[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-81837-en":3,"doc-seo-81837-105":31,"detail-sidebar-cat-0-en-105":92},{"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":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":28,"seo_description":14,"update_tm":29,"read_time":30},81837,4398048950312,"Violet","https://ap-avatar.wpscdn.com/avatar/400002538284de19e3c?_k=1778320343897328908",8,"Research & Report","An Algorithmic Approach for Computing Fundamental Domains of Crystallographic Groups","A crystallographic group Γ is an infinite discrete subgroup of the Euclidean group E(n) that admits a compact fundamental domain, but computing such domains algorithmically is challenging. The work focuses on computing Dirichlet cells that form fundamental domains by deriving the defining half-spaces from group elements acting on Rn. These elements are obtained via bounded-length words in a suitable generating set, yielding an algorithm and enabling construction of topological interlocking assemblies.","arXiv :2607 .02130v1 [math .MG] 2 Jul 2026  \nAn algorithmic approach for computing fundamental domains of  \ncrystallographic groups  \nReymond Akpanya∗† Alice C. Niemeyer‡ Lukas Schnelle §  \nAbstract  \nA crystallographic group is a discrete subgroup of the Euclidean group E(n) that has a compact fundamental domain. Since such a crystallographic group Γ is infinite, computing fundamental domains of Γ is algorithmically challenging. We address this difficulty by targeting the computation of Dirichlet cells that can form fundamental domains of Γ . We show that the half-spaces defining such a Dirichlet cell can be derived from elements of Γ acting on Rn that can be expressed as words of bounded length in a suitable generating set. Based on these results, we design an algorithm for the computation of fundamental domains of crystallographic groups and exploit it to study the construction of topological interlocking assemblies.  \nKeywords: Crystallographic groups, algorithmic group theory, fundamental domains, topological interlocking  \nMSC: 20H15, 52B55, 52C22, 68U05, 20F65  \n1 Introduction  \nOne definition of a crystallographic group Γ ≤ E(n) is that there exists a compact fundamental domain F ⊂ Rn of Γ, a compact subset which contains a system of representatives F ◦ ⊆ V ⊆ F for the natural action of Γ on Rn. Hence, a fundamental domain F of Γ allows us to tile the Euclidean space Rn through the action of Γ on F. This property makes crystallographic groups interesting not only from a purely mathematical perspective but also from an artistic one. For instance, the Dutch artist M. C. Escher exploited crystallographic groups in dimension 2 to create artworks that continue to fascinate both mathematicians and artists to this day (see [9] for examples of his work) . In 1912, Bieberbach proved that there is only a finite number of crystallographic groups Γ ≤ E(n) for every n ∈ N, see [2] . Furthermore, for n ≤ 6 all crystallographic groups of dimension n are known and enumerated (up to isomorphism) . These groups, together with additional structural information such as standard generating sets, can be found in [3] . For instance, for n = 2 there exist exactly 17 crystallographic groups (wallpaper groups) and for n = 3 there are exactly 230 crystallographic groups (space groups) .  \nSince the definition of a crystallographic group requires the existence of a compact fundamental domain, it is natural to ask how such a fundamental domain can be computed for a given crystallographic group Γ ≤ E(n) . This problem is the focus of this paper. If Γ arises from a suitable generating set, we are able to answer this question by presenting an algorithm that computes Dirichlet cells forming fundamental domains of Γ . (The choice of a suitable generating set of Γ is described in Remark 3.5.) ADirichlet cell (also referred to as Voronoi domain) is a convex set in Rn that can be constructed as the  \n∗ RWTH Aachen University, Chair of Algebra and Representation Theory, Pontdriesch 10-12, 52062 Aachen, Germany. Email: [reymond.akpanya@rwth-aachen.de](reymond.akpanya@rwth-aachen.de)  \n†School of Mathematics and Statistics, The University of Sydney, Carslaw Building F07, Camperdown NSW 2006, Australia. E-mail: [reymond.akpanya@sydney.edu.au](reymond.akpanya@sydney.edu.au).  \n‡RWTH Aachen University, Chair of Algebra and Representation Theory, Pontdriesch 10-12, 52062 Aachen, Germany. Email: [alice.niemeyer@momo.math.rwth-aachen.de](alice.niemeyer@momo.math.rwth-aachen.de)  \n§ RWTH Aachen University, Chair of Algebra and Representation Theory, Pontdriesch 10-12, 52062 Aachen, Germany. Email: [lukas.schnelle1@rwth-aachen.de](lukas.schnelle1@rwth-aachen.de), corresponding author  \nintersection of half spaces H + (u, w) := {v ∈ Rn | ∥u − v∥ ≤ ∥w − v∥} between points u and w in Rn , see Definition 2.1 . Loosely speaking, H + (u, w) contains every point v ∈ Rn that is closer to u than tow with respect to the Euclidean norm. For a given crystallographic group Γ ≤ E","cbCaisKuiIbuxH7r","https://ap.wps.com/l/cbCaisKuiIbuxH7r","pdf",6196442,4,1,16,"English","en",105,"# Introduction\n## Dirichlet cells and fundamental domains\n## Algorithmic strategy and key theorem","[{\"question\":\"Why is computing fundamental domains of crystallographic groups algorithmically difficult?