{"id":"https://openalex.org/W7206189573","doi":"https://doi.org/10.48550/arxiv.2609.01270","title":"Farey Structure in Modulo Krinkle Tilings: Mediant Splicing and Generation of Prototiles from a Single Edge","display_name":"Farey Structure in Modulo Krinkle Tilings: Mediant Splicing and Generation of Prototiles from a Single Edge","publication_year":2026,"publication_date":"2026-09-01","ids":{"openalex":"https://openalex.org/W7206189573","doi":"https://doi.org/10.48550/arxiv.2609.01270"},"language":"en","primary_location":{"id":"pmh:oai:arXiv.org:2609.01270","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.01270","pdf_url":"https://arxiv.org/pdf/2609.01270","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","license_id":"https://openalex.org/licenses/cc-by","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/2609.01270","any_repository_has_fulltext":true},"authorships":[{"author_position":"first","author":{"id":"https://openalex.org/A5065883269","display_name":"\u5e79\u5f18 \u85e4\u539f","orcid":null,"observed_orcids":[]},"institutions":[],"countries":[],"is_corresponding":true,"raw_author_name":"Fujiwara, Mikihiro","raw_affiliation_strings":["Independent Researcher , Tokyo , Japan"],"raw_orcid":null,"affiliations":[{"raw_affiliation_string":"Independent Researcher , Tokyo , Japan","institution_ids":[]}]}],"institutions":[],"countries_distinct_count":0,"institutions_distinct_count":0,"corresponding_author_ids":["https://openalex.org/A5065883269"],"corresponding_institution_ids":[],"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/T12504","display_name":"Mathematics and Applications","score":0.42309999465942383,"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/T12504","display_name":"Mathematics and Applications","score":0.42309999465942383,"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/T11887","display_name":"Quasicrystal Structures and Properties","score":0.1687999963760376,"subfield":{"id":"https://openalex.org/subfields/2505","display_name":"Materials Chemistry"},"field":{"id":"https://openalex.org/fields/25","display_name":"Materials Science"},"domain":{"id":"https://openalex.org/domains/3","display_name":"Physical Sciences"}},{"id":"https://openalex.org/T10588","display_name":"Mathematical Dynamics and Fractals","score":0.16529999673366547,"subfield":{"id":"https://openalex.org/subfields/2610","display_name":"Mathematical Physics"},"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/aperiodic-tilings","display_name":"aperiodic tilings","score":0.9589999914169312},{"id":"https://openalex.org/keywords/prototiles","display_name":"prototiles","score":0.7400000095367432},{"id":"https://openalex.org/keywords/stern-brocot-tree","display_name":"Stern-Brocot tree","score":0.9120000004768372},{"id":"https://openalex.org/keywords/christoffel-words","display_name":"Christoffel words","score":0.9929999709129333}],"concepts":[{"id":"https://openalex.org/C93157843","wikidata":"https://www.wikidata.org/wiki/Q1396592","display_name":"Farey sequence","level":2,"score":0.8666999936103821},{"id":"https://openalex.org/C33923547","wikidata":"https://www.wikidata.org/wiki/Q395","display_name":"Mathematics","level":0,"score":0.8033000230789185},{"id":"https://openalex.org/C114614502","wikidata":"https://www.wikidata.org/wiki/Q76592","display_name":"Combinatorics","level":1,"score":0.7077999711036682},{"id":"https://openalex.org/C126875415","wikidata":"https://www.wikidata.org/wiki/Q203425","display_name":"Unit circle","level":2,"score":0.6705999970436096},{"id":"https://openalex.org/C54732982","wikidata":"https://www.wikidata.org/wiki/Q1415345","display_name":"Modulo","level":2,"score":0.5889000296592712},{"id":"https://openalex.org/C187834632","wikidata":"https://www.wikidata.org/wiki/Q188804","display_name":"Factorization","level":2,"score":0.5476999878883362},{"id":"https://openalex.org/C62354387","wikidata":"https://www.wikidata.org/wiki/Q875399","display_name":"Boundary (topology)","level":2,"score":0.5297999978065491},{"id":"https://openalex.org/C118615104","wikidata":"https://www.wikidata.org/wiki/Q121416","display_name":"Discrete mathematics","level":1,"score":0.42250001430511475},{"id":"https://openalex.org/C45340560","wikidata":"https://www.wikidata.org/wiki/Q215382","display_name":"Disjoint sets","level":2,"score":0.36559998989105225},{"id":"https://openalex.org/C168773036","wikidata":"https://www.wikidata.org/wiki/Q264164","display_name":"Recursion (computer science)","level":2,"score":0.35019999742507935},{"id":"https://openalex.org/C74050887","wikidata":"https://www.wikidata.org/wiki/Q848368","display_name":"Rotation (mathematics)","level":2,"score":0.3434000015258789},{"id":"https://openalex.org/C122637931","wikidata":"https://www.wikidata.org/wiki/Q118084","display_name":"Unit (ring theory)","level":2,"score":0.3352999985218048},{"id":"https://openalex.org/C13355873","wikidata":"https://www.wikidata.org/wiki/Q2920850","display_name":"Connection (principal bundle)","level":2,"score":0.33469998836517334},{"id":"https://openalex.org/C2777735758","wikidata":"https://www.wikidata.org/wiki/Q817765","display_name":"Path (computing)","level":2,"score":0.32269999384880066},{"id":"https://openalex.org/C113174947","wikidata":"https://www.wikidata.org/wiki/Q2859736","display_name":"Tree (set theory)","level":2,"score":0.2994000017642975},{"id":"https://openalex.org/C70026782","wikidata":"https://www.wikidata.org/wiki/Q16726164","display_name":"Clockwise","level":3,"score":0.29840001463890076},{"id":"https://openalex.org/C2781089630","wikidata":"https://www.wikidata.org/wiki/Q21856745","display_name":"Realization (probability)","level":2,"score":0.2906999886035919},{"id":"https://openalex.org/C2780012671","wikidata":"https://www.wikidata.org/wiki/Q1343870","display_name":"Corollary","level":2,"score":0.28859999775886536},{"id":"https://openalex.org/C77553402","wikidata":"https://www.wikidata.org/wiki/Q13222579","display_name":"Upper and lower bounds","level":2,"score":0.2872999906539917},{"id":"https://openalex.org/C184720557","wikidata":"https://www.wikidata.org/wiki/Q7825049","display_name":"Topology (electrical circuits)","level":2,"score":0.28139999508857727},{"id":"https://openalex.org/C165263651","wikidata":"https://www.wikidata.org/wiki/Q2443460","display_name":"Topological conjugacy","level":2,"score":0.2687000036239624},{"id":"https://openalex.org/C6026789","wikidata":"https://www.wikidata.org/wiki/Q948010","display_name":"Primitive root modulo n","level":3,"score":0.2653000056743622},{"id":"https://openalex.org/C202444582","wikidata":"https://www.wikidata.org/wiki/Q837863","display_name":"Pure mathematics","level":1,"score":0.26269999146461487},{"id":"https://openalex.org/C17825722","wikidata":"https://www.wikidata.org/wiki/Q17285","display_name":"Plane (geometry)","level":2,"score":0.25519999861717224},{"id":"https://openalex.org/C9767117","wikidata":"https://www.wikidata.org/wiki/Q12510","display_name":"Torus","level":2,"score":0.2549000084400177},{"id":"https://openalex.org/C97137487","wikidata":"https://www.wikidata.org/wiki/Q729138","display_name":"Integer (computer science)","level":2,"score":0.2547999918460846},{"id":"https://openalex.org/C162329640","wikidata":"https://www.wikidata.org/wiki/Q332648","display_name":"Covering