[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-seo-136505-105":3,"detail-sidebar-cat-0-en-105":81,"doc-detail-136505-en":130},{"code":4,"msg":5,"data":6},0,"ok",{"site_id":7,"language":8,"slug":9,"title":10,"keywords":11,"description":12,"schema_data":13,"social_meta":74,"head_meta":76,"extra_data":78,"updated_unix":80},105,"en","intersecting-the-twin-dragon-with-rational-lines","INTERSECTING THE TWIN DRAGON WITH RATIONAL LINES","","Investigates intersections of the Knuth Twin Dragon with rational lines in the complex plane. While generic lines intersect the fractal boundary with Hausdorff dimension s−1, the paper proves that this drop never occurs for lines whose parameters are rational. It characterizes intersections with specific diagonals and axis-parallel lines, and formulates an automaton-based description for the α-expansions of K∩Δp,q,r. The approach uses canonical number systems and Büchi automata to analyze Hausdorff dimensions.",{"@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/136505/",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/136505.png","ImageObject",300,407,{"name":42,"@type":43},"Bintang","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-09-18","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 Knuth Twin Dragon and its boundary dimension?","Question",{"text":63,"@type":64},"The Knuth Twin Dragon is a compact planar set with fractal boundary whose Hausdorff dimension is s = (log λ)/(log √2), where λ satisfies λ^3 = λ^2 + 2.","Answer",{"name":66,"@type":61,"acceptedAnswer":67},"How does the intersection dimension behave for generic versus rational lines?",{"text":68,"@type":64},"For Lebesgue-almost all lines, the intersection with the boundary has Hausdorff dimension s−1. The paper proves that for lines with rational parameters, this value never occurs.",{"name":70,"@type":61,"acceptedAnswer":71},"What tools are used to analyze intersections with rational lines?",{"text":72,"@type":64},"The analysis uses canonical number system representations of points in the Twin Dragon fundamental domain and then describes the relevant α-expansions via a finite Büchi automaton to determine Hausdorff dimensions of intersections.","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},136505,1787387914,{"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},962085564381,"https://ap-avatar.wpscdn.com/davatar_6f874abed73319feea01a86fa6f0fab8","arXiv :2402 . 18371v1 [math .MG] 28 Feb 2024  \nINTERSECTING THE TWIN DRAGON WITH RATIONAL  \nLINES  \nSHIGEKI AKIYAMA, PAUL GROSSKOPF, BENOˆIT LORIDANT AND WOLFGANG  \nSTEINER  \nDedicated to Professor J¨org Thuswaldner on the occasion of his 50th birthday  \nAbstract. The Knuth Twin Dragon is a compact subset of the plane with fractal boundary of Hausdorff dimension s = (log λ)/(log √2), λ3 = λ2 + 2.  \nAlthough 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.  \n1. Introduction  \nWe investigate the intersections of the Knuth Twin Dragon with rational lines. Let α = −1 + i, then  \nK = ( dαkk : dk ∈ {0, 1})  \nis the Knuth Twin Dragon. The Hausdorff dimension of its boundary ∂K iss = l~~l~~o~~o~~g~~g~~2 ≈ 1.5236, where λ is the real number satisfying λ3 = λ2 + 2 . For lines (1.1) ∆p,q,r = {x + iy ∈ C : px + qy = r}  \nwith p, q, r ∈ Z, we show that the α-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) ∈ R3 for which ∆p,q,r ∩ K  ∅, the exceptional cases form a 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 Shigeki Akiyama and Klaus Scheicher [1] and add uncountably many examples of horizontal, vertical, and diagonal lines.  \nWe mention that similar results were obtained in [4] for lines intersecting the Sierpinski carpet F. The set F has Hausdorff dimension ~~log 8~~log~~ ~~3 . Manning and Simon showed that, given a slope α ∈ Q, the intersection of F with the line y = αx + β is strictly less than ~~log 8~~log~~ ~~3 − 1 for Lebesgue almost every β .  \nDate: February 29, 2024 .  \nKey words and phrases. Number system, Hausdorff dimension.  \n2 S. AKIYAMA, P. GROSSKOPF, B. LORIDANT AND W. STEINER  \n2. Main statement and proof  \nWe first recall the notions of a canonical number system and its fundamental domain. Let β be an algebraic integer and N = {0, 1 ,..., |N(β)| − 1}, where N (x) denotes the norm of x over Q (β)/Q. The pair (β, N) is called a canonical number system (CNS) if each γ ∈ Z[β] admits a representation of the form  \nn  \n(2.1) γ = X dkβk , dk ∈ N.  \nk=0  \nWe call β 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 (α, N), where α = −1+i is the root of the polynomial x2 −2x−2 and N = {0, 1} . The fundamental domain of a CNS is the set of all numbers that can be expressed with purely negative exponents. Since α4 = −4, it is often useful to consider groups of four digits:  \n∞X dαkk = ∞X P3j=0d4kα4k−jαj = ∞X (~~ ~~k4)k ,  \nk=1 k=1 k=1  \nwith the possibilities for bk = P3j=0 d4k−jαj being  \n[0000]α = 0 , [0001]α = 1 , [0010]α = −1+i, [0011]α = i,[0100]α = −2i, [0101]α = 1−2i, [0110]α = −1−i, [0111]α = −i,[1000]α = 2+2i, [1001]α = 3+2i, [1010]α = 1+3i, [1011]α = 2+3i,[1100]α = 2 , [1101]α = 3 , [1110]α = 1+i, [1111]α = 2+i.  \nIn other words, we have  \nK = ( (~~ ~~k4)k : bk ∈ D) ,  \nwith  \nD = {−1−i,−1+i,−2i,−i,0, i,1−2i,1, 1+i,1+3i,2, 2+i,2+2i,2+3i,3, 3+2i} Points in the intersection of K with lines ∆p,q,r = {x+iy : px+qy = r} can now be characterized by their digit expansion in the following way.  \nLemma 2.1 . We have z ∈ K ∩ ∆p,q,r if and only if there is a digit sequence b 1 b2 · · · ∈ DN with  \nz = ∞X (~~ ~~k4)k~~ ~~ and r = ∞X pR(bk(~~ ~~qk~~ ~~I~~ ~~(bk) . k=1 k=1  \nHere, R (b) denotes the real part and I (b) denotes the imaginary part of b ∈ C. We will show that we can characterize the digit expansion of the points in the intersection ∆p,q,r ∩ K via","cbCaigdz7XpBDXsB","https://ap.wps.com/l/cbCaigdz7XpBDXsB","pdf",562936,11,"English","# Introduction\n## Intersection behavior for rational lines\n# Main statement and proof\n## Canonical number systems and the Twin Dragon\n## Büchi automata and ω-languages\n## Lemmas for characterizing intersections","[{\"question\":\"What is the Knuth Twin Dragon and its boundary dimension?\",\"answer\":\"The Knuth Twin Dragon is a compact planar set with fractal boundary whose Hausdorff dimension is s = (log λ)/(log √2), where λ satisfies λ^3 = λ^2 + 2.\"},{\"question\":\"How does the intersection dimension behave for generic versus rational lines?\",\"answer\":\"For Lebesgue-almost all lines, the intersection with the boundary has Hausdorff dimension s−1. The paper proves that for lines with rational parameters, this value never occurs.\"},{\"question\":\"What tools are used to analyze intersections with rational lines?\",\"answer\":\"The analysis uses canonical number system representations of points in the Twin Dragon fundamental domain and then describes the relevant α-expansions via a finite Büchi automaton to determine Hausdorff dimensions of intersections.\"}]","INTERSECTING THE TWIN DRAGON WITH RATIONAL LINES | PDF",28]