Polycyclic Geometric Realizations of the Gray Configuration
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866909502959255552 |
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| author | Berman, Leah Wrenn Gévay, Gábor Pisanski, Tomaž |
| author_facet | Berman, Leah Wrenn Gévay, Gábor Pisanski, Tomaž |
| contents | The Gray configuration is a (27_3) configuration which typically is realized as the points and lines of the 3 x 3 x 3 integer lattice. It occurs as a member of an infinite family of configurations defined by Bouwer in 1972. Since their discovery, both the Gray configuration and its Levi graph (i.e., its point-line incidence graph) have been the subject of intensive study. Its automorphism group contains cyclic subgroups isomorphic to Z_3 and Z_9, so it is natural to ask whether the Gray configuration can be realized in the plane with any of the corresponding rotational symmetry. In this paper, we show that there are two distinct polycyclic realizations with Z_3 symmetry. In contrast, the only geometric polycyclic realization with straight lines and Z_9 symmetry is only a "weak" realization, with extra unwanted incidences (in particular, the realization is actually a (27_4) configuration). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_14484 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Polycyclic Geometric Realizations of the Gray Configuration Berman, Leah Wrenn Gévay, Gábor Pisanski, Tomaž Combinatorics 05B30, 05C10, 05C60, 51A45, 51E30 The Gray configuration is a (27_3) configuration which typically is realized as the points and lines of the 3 x 3 x 3 integer lattice. It occurs as a member of an infinite family of configurations defined by Bouwer in 1972. Since their discovery, both the Gray configuration and its Levi graph (i.e., its point-line incidence graph) have been the subject of intensive study. Its automorphism group contains cyclic subgroups isomorphic to Z_3 and Z_9, so it is natural to ask whether the Gray configuration can be realized in the plane with any of the corresponding rotational symmetry. In this paper, we show that there are two distinct polycyclic realizations with Z_3 symmetry. In contrast, the only geometric polycyclic realization with straight lines and Z_9 symmetry is only a "weak" realization, with extra unwanted incidences (in particular, the realization is actually a (27_4) configuration). |
| title | Polycyclic Geometric Realizations of the Gray Configuration |
| topic | Combinatorics 05B30, 05C10, 05C60, 51A45, 51E30 |
| url | https://arxiv.org/abs/2502.14484 |