[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85961-en":3,"doc-seo-85961-105":30,"detail-sidebar-cat-0-en-105":91},{"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":13,"seo_description":14,"update_tm":28,"read_time":29},85961,13056703019404,"Miles","https://ap-avatar.wpscdn.com/davatar_29158cc5080c5b710cf443261637dec0",8,"Research & Report","Which Wallpaper Groups Arise from Tiled Games","Which discrete symmetry groups can arise from strategic interaction is answered through a tiling construction based on the support complex of a bimatrix game. The plane is tiled with copies of the game’s support complex under controlled boundary rules, producing covers on which all seventeen wallpaper groups act. Each realization is certified with machine-verified graph automorphisms and exact toroidal quotients, with symmetry types identified via crystallographic recognition and cross-checked in GAP. A key lemma links symmorphic versus non-symmorphic behavior to lattice translations: full tile-lattice translations occur exactly for the thirteen symmorphic groups, while the four non-symmorphic groups appear only at translation index two.","arXiv :2607 . 10552v 1 [ cs .GT] 12 Jul 2026  \nWhich Wallpaper Groups Arise from Tiled Games?  \nMatthew Fried  \nFarmingdale State College, SUNY  \n[friedm1@farmingdale. edu](friedm1@farmingdale. edu)  \nAbstract  \nWhich discrete symmetry groups can arise from strategic interaction? We tile the plane with copies of a bimatrix game’s support complex, joined by controlled boundary rules, and show that all seventeen wallpaper groups act on the resulting covers: explicit generators, each a machineverified graph automorphism, every realization certified as the exact toroidal quotient, with types identified by a crystallographic recognizer in exact rational arithmetic and cross-validated in GAP. A three-line lemma turns the classical symmorphic/non-symmorphic distinction into a lattice classification: realizations whose translations contain the full tile lattice exist precisely for the thirteen symmorphic groups, and the four non-symmorphic groups are realized at translationlattice index exactly two, the minimum possible: the tile is the glide’s half-step.  \nTwo computational tracks accompany the construction. On the graph track, quotientinga straight cover by its translations recovers the tile exactly, β 1 (M/T ) = β1 (K), and swap boundaries add exactly 􀀀 m2􀀁, independent of payoffs and of cover size. On the game track, detecting a duplicated-strategy cover is a linear-time payoff scan, one tile solution folds to a full translation orbit of cover equilibria, and the tiled correlated-equilibrium system has dimension exactly r(d−q)+q, with expansion impossible. The polymatrix cover then carries the symmetry outright: every wallpaper action, glides included, is a group of genuine game automorphisms, equilibria collapse along any symmetry subgroup to a folded fixed-point problem, and a decorated refinement has game automorphism group exactly the toroidal wallpaper group.  \n1 Introduction  \n1.1 The Realization Question  \nA bimatrix game (A, B) ∈ Rm×n × Rm×n has a natural combinatorial object associated with it: the support complex K (A, B), whose nodes are candidate strategy support pairs (Si, Sj) connected by single-edit (pivot-adjacency) edges. This complex is the Cartesian product of two per-player support-edit graphs, and its first Betti number β 1 (K), the dimension of its cycle space, is a payofffree invariant computed in closed form in Section 2; for generic payoffs the equilibrium-feasible support pairs form a discrete independent set inside K (Appendix E), so every β1 statement in this paper concerns the ambient arena and its symmetries. Separately, computing a Nash equilibrium of a bimatrix game, for which Lemke and Howson [9] gave the classical pivoting algorithm, is PPAD-complete [8, 7] .  \nThe forward problem is familiar: given a game, find its symmetries and exploit them. This paper studies the inverse problem: which symmetry groups are realizable by game-theoretic interaction at all? We answer it for periodic planar interaction. Tiling copies of K(A, B) into a multigame cover Mr,s, with boundary rules deciding how adjacent copies communicate, produces graphs on which wallpaper groups, the 17 crystallographic symmetry groups of the plane, act by automorphisms. The realization theory splits along the classical symmorphic/non-symmorphic line (a wallpaper  \ngroup is symmorphic when an origin can be chosen so that every symmetry is a rotation or reflection followed by a lattice translation; thirteen of the seventeen are, four are not), and the split isnot an accident of our constructions but a theorem about integral affine actions (Lemma 8): a wallpaper group acting by affine maps on the tile grid, with translation subgroup containing the full grid, must be symmorphic. All thirteen symmorphic groups are realized on plain straight covers with verified graph automorphisms (Theorem 7) . A non-symmorphic group must instead place its own translation lattice at proper index inside the tile grid, so that the tile itself serves as the g","cbCaivmnSAOYpt7x","https://ap.wps.com/l/cbCaivmnSAOYpt7x","pdf",457086,4,1,27,"English","en",105,"# Abstract\n# Introduction\n## The Realization Question\n## Symmetry That No Game Possesses","[{\"question\":\"What is the central question about symmetry in this work?