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The work analyzes intersections with the two diagonals and various axis-parallel lines, and shows how the boundary intersection behavior differs from the generic case.",{"@graph":14,"@context":73},[15,34,56],{"@type":16,"itemListElement":17},"BreadcrumbList",[18,23,27,31],{"item":19,"name":20,"@type":21,"position":22},"https://docshare.wps.com","Home","ListItem",1,{"item":24,"name":25,"@type":21,"position":26},"https://docshare.wps.com/document/","Document",2,{"item":28,"name":29,"@type":21,"position":30},"https://docshare.wps.com/document/research-report/","Research & Report",3,{"item":32,"name":10,"@type":21,"position":33},"https://docshare.wps.com/document/intersecting-the-twin-dragon-with-rational-lines/136507/",4,{"url":32,"name":10,"@type":35,"image":36,"author":41,"headline":10,"publisher":44,"fileFormat":47,"inLanguage":8,"description":12,"dateModified":48,"datePublished":49,"encodingFormat":47,"isAccessibleForFree":50,"interactionStatistic":51},"DigitalDocument",{"url":37,"@type":38,"width":39,"height":40},"https://docshare.wps.com/thumbnails/intersecting-the-twin-dragon-with-rational-lines/136507.png","ImageObject",300,407,{"name":42,"@type":43},"Melati","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-09-20","2026-08-22",true,{"@type":52,"interactionType":53,"userInteractionCount":55},"InteractionCounter",{"@type":54},"ViewAction",6,{"@type":57,"mainEntity":58},"FAQPage",[59,65,69],{"name":60,"@type":61,"acceptedAnswer":62},"What is the main geometric object studied in this paper?","Question",{"text":63,"@type":64},"The paper studies the Knuth Twin Dragon, a compact fractal subset of the plane with a fractal boundary.","Answer",{"name":66,"@type":61,"acceptedAnswer":67},"How do intersections with rational lines differ from generic lines?",{"text":68,"@type":64},"Although intersections with Lebesgue almost all lines typically have Hausdorff dimension reduced by 1, the paper proves that this dimension is never attained for lines whose parameters are rational.",{"name":70,"@type":61,"acceptedAnswer":71},"How is the intersection with rational lines analyzed?",{"text":72,"@type":64},"The authors characterize the digit expansions of points in the intersection and use a Büchi automaton to describe which expansions correspond to boundary intersection points, enabling Hausdorff-dimension calculations.","https://schema.org",{"og:url":32,"og:type":75,"og:title":10,"og:site_name":45,"og:description":12},"article",{"robots":77,"canonical":32},"index,follow",{"doc_id":79,"site_id":7},136507,1787387941,{"code":4,"msg":82,"data":83},"success",[84,88,92,96,101,105,110,114,119,122,126],{"id":22,"doc_module":4,"doc_module_name":25,"category_name":85,"show_sort_weight":86,"slug":87},"Story & Novel",90,"story-novel",{"id":26,"doc_module":4,"doc_module_name":25,"category_name":89,"show_sort_weight":90,"slug":91},"Literature",80,"literature",{"id":33,"doc_module":4,"doc_module_name":25,"category_name":93,"show_sort_weight":94,"slug":95},"Exam",70,"exam",{"id":97,"doc_module":4,"doc_module_name":25,"category_name":98,"show_sort_weight":99,"slug":100},5,"Comic",60,"comic",{"id":55,"doc_module":4,"doc_module_name":25,"category_name":102,"show_sort_weight":103,"slug":104},"Technology",50,"technology",{"id":106,"doc_module":4,"doc_module_name":25,"category_name":107,"show_sort_weight":108,"slug":109},7,"Healthcare",40,"healthcare",{"id":111,"doc_module":4,"doc_module_name":25,"category_name":29,"show_sort_weight":112,"slug":113},8,30,"research-report",{"id":115,"doc_module":4,"doc_module_name":25,"category_name":116,"show_sort_weight":117,"slug":118},9,"Religion & Spirituality",20,"religion-spirituality",{"id":117,"doc_module":4,"doc_module_name":25,"category_name":120,"show_sort_weight":117,"slug":121},"World Cup","world-cup",{"id":123,"doc_module":4,"doc_module_name":25,"category_name":124,"show_sort_weight":123,"slug":125},10,"Lifestyle","lifestyle",{"id":127,"doc_module":4,"doc_module_name":25,"category_name":128,"show_sort_weight":97,"slug":129},19,"General","general",{"code":4,"msg":82,"data":131},{"doc_id":79,"user_id":132,"nickname":42,"user_avatar":133,"doc_module":4,"category_id":111,"category_name":29,"doc_title":10,"doc_description":12,"doc_content":134,"file_id":135,"file_url":136,"file_type":137,"file_size":138,"view_count":55,"is_deleted":4,"is_public":22,"is_downloadable":22,"audit_status":22,"page_count":139,"language":140,"language_code":8,"site_id":7,"html_lang":8,"table_of_contents":141,"faqs":142,"seo_title":143,"seo_description":12,"update_tm":80,"read_time":144},962085570644,"https://ap-avatar.wpscdn.com/davatar_994ba38a5ba835b3df7d355c54d3ed8d","J. Fractal Geom. 11 (2024), 205–217 DOI 10.4171/JFG/149  \n© 2024 European Mathematical Society  \nPublished by EMS Press This work is licensed under a CC BY 4 .0 license  \nIntersecting the Twin Dragon with rational lines  \nShigeki Akiyama, Paul Großkopf, Benoît Loridant, and Wolfgang Steiner  \nAbstract. The Knuth Twin Dragon is a compact subset of the plane with fractal boundary of Hausdorff dimension s D .log 􀀕/=.log p2/, 􀀕 3 D 􀀕 2 C 2. Although the intersection with a generic line has Hausdorff dimension s 􀀀 1, we prove that this does not occur for lines with rational parameters. We further describe the intersection of the Twin Dragon with the two diagonals as well as with various axis parallel lines.  \nDedicated to Professor Jörg Thuswaldner on the occasion of his 50th birthday  \n1. Introduction  \nWe investigate the intersections of the Knuth Twin Dragon with rational lines. Let ˛ D 􀀀 1 C i, then  \nK D ²1X d˛kk W dk 2 ¹0; 1º³  \nk D 1  \nis the Knuth Twin Dragon. The Hausdorff dimension of its boundary @K is s Dlogg􀀕p2 􀀙 1:5236, where 􀀕 is the real number satisfying 􀀕3 D 􀀕 2 C 2. For lines  \n􀂁p;q;r D ¹x C iy 2 C W px C qy D rº (1.1)  \nwith p; q; r 2 Z, we show that the ˛-expansions of K \\ 􀂁p;q;r are recognized by a finite automaton.  \nBy a result of John Marstrand [5], the intersection of @K with Lebesgue almost all lines going through K has Hausdorff dimension s 􀀀 1, meaning that in the set of all parameter triples .p; q; r/ 2 R3 for which 􀂁p;q;r \\ K ¤ ;, the exceptional cases forma Lebesgue null set. We obtain here that the Hausdorff dimension of the intersection of the boundary of the Twin Dragon with rational lines is never equal to s 􀀀 1.  \nFurther, we revisit results by Akiyama and Scheicher [1] and add uncountably many examples of horizontal, vertical, and diagonal lines.  \nMathematics Subject Classification 2020: 52C20 (primary); 28A80 (secondary) .  \nKeywords: number system, Hausdorff dimension.  \nS. Akiyama, P. Großkopf, B. Loridant, and W. Steiner 206  \nWe mention that similar results were obtained in [4] for lines intersecting the Sierpinski carpet F . The set F has Hausdorff dimension log~~ ~~8~~l~~og~~ ~~3 . Manning and Simon showed that, given a slope ˛ 2 Q, the intersection of F with the line y D ˛x C ˇ is strictly less than log~~ ~~8~~l~~og~~ ~~3 􀀀 1 for Lebesgue almost every ˇ .  \n2. Main statement and proof  \nWe first recall the notions of a canonical number system and its fundamental domain. Let ˇ be an algebraic integer and N D ¹0; 1; : : : ; jN.ˇ/j 􀀀 1º, where N.x/ denotes the norm of x over Q.ˇ/=Q. The pair .ˇ; N / is called a canonical number system (CNS) if each 􀀍 2 ZŒˇ􀂍 admits a representation of the form  \nn  \n􀀍 D X dk ˇk ; dk 2 N: (2.1)  \nk D0  \nWe call ˇ the radix or base and N the set of digits. The representation (2.1) is unique up to leading zeros.  \nThe Knuth Twin Dragon K appears as the fundamental domain of the CNS .˛; N /, where ˛ D 􀀀 1 C i is the root of the polynomial x 2 􀀀 2x 􀀀 2 and N D ¹0; 1º . The fundamental domain of a CNS is the set of all numbers that can be expressed with purely negative exponents. Since ˛ 4 D 􀀀 4, it is often useful to consider groups of four digits:  \n1  \nX  \nk D 1  \nd˛kk D 1X P3jD0d˛44kk􀀀j˛jk D 1  \nD  \n1  \nX  \nk D 1  \nbk  \n.􀀀4/k ;  \nwith the possibilities for bk D P3jD0 d4k􀀀j˛j being  \nŒ0000􀂍 ˛ D 0; Œ0001􀂍 ˛ D 1; Œ0010􀂍 ˛ D 􀀀 1Ci; Œ0011􀂍˛ D i;  \nŒ0100􀂍 ˛ D 􀀀 2i; Œ0101􀂍 ˛ D 1􀀀2i; Œ0110􀂍˛ D 􀀀 1􀀀i; Œ0111􀂍˛ D 􀀀 i;  \nŒ1000􀂍 ˛ D 2C2i; Œ1001􀂍˛ D 3C2i; Œ1010􀂍˛ D 1C3i; Œ1011􀂍˛ D 2C3i;  \nŒ1100􀂍 ˛ D 2; Œ1101􀂍 ˛ D 3; Œ1110􀂍 ˛ D 1Ci; Œ1111􀂍˛ D 2Ci: In other words, we have  \nK D ²1X .k4/k W bk 2 D³ ;  \nk D 1  \nwith  \nD D ¹􀀀1􀀀i; 􀀀1Ci; 􀀀2i; 􀀀i; 0; i; 1􀀀2i; 1; 1Ci; 1C3i; 2; 2Ci; 2C2i; 2C3i; 3; 3C2iº:  \nPoints in the intersection of K with lines 􀂁p;q;r D ¹x C iy W px C qy D rº can now be characterized by their digit expansion in the following way.  \nIntersecting the Twin Dragon with rational lines 207  \nFigure 1. An automaton characterizing @K (in base ˛), where all states are i","cbCaiuOV8ZFLyEXo","https://ap.wps.com/l/cbCaiuOV8ZFLyEXo","pdf",483789,13,"English","# Introduction\n# Main statement and proof\n## Canonical number systems and the Knuth Twin Dragon\n## Digit expansions and rational line intersections\n## Büchi automata and boundary characterization","[{\"question\":\"What is the main geometric object studied in this paper?\",\"answer\":\"The paper studies the Knuth Twin Dragon, a compact fractal subset of the plane with a fractal boundary.\"},{\"question\":\"How do intersections with rational lines differ from generic lines?\",\"answer\":\"Although intersections with Lebesgue almost all lines typically have Hausdorff dimension reduced by 1, the paper proves that this dimension is never attained for lines whose parameters are rational.\"},{\"question\":\"How is the intersection with rational lines analyzed?\",\"answer\":\"The authors characterize the digit expansions of points in the intersection and use a Büchi automaton to describe which expansions correspond to boundary intersection points, enabling Hausdorff-dimension calculations.\"}]","Intersecting the Twin Dragon with rational lines | PDF",33]