 3$$ and k odd, all cyclic $$(2m_{3})$$ configurations are realizable as geometric chiral astral configurations using the methods described in this paper. | ||
| 650 | 4 | |a Cyclic configurations | |
| 650 | 4 | |a Combinatorial configurations | |
| 650 | 4 | |a Geometric configurations | |
| 650 | 4 | |a Astral configurations | |
| 650 | 4 | |a Realizations with symmetry | |
| 650 | 4 | |a Haar graphs | |
| 700 | 1 | |a DeOrsey, Philip |4 aut | |
| 700 | 1 | |a Faudree, Jill R. |4 aut | |
| 700 | 1 | |a Pisanski, Tomaž |4 aut | |
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10.1007/s00454-020-00203-1 doi (DE-627)OLC2118740980 (DE-He213)s00454-020-00203-1-p DE-627 ger DE-627 rakwb eng 004 VZ 510 004 VZ 17,1 ssgn Berman, Leah Wrenn verfasserin aut Chiral Astral Realizations of Cyclic 3-Configurations 2020 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science+Business Media, LLC, part of Springer Nature 2020 Abstract A cyclic $$(n_{3})$$ configuration is a combinatorial configuration whose automorphism group contains a cyclic permutation of the points of the configuration; that is, the points of the configuration may be considered to be elements of $${\mathbb {Z}}_{n}$$, and the lines of the configuration as cyclic shifts of a single fixed starting block [0, a, b], where $$a, b \in {\mathbb {Z}}_{n}$$. We denote such configurations as $$\mathrm{Cyc}_{n}(0,a,b)$$. One of the fundamental questions in the study of configurations is that of geometric realizability. In the case where $$n = 2m$$, it is combinatorially possible to divide the points and lines of the configuration into two classes according to parity, so it is natural to ask whether the configuration can be realized using those classes. We provide methods for producing geometric realizations of configurations $$\mathrm{Cyc}_{2m}(0,a,b)$$ that have two symmetry classes under the maximal rotational subgroup of the geometric symmetry group (that is, chiral astral realizations), and we provide a number of constraints on a and b that guarantee such a realization exists. Experiments on up to 500 points suggest that, with the exception of some small sporadic examples and a single infinite family $$\mathrm{Cyc}_{2(k+1)}(0,1,k)$$, $$k \ge 3$$ and k odd, all cyclic $$(2m_{3})$$ configurations are realizable as geometric chiral astral configurations using the methods described in this paper. Cyclic configurations Combinatorial configurations Geometric configurations Astral configurations Realizations with symmetry Haar graphs DeOrsey, Philip aut Faudree, Jill R. aut Pisanski, Tomaž aut Žitnik, Arjana (orcid)0000-0001-7737-1836 aut Enthalten in Discrete & computational geometry Springer US, 1986 64(2020), 2 vom: 27. Apr., Seite 542-565 (DE-627)129197556 (DE-600)53957-0 (DE-576)01445694X 0179-5376 nnns volume:64 year:2020 number:2 day:27 month:04 pages:542-565 https://doi.org/10.1007/s00454-020-00203-1 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_2018 GBV_ILN_2409 AR 64 2020 2 27 04 542-565 |
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10.1007/s00454-020-00203-1 doi (DE-627)OLC2118740980 (DE-He213)s00454-020-00203-1-p DE-627 ger DE-627 rakwb eng 004 VZ 510 004 VZ 17,1 ssgn Berman, Leah Wrenn verfasserin aut Chiral Astral Realizations of Cyclic 3-Configurations 2020 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science+Business Media, LLC, part of Springer Nature 2020 Abstract A cyclic $$(n_{3})$$ configuration is a combinatorial configuration whose automorphism group contains a cyclic permutation of the points of the configuration; that is, the points of the configuration may be considered to be elements of $${\mathbb {Z}}_{n}$$, and the lines of the configuration as cyclic shifts of a single fixed starting block [0, a, b], where $$a, b \in {\mathbb {Z}}_{n}$$. We denote such configurations as $$\mathrm{Cyc}_{n}(0,a,b)$$. One of the fundamental questions in the study of configurations is that of geometric realizability. In the case where $$n = 2m$$, it is combinatorially possible to divide the points and lines of the configuration into two classes according to parity, so it is natural to ask whether the configuration can be realized using those classes. We provide methods for producing geometric realizations of configurations $$\mathrm{Cyc}_{2m}(0,a,b)$$ that have two symmetry classes under the maximal rotational subgroup of the geometric symmetry group (that is, chiral astral realizations), and