[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-seo-650975-105":3,"detail-sidebar-cat-0-en-105":80,"doc-detail-650975-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":73,"head_meta":75,"extra_data":77,"updated_unix":79},105,"en","three-dimensional-and-dynamic-constructions-based-on-leonardo-grids","Three-dimensional and dynamic constructions based on Leonardo grids","","Leonardo grids is a bar-grid construction system for generating dome and sphere forms from simple rod or beam elements using one connection rule. The paper extends the system beyond planar, static arrangements to explore non-planar and dynamic possibilities, including stable double-sphere constructions and interwoven patterns that produce real 3D Leonardo grids. It further investigates infinite double-layer space-frame structures with adjustable behavior, and introduces the concepts of rings and strings as larger composite units within the grid.",{"@graph":14,"@context":72},[15,34,55],{"@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/three-dimensional-and-dynamic-constructions-based-on-leonardo-grids/650975/",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/three-dimensional-and-dynamic-constructions-based-on-leonardo-grids/650975.png","ImageObject",300,407,{"name":42,"@type":43},"Ophelia","Person",{"url":19,"name":45,"@type":46},"DocShare","Organization","application/pdf","2026-10-29","2026-10-24",true,{"@type":52,"interactionType":53,"userInteractionCount":26},"InteractionCounter",{"@type":54},"ViewAction",{"@type":56,"mainEntity":57},"FAQPage",[58,64,68],{"name":59,"@type":60,"acceptedAnswer":61},"What is the Leonardo grids bar grid construction system?","Question",{"text":62,"@type":63},"It is a method for building structures from simple line-shaped elements by defining four connecting points on each element and using a strict rule for how endpoints connect to interior points.","Answer",{"name":65,"@type":60,"acceptedAnswer":66},"How does the paper extend Leonardo grids from 2D to 3D?",{"text":67,"@type":63},"It replaces planar-graph representations with non-planar graphs, presenting examples such as a stable sphere-in-sphere double layer and interwoven patterns that fold into spherical constructions.",{"name":69,"@type":60,"acceptedAnswer":70},"What are the infinite double layer structures discussed in the paper?",{"text":71,"@type":63},"They are space-frame-like constructions formed by systematically connecting cube-based Leonardo grids, resulting in a non-planar infinite system with dynamic properties such as sliding and adjustable pressing or stretching.","https://schema.org",{"og:url":32,"og:type":74,"og:title":10,"og:site_name":45,"og:description":12},"article",{"robots":76,"canonical":32},"index,follow",{"doc_id":78,"site_id":7},650975,1792855687,{"code":4,"msg":81,"data":82},"success",[83,87,91,95,100,105,110,114,119,122,126],{"id":22,"doc_module":4,"doc_module_name":25,"category_name":84,"show_sort_weight":85,"slug":86},"Story & Novel",90,"story-novel",{"id":26,"doc_module":4,"doc_module_name":25,"category_name":88,"show_sort_weight":89,"slug":90},"Literature",80,"literature",{"id":33,"doc_module":4,"doc_module_name":25,"category_name":92,"show_sort_weight":93,"slug":94},"Exam",70,"exam",{"id":96,"doc_module":4,"doc_module_name":25,"category_name":97,"show_sort_weight":98,"slug":99},5,"Comic",60,"comic",{"id":101,"doc_module":4,"doc_module_name":25,"category_name":102,"show_sort_weight":103,"slug":104},6,"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":96,"slug":129},19,"General","general",{"code":4,"msg":81,"data":131},{"doc_id":78,"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":26,"is_deleted":4,"is_public":22,"is_downloadable":22,"audit_status":22,"page_count":111,"language":139,"language_code":8,"site_id":7,"html_lang":8,"table_of_contents":140,"faqs":141,"seo_title":142,"seo_description":12,"update_tm":79,"read_time":117},7971461741311,"https://ap-avatar.wpscdn.com/avatar/74000253aff267980c6?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779345379180704826","Three-dimensional and dynamic constructions based on Leonardo grids  \nRinus Roelofs  \nSculptor  \nLansinkweg 28  \n7553AL Hengelo, the Netherlands  \nE-mail: [rinusroelofs@hetnet.nl](rinusroelofs@hetnet.nl)  \nAbstract  \nLeonardo grids is the name I gave to my bar grid construction system with which I was able to construct domes and spheres out of simple elements using one constructing rule. Most of the constructions I made where planar and static.  \nIn this paper I want to focus on the nonplanar and dynamic possibilities of the Leonardo grids.  \n1. Definition of the bar grid system  \nThe Leonardo grid construction system is a way to make constructions using simple line shaped elements, like rods or beams. The system can be described as follows: on each element we define four connecting points, two at or near the ends and the other two somewhere in the middle at a certain distance from eachother. We call these points respectively endpoints and interior points. In this kind of constructions endpoints of one rod may only be connected to midpoints of another rod and vice versa. An example can be seen in Figure 2, a dome structure also represented in the drawing by Leonardo da Vinci in Figure 1 atthe left. The drawing of the stucture (Figure 3), which we shall call a Leonardo grid, can be seen as a graph. And this graph has the property that every vertex has degree 3. For the domes the representing graph is always a planar graph. The name Leonardo grid was chosen because his drawings are the only references I found.  \nFigure 1: Drawing by L. da Vinci. Figure 2: Dome construction. Figure 3: Graph.  \n2. From 2D to 3D  \nAlso the spheres built with the Leonardo grid construction system can be represented by a planar graph. So in fact the sphere is a 2D object. It is no more than a bent surface. In the example (Figure 4) you can see that the elements don't have to be straight sticks. The four connecting points and the connection rule can be recognized in the model, of which the graph is presented in Figure 5. I found many patterns that can be used as Leonardo grids and all these patterns can be used as basic designs for domes and spheres.  \nFigure 4: Sphere-12 elements. Figure 5: Graph. Figure 6: Sphere in sphere.  \nThe question arose whether it would be possible to make real 3D constructions based on the Leonardo grids. Is it possible to make multi layer objects using the Leonardo grids? Or even can we make 3D space frames built out of simple elements just like in the domes? So mathematically speaking instead of planar graphs I had to look for non-planar graphs now.  \nA first example of a real 3D construction is the double sphere in Figure 6. In this object you can see a sphere within a sphere. Each of the 24 elements is half in the inner sphere and half in the outer sphere. The design was made by starting with two layers of Leonardo grids. The layers were placed above eachother in such a way that after cutting each element in two parts, half elements from the upper layer could be connected to half elements of the under layer. The resulting structure again has all the properties needed for a Leonardo grid: each element is connected to 4 other elements in the right way: all endpoint connected to midpoints and all midpoints connected to endpoints. Half of the connecting points are on the surface of the inner sphere, the other half on the surface of the outer sphere. A first non-planar or real 3D result. And surprisingly enough the total structure appeared to be stable.  \nFigure 7: Three interwoven patterns. Figure 8: Icosaherdon-plan.  \nAnother approach is to start with interwoven patterns (Figure 7) . A simple way to transform a flat pattern into a spherical construction is to use a polyhedron. When the pattern is hexagonal, the net of a icosahedron can be used. In the special case of the three interwoven patterns of Figure 8 the cut off elements of pattern A will be connected to the cut off elements of pattern B when folding the net to an","cbCaifrAO10IXhOo","https://ap.wps.com/l/cbCaifrAO10IXhOo","pdf",678244,"English","# Abstract\n# Definition of the bar grid system\n# From 2D to 3D\n# Infinite double layer structures\n# Rings and strings","[{\"question\":\"What is the Leonardo grids bar grid construction system?\",\"answer\":\"It is a method for building structures from simple line-shaped elements by defining four connecting points on each element and using a strict rule for how endpoints connect to interior points.\"},{\"question\":\"How does the paper extend Leonardo grids from 2D to 3D?\",\"answer\":\"It replaces planar-graph representations with non-planar graphs, presenting examples such as a stable sphere-in-sphere double layer and interwoven patterns that fold into spherical constructions.\"},{\"question\":\"What are the infinite double layer structures discussed in the paper?\",\"answer\":\"They are space-frame-like constructions formed by systematically connecting cube-based Leonardo grids, resulting in a non-planar infinite system with dynamic properties such as sliding and adjustable pressing or stretching.\"}]","Three-dimensional and dynamic constructions based on Leonardo grids | PDF"]