{"paper":{"title":"Highly symmetric fundamental domains for lattices in R^2 and R^3","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MG"],"primary_cat":"math.CO","authors_text":"Dirk Frettl\\\"oh, Joseph Ray Clarence G. Damasco, Manuel Joseph C. Loquias","submitted_at":"2013-05-08T12:46:38Z","abstract_excerpt":"It is shown that most lattices $\\Gamma$ in $\\mathbb{R}^2$ and $\\mathbb{R}^3$ possess a fundamental domain $F$ for the action of $\\Gamma$ on $\\mathbb{R}^2$, respectively $\\mathbb{R}^3$, having more symmetries than the point group $P(\\Gamma)$, i.e., the group $P (\\Gamma) \\subset O(d)$ fixing $\\Gamma$. In particular, $P (\\Gamma)$ is a subgroup of the symmetry group $S(F)$ of $F$ of index 2 in these cases. Exceptions are cubic lattices in the three-dimensional case, where such an $F$ does not exist. Possible exceptions are rhombic lattices in the plane case, where the constructions presented here "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.1798","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}