[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-82430-en":3,"doc-seo-82430-105":29,"detail-sidebar-cat-0-en-105":90},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},82430,13056703020460,"Valentina","https://ap-avatar.wpscdn.com/avatar/be000253dac470eee5d?_k=1778207105932848923",8,"Research & Report","Overlapping Unfoldings of Cones and Convex Polyhedra","Research on Dürer’s problem studies edge unfoldings of convex polyhedra that avoid overlap. This work reverses the objective by constructing unfoldings that overlap by any prescribed thickness t. Two main results are proved via explicit geometric constructions: with general (non-edge) cuts, some convex polyhedron admits overlap thickness t for all t∈N; with edge unfoldings, for each t there exists a convex polyhedron whose edge unfolding overlaps with thickness t.","Overlapping Unfoldings of Cones and Convex Polyhedra  \nMIT CompGeom Group∗  \nHugo A. Akitaya† Anna Lubiw ‖  \nErik D. Demaine‡ Fabian Frei§ Joseph O’Rourke∗∗  \nStefan Langerman¶  \narXiv :2607 .09606v1 [ cs .CG] 10 Jul 2026  \nAbstract  \nResearch on D¨urer’s problem focuses on edge unfoldingsof convex polyhedra that avoid overlap. We invert the goal and find unfoldings that overlap at some point to any given thickness t.  \nWe have two main results. The first is that, if we allow unfolding cuts that do not follow polyhedron edges, then there is a convex polyhedron that can unfold with overlap of any given thickness. The second result is that for any given thickness, there is a convex polyhedron with an edge unfolding that overlaps to that thickness.  \n1 Introduction  \nD¨urer’s problem [2, Open Problem 21.1] goes back toa treatise by Albrecht D¨urer [3] from 1525, was fully formalized by Shephard [8] in 1975, and has remained unsolved. It asks whether every convex polyhedron hasan edge unfolding (i.e., an unfolding where only cuts along edges are allowed) to a net, that is, to a simple (i.e., nonoverlapping) planar polygon.  \nAmong the strongest positive results is Ghomi’s proof that every convex polyhedron can be stretched affinely into a new polyhedron that has an edge unfolding to anet. A type of inverse to Ghomi’s theorem is the result that every combinatorial convex polyhedron has a metric realization that has an edge unfolding that overlaps [7] .  \nThere are convex polyhedra for which every edge unfolding provides a net—for example, the Platonic solids [4] . On the other hand, there are polyhedra with overlapping edge unfoldings, for example a skinny tetrahedron [6] (or see [2]), and a recently discovered family of convex polyhedra with regular faces [9] . In these examples at most two layers overlap at any point.  \nIn a general unfolding the cuts are not restricted to edges of the polyhedron; the unfolded surface is still required to be flat and connected; equivalently, the cuts  \n∗ Artificial first author to highlight that the other authors (in alphabetical order) worked as an equal group.  \n†U. Mass. Lowell, hugo [akitaya@uml.edu](akitaya@uml.edu)[ ](akitaya@uml.edu)‡MIT, [edemaine@mit.edu](edemaine@mit.edu)  \n§ MIT, [ffrei@mit.edu](ffrei@mit.edu)  \n¶ U. Libre de Bruxelles, [sl@slef.org](sl@slef.org)[ ](sl@slef.org)‖U. Waterloo, [alubiw@uwaterloo.ca](alubiw@uwaterloo.ca)[ ](alubiw@uwaterloo.ca)∗∗ Smith College, [jorourke@smith.edu](jorourke@smith.edu)  \nform a tree that includes all the vertices of the polyhedron. Every convex polyhedron has a non-overlapping general unfolding, specifically, the source or star unfolding [2] .  \nIn this paper, we examine whether overlaps of larger thickness are possible for edge or for general unfoldingsand, if so, what the limits on overlap thickness are. The first question is answered positively by the unfolding of a convex polyhedron in Fig. 1. Before addressing the second question, we define the concept of overlap thickness more formally.  \nGiven an unfolding of a polyhedron to the Euclidean plane, define this unfolding’s thickness of overlap ata point of the plane as the number of distinct, nonboundary points of the surface unfolded to this point (i.e., the number of layers or plies of the unfolding at this point) . More precisely, thickness t at a point p means that a line orthogonal to the plane pierces the interior of the unfolded surface t times. Thickness 0 at a point means that the line misses the interior of the unfolding. The thickness of an unfolding is its maximum thickness across all points. An unfolding of thickness 1 has no overlap, the goal of D¨urer’s problem for any given polyhedron.  \nFigure 1: A rectangular prismoid shown from the top with the red lines showing a cut tree and the resulting unfolding with triple overlap in the yellow diamond. The bottom face is omitted in the picture.  \nWe have two main results: The first concerns general unfoldings of a convex polyhed","cbCaicbW3kZLGjGB","https://ap.wps.com/l/cbCaicbW3kZLGjGB","pdf",9359255,1,7,"English","en",105,"# Introduction\n## Dürer’s problem and edge unfoldings\n## Thickness of overlap\n# Cone Overlap\n## Cone constructions for overlap thickness\n## Lemma 2 and Lemma 3\n# Main Results\n## General unfoldings with arbitrary thickness\n## Edge unfoldings with tight bounds","[{\"question\":\"What is Dürer’s problem in this context?