[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"detail-sidebar-cat-0-en-105":3,"doc-seo-142610-105":59,"doc-detail-142610-en":124},{"code":4,"msg":5,"data":6},0,"success",[7,13,18,23,28,33,38,43,48,51,55],{"id":8,"doc_module":4,"doc_module_name":9,"category_name":10,"show_sort_weight":11,"slug":12},1,"Document","Story & Novel",90,"story-novel",{"id":14,"doc_module":4,"doc_module_name":9,"category_name":15,"show_sort_weight":16,"slug":17},2,"Literature",80,"literature",{"id":19,"doc_module":4,"doc_module_name":9,"category_name":20,"show_sort_weight":21,"slug":22},4,"Exam",70,"exam",{"id":24,"doc_module":4,"doc_module_name":9,"category_name":25,"show_sort_weight":26,"slug":27},5,"Comic",60,"comic",{"id":29,"doc_module":4,"doc_module_name":9,"category_name":30,"show_sort_weight":31,"slug":32},6,"Technology",50,"technology",{"id":34,"doc_module":4,"doc_module_name":9,"category_name":35,"show_sort_weight":36,"slug":37},7,"Healthcare",40,"healthcare",{"id":39,"doc_module":4,"doc_module_name":9,"category_name":40,"show_sort_weight":41,"slug":42},8,"Research & Report",30,"research-report",{"id":44,"doc_module":4,"doc_module_name":9,"category_name":45,"show_sort_weight":46,"slug":47},9,"Religion & Spirituality",20,"religion-spirituality",{"id":46,"doc_module":4,"doc_module_name":9,"category_name":49,"show_sort_weight":46,"slug":50},"World Cup","world-cup",{"id":52,"doc_module":4,"doc_module_name":9,"category_name":53,"show_sort_weight":52,"slug":54},10,"Lifestyle","lifestyle",{"id":56,"doc_module":4,"doc_module_name":9,"category_name":57,"show_sort_weight":24,"slug":58},19,"General","general",{"code":4,"msg":60,"data":61},"ok",{"site_id":62,"language":63,"slug":64,"title":65,"keywords":66,"description":67,"schema_data":68,"social_meta":117,"head_meta":119,"extra_data":121,"updated_unix":123},105,"en","bridges-2024-conference-proceedings-3d-weaving-patterns-and-modular-kirigami","Bridges 2024 Conference Proceedings - 3D Weaving Patterns and Modular Kirigami","","The work extends weaving from two-dimensional fabric to three-dimensional space by defining and constructing infinite 3D weaving patterns using skew polygons and systematic translations. It presents a formal framework for when such polygons generate valid patterns with regular over/under interlacing and fully connected structure. The paper also introduces modular kirigami techniques that turn the weaving patterns into templates for building intricate 3D models and structures.",{"@graph":69,"@context":116},[70,84,99],{"@type":71,"itemListElement":72},"BreadcrumbList",[73,77,79,82],{"item":74,"name":75,"@type":76,"position":8},"https://docshare.wps.com","Home","ListItem",{"item":78,"name":9,"@type":76,"position":14},"https://docshare.wps.com/document/",{"item":80,"name":40,"@type":76,"position":81},"https://docshare.wps.com/document/research-report/",3,{"item":83,"name":65,"@type":76,"position":19},"https://docshare.wps.com/document/bridges-2024-conference-proceedings-3d-weaving-patterns-and-modular-kirigami/142610/",{"url":83,"name":65,"@type":85,"author":86,"headline":65,"publisher":89,"fileFormat":92,"inLanguage":63,"description":67,"dateModified":93,"datePublished":93,"encodingFormat":92,"isAccessibleForFree":94,"interactionStatistic":95},"DigitalDocument",{"name":87,"@type":88},"Riley","Person",{"url":74,"name":90,"@type":91},"DocShare","Organization","application/pdf","2026-08-25",true,{"@type":96,"interactionType":97,"userInteractionCount":4},"InteractionCounter",{"@type":98},"ViewAction",{"@type":100,"mainEntity":101},"FAQPage",[102,108,112],{"name":103,"@type":104,"acceptedAnswer":105},"What is the main idea behind extending weaving from 2D to 3D?","Question",{"text":106,"@type":107},"The paper generalizes flat weaving by using skew polygons whose edges generate 3D pattern lines via translations. Threads following those lines can interlace over/under in different planes and connect through shared intersections.","Answer",{"name":109,"@type":104,"acceptedAnswer":110},"How does the paper define a 3D weaving pattern in terms of a generating polygon?",{"text":111,"@type":107},"It defines a set of lines closed under “pattern translations” for the edges of a skew polygon P with a parameter k>1. The polygon generates a valid pattern only when intersections occur exactly according to adjacency of edges.",{"name":113,"@type":104,"acceptedAnswer":114},"How are modular kirigami methods used in relation to the weaving patterns?",{"text":115,"@type":107},"In the presented approach, threads are replaced by modular pieces of the same shape. The resulting weaving pattern acts as a template for assembling complex modular 3D structures.","https://schema.org",{"og:url":83,"og:type":118,"og:title":65,"og:site_name":90,"og:description":67},"article",{"robots":120,"canonical":83},"index,follow",{"doc_id":122,"site_id":62},142610,1787684954,{"code":4,"msg":5,"data":125},{"doc_id":122,"user_id":126,"nickname":87,"user_avatar":127,"doc_module":4,"category_id":39,"category_name":40,"doc_title":65,"doc_description":67,"doc_content":128,"file_id":129,"file_url":130,"file_type":131,"file_size":132,"view_count":4,"is_deleted":4,"is_public":8,"is_downloadable":8,"audit_status":8,"page_count":39,"language":133,"language_code":63,"site_id":62,"html_lang":63,"table_of_contents":134,"faqs":135,"seo_title":136,"seo_description":67,"update_tm":123,"read_time":46},1374391975076,"https://ap-avatar.wpscdn.com/avatar/14000253ca4ec9f6853?x-image-process=image/resize,m_fixed,w_180,h_180&k=1783305029341752051","Bridges 2024 Conference Proceedings  \n3D Weaving Patterns and Modular Kirigami  \nMircea Draghicescu  \nPortland, Oregon, USA; [mircead3@gmail.com.com](mircead3@gmail.com.com)  \nAbstract  \nWe extend the concept of weaving from the plane to the three dimensional space, describe several in􀀜nite weaving patterns and illustrate them with many models. We also describe modular kirigami techniques that can be used to build interesting 3D structures based on these patterns.  \nIntroduction  \nIn weaving, threads parallel to two or more axes in the plane are interlaced to form a fabric. Two such weaving patterns are known: the biaxial (orplain weave)pattern, with two axes and the triaxial (or hexagonal, kagome) pattern with three axes, see Figure 1 (a),(b) . Di􀀛erent designs can be obtained by changing the way in which the threads cross each other; in the simplest case, crossings alternate in an over/under manner.  \nThere are many ways to extend the concept of weaving from 2D to 3D. A 􀀝at weave can be bent to make baskets and other three-dimensional objects. More generally,[1] shows how the kagome pattern can be used to weave complex, three-dimensional surfaces, such as the ones showcased in [9] . Weaving polyhedra surfaces that are topological spheres is explored in [4] and [5] . The surface of a polyhedron whose vertices have degree 4 can be woven with threads that follow the edges of the polyhedron; the threads can always be arranged to cross each other alternately over/under. In [2] we examine a class of polyhedra that can be woven in this way with a single, self-intersecting, loop. Related to 3D weaving is the crystallography topic of cylinder (or rod) packing, see, for example,[10],[11] . Symmetric 3D weaves of helices are described in [14]; two examples are shown in [13] . A technique for generating 3D weaves by projecting line arrays in the hyperbolic plane onto triply periodic minimal surfaces is described in [3] .  \nNew, evolving, manufacturing methods can stitch together several layers of fabric into a 􀀐thick􀀑 weave by adding vertical strands to the warp and weft strands ofthe 2D weaving. These technologies have engineering applications that extend beyond the textile industry, see, for example,[7], [8],[12] .  \nWe will de􀀜ne3D weaving herein away that directly generalizes􀀝at weaving. While the lines abstracting the threads of the two 2D weaving patterns can be generated by translating a (􀀝at) polygon, the lines of a 3D weaving pattern are generated by translating a skew polygon (a polygon whose vertices do not lie in a plane) . Threads following these lines can be interlaced, over/under, in di􀀛erent planes and intersecting planes are connected by their common thread. The result is a fully connected, conceptually space-􀀜lling, weave.  \nWe illustrate the 3D weaving patterns with various models, including paper models built using modular kirigami methods. In one of these methods, threads are replaced with same shape pieces and a weaving pattern becomes a template for building complex, modular 3D structures, see Figure 1 (c) .  \nWeaving Patterns  \nInformally, a weaving pattern is an in􀀜nite set of lines that are parallel to a (􀀜nite) number of axes and intersect each other in a regular manner. A weaving pattern will be de􀀜ned with reference to a generating polygon. Not all polygons generate weaving patterns; we will impose a condition that ensures that only two lines meet at each intersection point and no pattern lines meet at some 􀀐extraneous􀀑 intersection point.  \n(a) 2D biaxial (b) 2D triaxial, image from [1] (c) Modular kirigami model based  \non a 3D four-axial pattern  \nFigure 1: Weaving patterns.  \nFormally, let P be a skew polygon such that any two parallel edges, if they exist, have the same length and let k > 1. For an edge e of P denote by |e| the length of e and by  the line extending e. A translation along , in either direction, by distance k |e| will be called a pattern translation; let L (P, k) be the closure, under pa","cbCaib6qRSedDjeh","https://ap.wps.com/l/cbCaib6qRSedDjeh","pdf",3263521,"English","# Abstract\n# Introduction\n# Weaving Patterns","[{\"question\":\"What is the main idea behind extending weaving from 2D to 3D?\",\"answer\":\"The paper generalizes flat weaving by using skew polygons whose edges generate 3D pattern lines via translations. Threads following those lines can interlace over/under in different planes and connect through shared intersections.\"},{\"question\":\"How does the paper define a 3D weaving pattern in terms of a generating polygon?\",\"answer\":\"It defines a set of lines closed under “pattern translations” for the edges of a skew polygon P with a parameter k\\u003e1. The polygon generates a valid pattern only when intersections occur exactly according to adjacency of edges.\"},{\"question\":\"How are modular kirigami methods used in relation to the weaving patterns?\",\"answer\":\"In the presented approach, threads are replaced by modular pieces of the same shape. The resulting weaving pattern acts as a template for assembling complex modular 3D structures.\"}]","Bridges 2024 Conference Proceedings - 3D Weaving Patterns and Modular Kirigami | PDF"]