we provide a number of constraints on a and b that guarantee such a realization exists. Experiments on up to 500 points suggest that, with the exception of some small sporadic examples and a single infinite family $$\mathrm{Cyc}_{2(k+1)}(0,1,k)$$, $$k \ge 3$$ and k odd, all cyclic $$(2m_{3})$$ configurations are realizable as geometric chiral astral configurations using the methods described in this paper. Cyclic configurations Combinatorial configurations Geometric configurations Astral configurations Realizations with symmetry Haar graphs DeOrsey, Philip aut Faudree, Jill R. aut Pisanski, Tomaž aut Žitnik, Arjana (orcid)0000-0001-7737-1836 aut Enthalten in Discrete & computational geometry Springer US, 1986 64(2020), 2 vom: 27. Apr., Seite 542-565 (DE-627)129197556 (DE-600)53957-0 (DE-576)01445694X 0179-5376 nnns volume:64 year:2020 number:2 day:27 month:04 pages:542-565 https://doi.org/10.1007/s00454-020-00203-1 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_2018 GBV_ILN_2409 AR 64 2020 2 27 04 542-565 |
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10.1007/s00454-020-00203-1 doi (DE-627)OLC2118740980 (DE-He213)s00454-020-00203-1-p DE-627 ger DE-627 rakwb eng 004 VZ 510 004 VZ 17,1 ssgn Berman, Leah Wrenn verfasserin aut Chiral Astral Realizations of Cyclic 3-Configurations 2020 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science+Business Media, LLC, part of Springer Nature 2020 Abstract A cyclic $$(n_{3})$$ configuration is a combinatorial configuration whose automorphism group contains a cyclic permutation of the points of the configuration; that is, the points of the configuration may be considered to be elements of $${\mathbb {Z}}_{n}$$, and the lines of the configuration as cyclic shifts of a single fixed starting block [0, a, b], where $$a, b \in {\mathbb {Z}}_{n}$$. We denote such configurations as $$\mathrm{Cyc}_{n}(0,a,b)$$. One of the fundamental questions in the study of configurations is that of geometric realizability. In the case where $$n = 2m$$, it is combinatorially possible to divide the points and lines of the configuration into two classes according to parity, so it is natural to ask whether the configuration can be realized using those classes. We provide methods for producing geometric realizations of configurations $$\mathrm{Cyc}_{2m}(0,a,b)$$ that have two symmetry classes under the maximal rotational subgroup of the geometric symmetry group (that is, chiral astral realizations), and we provide a number of constraints on a and b that guarantee such a realization exists. Experiments on up to 500 points suggest that, with the exception of some small sporadic examples and a single infinite family $$\mathrm{Cyc}_{2(k+1)}(0,1,k)$$, $$k \ge 3$$ and k odd, all cyclic $$(2m_{3})$$ configurations are realizable as geometric chiral astral configurations using the methods described in this paper. Cyclic configurations Combinatorial configurations Geometric configurations Astral configurations Realizations with symmetry Haar graphs DeOrsey, Philip aut Faudree, Jill R. aut Pisanski, Tomaž aut Žitnik, Arjana (orcid)0000-0001-7737-1836 aut Enthalten in Discrete & computational geometry Springer US, 1986 64(2020), 2 vom: 27. Apr., Seite 542-565 (DE-627)129197556 (DE-600)53957-0 (DE-576)01445694X 0179-5376 nnns volume:64 year:2020 number:2 day:27 month:04 pages:542-565 https://doi.org/10.1007/s00454-020-00203-1 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_2018 GBV_ILN_2409 AR 64 2020 2 27 04 542-565 |
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Chiral Astral Realizations of Cyclic 3-Configurations |
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Chiral Astral Realizations of Cyclic 3-Configurations |
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Berman, Leah Wrenn |
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Berman, Leah Wrenn DeOrsey, Philip Faudree, Jill R. Pisanski, Tomaž Žitnik, Arjana |
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chiral astral realizations of cyclic 3-configurations |
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Chiral Astral Realizations of Cyclic 3-Configurations |
