{"id":"https://openalex.org/W7220705376","doi":"https://doi.org/10.48550/arxiv.2610.05438","title":"Area and diameter gaps for hyperbolic monotiles","display_name":"Area and diameter gaps for hyperbolic monotiles","publication_year":2026,"publication_date":"2026-10-04","ids":{"openalex":"https://openalex.org/W7220705376","doi":"https://doi.org/10.48550/arxiv.2610.05438"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2610.05438","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.05438","pdf_url":"https://arxiv.org/pdf/2610.05438","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"volume":null,"type":"preprint","is_traditional":true,"indexed_in":["arxiv","datacite"],"open_access":{"is_oa":true,"oa_status":"green","oa_url":"https://arxiv.org/pdf/2610.05438","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5029619165","display_name":"Yixi Liao","orcid":"https://orcid.org/0000-0002-3795-0021","observed_orcids":["https://orcid.org/0000-0002-3795-0021"]},"institutions":[{"id":"https://openalex.org/I135237710","display_name":"Zhejiang Normal University","ror":"https://ror.org/01vevwk45","country_code":"CN","type":"education","lineage":["https://openalex.org/I135237710"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Liao, Yixi","raw_affiliation_strings":["Zhejiang Normal University"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Zhejiang Normal University","institution_ids":["https://openalex.org/I135237710"]}]},{"author_position":"last","author":{"id":"https://openalex.org/A5022227739","display_name":"Erxiao Wang","orcid":"https://orcid.org/0000-0002-8988-8733","observed_orcids":["https://orcid.org/0000-0002-8988-8733"]},"institutions":[{"id":"https://openalex.org/I135237710","display_name":"Zhejiang Normal University","ror":"https://ror.org/01vevwk45","country_code":"CN","type":"education","lineage":["https://openalex.org/I135237710"]}],"countries":["CN"],"is_corresponding":false,"raw_author_name":"Wang, Erxiao","raw_affiliation_strings":["Zhejiang Normal University"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Zhejiang Normal University","institution_ids":["https://openalex.org/I135237710"]}]}],"institutions":[],"countries_distinct_count":1,"institutions_distinct_count":1,"corresponding_author_ids":[],"corresponding_institution_ids":["https://openalex.org/I135237710"],"apc_list":null,"apc_paid":null,"fwci":null,"has_fulltext":true,"cited_by_count":0,"citation_normalized_percentile":null,"cited_by_percentile_year":null,"biblio":{"volume":null,"issue":null,"first_page":null,"last_page":null},"is_retracted":false,"is_paratext":false,"is_xpac":false,"primary_topic":{"id":"https://openalex.org/T10304","display_name":"Geometric and Algebraic Topology","score":0.4503999948501587,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},"topics":[{"id":"https://openalex.org/T10304","display_name":"Geometric and Algebraic Topology","score":0.4503999948501587,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T12504","display_name":"Mathematics and Applications","score":0.4487000107765198,"subfield":{"id":"https://openalex.org/subfields/2608","display_name":"Geometry and Topology"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10229","display_name":"Geometric Analysis and Curvature Flows","score":0.029600000008940697,"subfield":{"id":"https://openalex.org/subfields/2604","display_name":"Applied Mathematics"},"field":{"id":"https://openalex.org/fields/26","display_name":"Mathematics"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}}],"keywords":[{"id":"https://openalex.org/keywords/hyperbolic-plane","display_name":"hyperbolic plane","score":0.8009999990463257}],"concepts":[{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8478000164031982},{"id":"https://openalex.org/C165818556","wikidata":"https://www.wikidata.org/wiki/Q213488","display_name":"Geodesic","level":2,"score":0.6682000160217285},{"id":"https://openalex.org/C112680207","wikidata":"https://www.wikidata.org/wiki/Q714886","display_name":"Regular polygon","level":2,"score":0.616599977016449},{"id":"https://openalex.org/C122041747","wikidata":"https://www.wikidata.org/wiki/Q838611","display_name":"Ball (mathematics)","level":2,"score":0.5842999815940857},{"id":"https://openalex.org/C145691206","wikidata":"https://www.wikidata.org/wiki/Q747980","display_name":"Polytope","level":2,"score":0.5734000205993652},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.5701000094413757},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.512499988079071},{"id":"https://openalex.org/C80899671","wikidata":"https://www.wikidata.org/wiki/Q1304193","display_name":"Vertex (graph theory)","level":3,"score":0.4999000132083893},{"id":"https://openalex.org/C82673646","wikidata":"https://www.wikidata.org/wiki/Q2246553","display_name":"Hyperbolic triangle","level":4,"score":0.4706999957561493},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.4700999855995178},{"id":"https://openalex.org/C206352148","wikidata":"https://www.wikidata.org/wiki/Q209306","display_name":"Hyperbolic geometry","level":3,"score":0.4456000030040741},{"id":"https://openalex.org/C62354387","wikidata":"https://www.wikidata.org/wiki/Q875399","display_name":"Boundary (topology)","level":2,"score":0.4108999967575073},{"id":"https://openalex.org/C2524010","wikidata":"https://www.wikidata.org/wiki/Q8087","display_name":"Geometry","level":1,"score":0.4023999869823456},{"id":"https://openalex.org/C2780728851","wikidata":"https://www.wikidata.org/wiki/Q468402","display_name":"Tile","level":2,"score":0.3937999904155731},{"id":"https://openalex.org/C134306372","wikidata":"https://www.wikidata.org/wiki/Q7754","display_name":"Mathematical analysis","level":1,"score":0.33000001311302185},{"id":"https://openalex.org/C2777027219","wikidata":"https://www.wikidata.org/wiki/Q1284190","display_name":"Constant (computer programming)","level":2,"score":0.32589998841285706},{"id":"https://openalex.org/C2780586882","wikidata":"https://www.wikidata.org/wiki/Q7520643","display_name":"Simple (philosophy)","level":2,"score":0.3228999972343445},{"id":"https://openalex.org/C110943637","wikidata":"https://www.wikidata.org/wiki/Q4129097","display_name":"Convex polytope","level":5,"score":0.3050999939441681},{"id":"https://openalex.org/C134912446","wikidata":"https://www.wikidata.org/wiki/Q5166516","display_name":"Convex body","level":4,"score":0.29750001430511475},{"id":"https://openalex.org/C54829058","wikidata":"https://www.wikidata.org/wiki/Q172937","display_name":"Polyhedron","level":2,"score":0.29409998655319214},{"id":"https://openalex.org/C199068039","wikidata":"https://www.wikidata.org/wiki/Q574523","display_name":"Immersion (mathematics)","level":2,"score":0.2922999858856201},{"id":"https://openalex.org/C162838799","wikidata":"https://www.wikidata.org/wiki/Q596077","display_name":"Counterexample","level":2,"score":0.2849999964237213},{"id":"https://openalex.org/C105313907","wikidata":"https://www.wikidata.org/wiki/Q1053857","display_name":"Dual polyhedron","level":2,"score":0.2833000123500824},{"id":"https://openalex.org/C39308205","wikidata":"https://www.wikidata.org/wiki/Q7880612","display_name":"Ultraparallel theorem","level":5,"score":0.2728999853134155},{"id":"https://openalex.org/C2778023678","wikidata":"https://www.wikidata.org/wiki/Q554403","display_name":"Duality (order theory)","level":2,"score":0.2581000030040741}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2610.05438","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.05438","pdf_url":"https://arxiv.org/pdf/2610.05438","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2610.05438","doi":"https://doi.org/10.48550/arxiv.2610.05438","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2610.05438","pdf_url":null,"source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":"cc-by-nc-nd","license_id":"https://openalex.org/licenses/cc-by-nc-nd","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2610.05438","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2610.05438","pdf_url":"https://arxiv.org/pdf/2610.05438","source":{"id":"https://openalex.org/S4306400194","display_name":"arXiv (Cornell University)","issn_l":"2331-8422","issn":["2331-8422"],"is_oa":true,"is_in_doaj":false,"is_core":false,"listed_in":[],"host_organization":"https://openalex.org/I205783295","host_organization_name":"Cornell University","host_organization_lineage":["https://openalex.org/I205783295"],"host_organization_lineage_names":[],"type":"repository"},"license":null,"license_id":null,"version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[{"id":"https://openalex.org/G5817270534","display_name":"\u7528\u5168\u7b49\u56db\u8fb9\u5f62\"\u8fb9\u5bf9\u8fb9\"\u5bc6\u94fa\u7403\u9762\u7684\u5b8c\u6574\u5206\u7c7b","funder_award_id":"LZ22A010003","funder_id":"https://openalex.org/F4320338464","funder_display_name":"Natural