\",\"answer\":\"Because a crystallographic group is infinite, the set of points and group elements that must be considered to determine a fundamental domain grows without bound, making direct computation hard.\"},{\"question\":\"What geometric object does the paper compute instead of the whole fundamental domain directly?\",\"answer\":\"The paper computes Dirichlet cells, defined as intersections of half-spaces, and uses them to obtain fundamental domains of the crystallographic group.\"},{\"question\":\"How does the algorithm identify which group elements are needed to determine a Dirichlet cell?\",\"answer\":\"It bounds the necessary elements using a suitable generating set, showing that the relevant half-space-defining elements can be expressed as words of bounded length.\"}]","An Algorithmic Approach for Computing Fundamental Domains of Crystallographic Groups | PDF",1784176548,40,{"code":4,"msg":32,"data":33},"ok",{"site_id":25,"language":24,"slug":34,"title":13,"keywords":35,"description":14,"schema_data":36,"social_meta":87,"head_meta":89,"extra_data":91,"updated_unix":29},"an-algorithmic-approach-for-computing-fundamental-domains-of-crystallographic-groups","",{"@graph":37,"@context":86},[38,54,69],{"@type":39,"itemListElement":40},"BreadcrumbList",[41,45,49,52],{"item":42,"name":43,"@type":44,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":46,"name":47,"@type":44,"position":48},"https://docshare.wps.com/document/","Document",2,{"item":50,"name":12,"@type":44,"position":51},"https://docshare.wps.com/document/research-report/",3,{"item":53,"name":13,"@type":44,"position":20},"https://docshare.wps.com/document/an-algorithmic-approach-for-computing-fundamental-domains-of-crystallographic-groups/81837/",{"url":53,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":24,"description":14,"dateModified":62,"datePublished":63,"encodingFormat":61,"isAccessibleForFree":64,"interactionStatistic":65},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":42,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-07-29","2026-07-16",true,{"@type":66,"interactionType":67,"userInteractionCount":20},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72,78,82],{"name":73,"@type":74,"acceptedAnswer":75},"Why is computing fundamental domains of crystallographic groups algorithmically difficult?","Question",{"text":76,"@type":77},"Because a crystallographic group is infinite, the set of points and group elements that must be considered to determine a fundamental domain grows without bound, making direct computation hard.","Answer",{"name":79,"@type":74,"acceptedAnswer":80},"What geometric object does the paper compute instead of the whole fundamental domain directly?",{"text":81,"@type":77},"The paper computes Dirichlet cells, defined as intersections of half-spaces, and uses them to obtain fundamental domains of the crystallographic group.",{"name":83,"@type":74,"acceptedAnswer":84},"How does the algorithm identify which group elements are needed to determine a Dirichlet cell?",{"text":85,"@type":77},"It bounds the necessary elements using a suitable generating set, showing that the relevant half-space-defining elements can be expressed as words of bounded length.","https://schema.org",{"og:url":53,"og:type":88,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":90,"canonical":53},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":93},[94,98,102,106,111,116,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":47,"category_name":95,"show_sort_weight":96,"slug":97},"Story & Novel",90,"story-novel",{"id":48,"doc_module":4,"doc_module_name":47,"category_name":99,"show_sort_weight":100,"slug":101},"Literature",80,"literature",{"id":20,"doc_module":4,"doc_module_name":47,"category_name":103,"show_sort_weight":104,"slug":105},"Exam",70,"exam",{"id":107,"doc_module":4,"doc_module_name":47,"category_name":108,"show_sort_weight":109,"slug":110},5,"Comic",60,"comic",{"id":112,"doc_module":4,"doc_module_name":47,"category_name":113,"show_sort_weight":114,"slug":115},6,"Technology",50,"technology",{"id":117,"doc_module":4,"doc_module_name":47,"category_name":118,"show_sort_weight":30,"slug":119},7,"Healthcare","healthcare",{"id":11,"doc_module":4,"doc_module_name":47,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":47,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":47,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":47,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":47,"category_name":137,"show_sort_weight":107,"slug":138},19,"General","general"]