space","level":2,"score":0.2502000033855438}],"mesh":[],"locations_count":2,"locations":[{"id":"pmh:oai:arXiv.org:2609.01270","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.01270","pdf_url":"https://arxiv.org/pdf/2609.01270","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","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},{"id":"doi:10.48550/arxiv.2609.01270","doi":"https://doi.org/10.48550/arxiv.2609.01270","is_oa":true,"landing_page_url":"https://doi.org/10.48550/arxiv.2609.01270","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","license_id":"https://openalex.org/licenses/cc-by","version":null,"is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"Preprint"}],"best_oa_location":{"id":"pmh:oai:arXiv.org:2609.01270","doi":null,"is_oa":true,"landing_page_url":"https://arxiv.org/abs/2609.01270","pdf_url":"https://arxiv.org/pdf/2609.01270","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","license_id":"https://openalex.org/licenses/cc-by","version":"submittedVersion","is_accepted":false,"is_published":false,"raw_source_name":null,"raw_type":"text"},"sustainable_development_goals":[],"study_designs":[],"awards":[],"funders":[],"has_content":{"grobid_xml":true,"pdf":true},"content_urls":{"pdf":"https://content.openalex.org/works/W7206189573.pdf","grobid_xml":"https://content.openalex.org/works/W7206189573.grobid-xml"},"referenced_works_count":1,"referenced_works":["https://openalex.org/W4417134187"],"related_works":["https://openalex.org/W2002263906","https://openalex.org/W6929795360","https://openalex.org/W2270798562","https://openalex.org/W6903729615","https://openalex.org/W7222961673","https://openalex.org/W2887116182","https://openalex.org/W7099500427","https://openalex.org/W6888855267","https://openalex.org/W6891834509","https://openalex.org/W4415641343"],"abstract_inverted_index":{"The":[0,105],"Modulo":[1,138],"Krinkle":[2,139],"tilings":[3,14],"of":[4,10,34,51,68,108,116,170,188],"Imura":[5],"(arXiv:2506.07638)":[6],"form":[7],"a":[8,17,101,148,168],"family":[9],"non-periodic,":[11],"spiral":[12,192],"monohedral":[13],"parametrized":[15],"by":[16,40,56],"reduced":[18],"fraction":[19],"$m/k$":[20],"and":[21,93,131,190,196],"an":[22,41,63],"integer":[23],"$t\\ge":[24],"2$.":[25],"We":[26],"show":[27],"that":[28,152],"the":[29,47,52,58,87,94,97,109,112,119,132,137,142,157,175,182,186],"Farey":[30,177],"sum":[31],"(mediant)":[32],"$(m_1+m_2)/(k_1+k_2)$":[33],"two":[35,176],"Farey-adjacent":[36],"parameters":[37,83],"is":[38,54,111,122,156],"realized":[39],"exact":[42,124],"geometric":[43],"operation":[44,110],"on":[45,127],"prototiles:":[46],"lower":[48,60],"boundary":[49],"path":[50],"$(m_1+m_2,k_1+k_2)$-prototile":[53],"obtained":[55],"concatenating":[57],"parents'":[59],"paths":[61],"after":[62],"edge-length-preserving":[64],"progressive":[65],"rotation":[66],"(fan-twist)":[67],"their":[69],"edges.":[70,166],"Conversely,":[71],"every":[72,153],"prototile":[73,155],"admits":[74],"exactly":[75],"one":[76],"fan-twist":[77],"splice":[78,85],"decomposition":[79],"--":[80,86],"no":[81],"non-adjacent":[82],"ever":[84],"cut":[88],"position":[89],"being":[90],"$k_1=m^{-1}\\bmod":[91],"k$,":[92],"recursion":[95],"descends":[96],"Stern-Brocot":[98],"tree":[99],"to":[100],"single":[102],"unit":[103],"edge.":[104],"combinatorial":[106],"core":[107,160],"classical":[113],"standard":[114],"factorization":[115],"Christoffel":[117],"words;":[118],"contribution":[120],"here":[121],"its":[123],"edge-isometric":[125],"realization":[126],"circular":[128],"direction":[129],"systems":[130],"resulting":[133],"structure":[134],"theory":[135],"for":[136],"family,":[140],"including":[141],"recently":[143],"introduced":[144],"variants:":[145],"we":[146],"prove":[147],"separation":[149],"theorem":[150],"stating":[151],"variant":[154],"common":[158],"recursively-generated":[159],"plus":[161],"finitely":[162],"many":[163],"direction-invariant":[164],"decoration":[165],"As":[167],"corollary":[169],"Imura's":[171],"spiral-arm":[172],"count":[173],"formula,":[174],"parents":[178],"are":[179,194],"visible":[180],"in":[181],"offset-free":[183],"tiling":[184],"itself:":[185],"numbers":[187],"counterclockwise":[189],"clockwise":[191],"arms":[193],"$tk_1$":[195],"$tk_2$.":[197]},"counts_by_year":[],"updated_date":"2026-11-07T11:25:07.637180","created_date":"2026-09-03T00:00:00"}