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Abstract A cyclic $$(n_{3})$$ configuration is a combinatorial configuration whose automorphism group contains a cyclic permutation of the points of the configuration; that is, the points of the configuration may be considered to be elements of $${\mathbb {Z}}_{n}$$, and the lines of the configuration as cyclic shifts of a single fixed starting block [0, a, b], where $$a, b \in {\mathbb {Z}}_{n}$$. We denote such configurations as $$\mathrm{Cyc}_{n}(0,a,b)$$. One of the fundamental questions in the study of configurations is that of geometric realizability. In the case where $$n = 2m$$, it is combinatorially possible to divide the points and lines of the configuration into two classes according to parity, so it is natural to ask whether the configuration can be realized using those classes. We provide methods for producing geometric realizations of configurations $$\mathrm{Cyc}_{2m}(0,a,b)$$ that have two symmetry classes under the maximal rotational subgroup of the geometric symmetry group (that is, chiral astral realizations), and we provide a number of constraints on a and b that guarantee such a realization exists. Experiments on up to 500 points suggest that, with the exception of some small sporadic examples and a single infinite family $$\mathrm{Cyc}_{2(k+1)}(0,1,k)$$, $$k \ge 3$$ and k odd, all cyclic $$(2m_{3})$$ configurations are realizable as geometric chiral astral configurations using the methods described in this paper. © Springer Science+Business Media, LLC, part of Springer Nature 2020 |
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Abstract A cyclic $$(n_{3})$$ configuration is a combinatorial configuration whose automorphism group contains a cyclic permutation of the points of the configuration; that is, the points of the configuration may be considered to be elements of $${\mathbb {Z}}_{n}$$, and the lines of the configuration as cyclic shifts of a single fixed starting block [0, a, b], where $$a, b \in {\mathbb {Z}}_{n}$$. We denote such configurations as $$\mathrm{Cyc}_{n}(0,a,b)$$. One of the fundamental questions in the study of configurations is that of geometric realizability. In the case where $$n = 2m$$, it is combinatorially possible to divide the points and lines of the configuration into two classes according to parity, so it is natural to ask whether the configuration can be realized using those classes. We provide methods for producing geometric realizations of configurations $$\mathrm{Cyc}_{2m}(0,a,b)$$ that have two symmetry classes under the maximal rotational subgroup of the geometric symmetry group (that is, chiral astral realizations), and we provide a number of constraints on a and b that guarantee such a realization exists. Experiments on up to 500 points suggest that, with the exception of some small sporadic examples and a single infinite family $$\mathrm{Cyc}_{2(k+1)}(0,1,k)$$, $$k \ge 3$$ and k odd, all cyclic $$(2m_{3})$$ configurations are realizable as geometric chiral astral configurations using the methods described in this paper. © Springer Science+Business Media, LLC, part of Springer Nature 2020 |
| abstract_unstemmed |
Abstract A cyclic $$(n_{3})$$ configuration is a combinatorial configuration whose automorphism group contains a cyclic permutation of the points of the configuration; that is, the points of the configuration may be considered to be elements of $${\mathbb {Z}}_{n}$$, and the lines of the configuration as cyclic shifts of a single fixed starting block [0, a, b], where $$a, b \in {\mathbb {Z}}_{n}$$. We denote such configurations as $$\mathrm{Cyc}_{n}(0,a,b)$$. One of the fundamental questions in the study of configurations is that of geometric realizability. In the case where $$n = 2m$$, it is combinatorially possible to divide the points and lines of the configuration into two classes according to parity, so it is natural to ask whether the configuration can be realized using those classes. We provide methods for producing geometric realizations of configurations $$\mathrm{Cyc}_{2m}(0,a,b)$$ that have two symmetry classes under the maximal rotational subgroup of the geometric symmetry group (that is, chiral astral realizations), and we provide a number of constraints on a and b that guarantee such a realization exists. Experiments on up to 500 points suggest that, with the exception of some small sporadic examples and a single infinite family $$\mathrm{Cyc}_{2(k+1)}(0,1,k)$$, $$k \ge 3$$ and k odd, all cyclic $$(2m_{3})$$ configurations are realizable as geometric chiral astral configurations using the methods described in this paper. © Springer Science+Business Media, LLC, part of Springer Nature 2020 |
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