Science Foundation of Zhejiang Province"}],"funders":[{"id":"https://openalex.org/F4320321001","display_name":"National Natural Science Foundation of China","ror":"https://ror.org/01h0zpd94"},{"id":"https://openalex.org/F4320322819","display_name":"Zhejiang Normal University","ror":"https://ror.org/01vevwk45"},{"id":"https://openalex.org/F4320338464","display_name":"Natural Science Foundation of Zhejiang Province","ror":"https://ror.org/01h0zpd94"}],"has_content":{"pdf":true,"grobid_xml":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7220705376.pdf","grobid_xml":"https://content.openalex.org/works/W7220705376.grobid-xml"},"referenced_works_count":23,"referenced_works":["https://openalex.org/W1565649417","https://openalex.org/W1622327413","https://openalex.org/W1968660981","https://openalex.org/W1976591270","https://openalex.org/W2026880914","https://openalex.org/W2041591002","https://openalex.org/W2048190114","https://openalex.org/W2073943587","https://openalex.org/W2076508310","https://openalex.org/W2116827925","https://openalex.org/W2246805282","https://openalex.org/W2328980644","https://openalex.org/W2963680558","https://openalex.org/W2982653103","https://openalex.org/W2985771827","https://openalex.org/W3186676858","https://openalex.org/W4292379800","https://openalex.org/W4293217885","https://openalex.org/W4386517676","https://openalex.org/W4390214908","https://openalex.org/W4401363286","https://openalex.org/W7125954547","https://openalex.org/W7157676559"],"related_works":["https://openalex.org/W2985771827","https://openalex.org/W2982653103","https://openalex.org/W7148178149","https://openalex.org/W1501316957","https://openalex.org/W4288017058","https://openalex.org/W4400687219","https://openalex.org/W4205805488","https://openalex.org/W7157676559","https://openalex.org/W7220650358","https://openalex.org/W2883408634"],"abstract_inverted_index":{"For":[0],"each":[1,139],"fixed":[2],"$n$,":[3],"we":[4,92,136],"prove":[5,26],"a":[6,14,27,36,39,44,97,126,142],"positive":[7,28],"lower":[8,29],"bound":[9,30],"for":[10,31,60,101,138],"the":[11,21,32,50,118,131],"diameter":[12,149],"of":[13,20,34,43,49,58],"compact":[15],"simple":[16],"geodesic":[17,143],"$n$-gonal":[18],"monotile":[19],"hyperbolic":[22,46],"plane.":[23],"We":[24],"also":[25],"area":[33],"such":[35],"tile":[37,67,132],"in":[38,63,117],"finite":[40],"monohedral":[41],"tiling":[42],"closed":[45,115],"surface,":[47],"independent":[48,57],"topology":[51],"and":[52,84,122,148],"metric.":[53],"Both":[54],"bounds":[55],"become":[56],"$n$":[59],"vertex-proper":[61],"tilings,":[62],"which":[64],"every":[65],"genuine":[66],"vertex":[68],"belongs":[69],"to":[70,168,181],"at":[71],"least":[72],"three":[73],"distinct":[74],"tiles.":[75],"This":[76],"includes":[77],"tilings":[78],"by":[79],"convex":[80],"polygons.":[81],"Reflex":[82],"angles":[83],"non-edge-to-edge":[85],"incidences":[86],"are":[87,175,178],"allowed.":[88],"After":[89],"qualitative":[90],"proofs,":[91],"obtain":[93],"explicit":[94],"constants":[95,174],"from":[96],"sharp":[98],"gap":[99],"estimate":[100],"packing":[102],"polytopes":[103],"with":[104,145],"arbitrarily":[105],"coupled":[106],"nonnegative":[107],"integer":[108,140],"constraints.":[109],"Exact":[110],"corner":[111],"balance":[112],"applies":[113],"on":[114,130],"surfaces;":[116],"plane,":[119],"covering":[120],"duality":[121],"ball":[123],"counts":[124,156],"give":[125],"boundary":[127],"factor":[128],"depending":[129],"diameter.":[133],"Following":[134],"Zare,":[135],"give,":[137],"$q\\ge2$,":[141],"construction":[144],"$2q+3$":[146],"sides":[147],"less":[150],"than":[151],"$3/q$,":[152],"showing":[153],"that":[154],"side":[155],"cannot":[157],"be":[158,182],"unrestricted":[159],"without":[160],"an":[161],"additional":[162],"condition.":[163],"The":[164,172],"closed-surface":[165],"results":[166],"extend":[167],"regular":[169],"curved":[170],"sides.":[171],"geometric":[173],"effective":[176],"but":[177],"not":[179],"claimed":[180],"optimal.":[183]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-10-07T00:00:00"}