\",\"answer\":\"Dürer’s problem asks whether every convex polyhedron has an edge unfolding to a net, meaning a planar simple polygon formed using cuts only along polyhedron edges.\"},{\"question\":\"How is overlap thickness defined for an unfolding?\",\"answer\":\"Overlap thickness at a point in the plane equals the number of distinct nonboundary points of the unfolded surface that map to that point, equivalently the number of layers pierced by a perpendicular line. The unfolding’s thickness is the maximum over all points.\"},{\"question\":\"What do the main theorems show about achievable overlap thickness?\",\"answer\":\"The results show that for any t∈N, there exists a convex polyhedron with a general unfolding having overlap thickness t. For edge unfoldings, for any t∈N there exists a convex polyhedron whose edge unfolding achieves overlap thickness at least t, with the vertex bound tight up to order via Ω(t) faces/vertices.\"}]",1784180346,18,{"code":4,"msg":30,"data":31},"ok",{"site_id":24,"language":23,"slug":32,"title":13,"keywords":33,"description":14,"schema_data":34,"social_meta":85,"head_meta":87,"extra_data":89,"updated_unix":27},"overlapping-unfoldings-of-cones-and-convex-polyhedra","",{"@graph":35,"@context":84},[36,53,67],{"@type":37,"itemListElement":38},"BreadcrumbList",[39,43,47,50],{"item":40,"name":41,"@type":42,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":44,"name":45,"@type":42,"position":46},"https://docshare.wps.com/document/","Document",2,{"item":48,"name":12,"@type":42,"position":49},"https://docshare.wps.com/document/research-report/",3,{"item":51,"name":13,"@type":42,"position":52},"https://docshare.wps.com/document/overlapping-unfoldings-of-cones-and-convex-polyhedra/82430/",4,{"url":51,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":23,"description":14,"dateModified":61,"datePublished":61,"encodingFormat":60,"isAccessibleForFree":62,"interactionStatistic":63},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":40,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-16",true,{"@type":64,"interactionType":65,"userInteractionCount":4},"InteractionCounter",{"@type":66},"ViewAction",{"@type":68,"mainEntity":69},"FAQPage",[70,76,80],{"name":71,"@type":72,"acceptedAnswer":73},"What is Dürer’s problem in this context?","Question",{"text":74,"@type":75},"Dürer’s problem asks whether every convex polyhedron has an edge unfolding to a net, meaning a planar simple polygon formed using cuts only along polyhedron edges.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"How is overlap thickness defined for an unfolding?",{"text":79,"@type":75},"Overlap thickness at a point in the plane equals the number of distinct nonboundary points of the unfolded surface that map to that point, equivalently the number of layers pierced by a perpendicular line. The unfolding’s thickness is the maximum over all points.",{"name":81,"@type":72,"acceptedAnswer":82},"What do the main theorems show about achievable overlap thickness?",{"text":83,"@type":75},"The results show that for any t∈N, there exists a convex polyhedron with a general unfolding having overlap thickness t. For edge unfoldings, for any t∈N there exists a convex polyhedron whose edge unfolding achieves overlap thickness at least t, with the vertex bound tight up to order via Ω(t) faces/vertices.","https://schema.org",{"og:url":51,"og:type":86,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":88,"canonical":51},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":91},[92,96,100,104,109,114,118,121,126,129,133],{"id":20,"doc_module":4,"doc_module_name":45,"category_name":93,"show_sort_weight":94,"slug":95},"Story & Novel",90,"story-novel",{"id":46,"doc_module":4,"doc_module_name":45,"category_name":97,"show_sort_weight":98,"slug":99},"Literature",80,"literature",{"id":52,"doc_module":4,"doc_module_name":45,"category_name":101,"show_sort_weight":102,"slug":103},"Exam",70,"exam",{"id":105,"doc_module":4,"doc_module_name":45,"category_name":106,"show_sort_weight":107,"slug":108},5,"Comic",60,"comic",{"id":110,"doc_module":4,"doc_module_name":45,"category_name":111,"show_sort_weight":112,"slug":113},6,"Technology",50,"technology",{"id":21,"doc_module":4,"doc_module_name":45,"category_name":115,"show_sort_weight":116,"slug":117},"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":45,"category_name":12,"show_sort_weight":119,"slug":120},30,"research-report",{"id":122,"doc_module":4,"doc_module_name":45,"category_name":123,"show_sort_weight":124,"slug":125},9,"Religion & Spirituality",20,"religion-spirituality",{"id":124,"doc_module":4,"doc_module_name":45,"category_name":127,"show_sort_weight":124,"slug":128},"World Cup","world-cup",{"id":130,"doc_module":4,"doc_module_name":45,"category_name":131,"show_sort_weight":130,"slug":132},10,"Lifestyle","lifestyle",{"id":134,"doc_module":4,"doc_module_name":45,"category_name":135,"show_sort_weight":105,"slug":136},19,"General","general"]