\",\"answer\":\"The work asks which discrete symmetry groups can be realized by game-theoretic interaction, specifically through periodic planar interaction constructed from tiled copies of a game's support complex.\"},{\"question\":\"How are wallpaper groups realized from the tiled game construction?\",\"answer\":\"Copies of the game’s support complex are tiled into a multigame cover with boundary rules that control how adjacent copies communicate, yielding graphs where wallpaper group actions act by automorphisms.\"},{\"question\":\"What distinguishes symmorphic from non-symmorphic wallpaper groups in the realizations?\",\"answer\":\"A realization whose translations contain the full tile lattice exists exactly for the thirteen symmorphic groups; the four non-symmorphic groups require a smaller translation lattice at index two, so the tile functions as the glide’s half-step.\"}]",1784207402,68,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":86,"head_meta":88,"extra_data":90,"updated_unix":28},"which-wallpaper-groups-arise-from-tiled-games","",{"@graph":36,"@context":85},[37,53,68],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":20},"https://docshare.wps.com/document/which-wallpaper-groups-arise-from-tiled-games/85961/",{"url":52,"name":13,"@type":54,"author":55,"headline":13,"publisher":57,"fileFormat":60,"inLanguage":24,"description":14,"dateModified":61,"datePublished":62,"encodingFormat":60,"isAccessibleForFree":63,"interactionStatistic":64},"DigitalDocument",{"name":9,"@type":56},"Person",{"url":41,"name":58,"@type":59},"DocShare","Organization","application/pdf","2026-07-26","2026-07-16",true,{"@type":65,"interactionType":66,"userInteractionCount":20},"InteractionCounter",{"@type":67},"ViewAction",{"@type":69,"mainEntity":70},"FAQPage",[71,77,81],{"name":72,"@type":73,"acceptedAnswer":74},"What is the central question about symmetry in this work?","Question",{"text":75,"@type":76},"The work asks which discrete symmetry groups can be realized by game-theoretic interaction, specifically through periodic planar interaction constructed from tiled copies of a game's support complex.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"How are wallpaper groups realized from the tiled game construction?",{"text":80,"@type":76},"Copies of the game’s support complex are tiled into a multigame cover with boundary rules that control how adjacent copies communicate, yielding graphs where wallpaper group actions act by automorphisms.",{"name":82,"@type":73,"acceptedAnswer":83},"What distinguishes symmorphic from non-symmorphic wallpaper groups in the realizations?",{"text":84,"@type":76},"A realization whose translations contain the full tile lattice exists exactly for the thirteen symmorphic groups; the four non-symmorphic groups require a smaller translation lattice at index two, so the tile functions as the glide’s half-step.","https://schema.org",{"og:url":52,"og:type":87,"og:title":13,"og:site_name":58,"og:description":14},"article",{"robots":89,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":92},[93,97,101,105,110,115,120,123,128,131,135],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":94,"show_sort_weight":95,"slug":96},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":98,"show_sort_weight":99,"slug":100},"Literature",80,"literature",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":102,"show_sort_weight":103,"slug":104},"Exam",70,"exam",{"id":106,"doc_module":4,"doc_module_name":46,"category_name":107,"show_sort_weight":108,"slug":109},5,"Comic",60,"comic",{"id":111,"doc_module":4,"doc_module_name":46,"category_name":112,"show_sort_weight":113,"slug":114},6,"Technology",50,"technology",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":121,"slug":122},30,"research-report",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":126,"slug":127},9,"Religion & Spirituality",20,"religion-spirituality",{"id":126,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":126,"slug":130},"World Cup","world-cup",{"id":132,"doc_module":4,"doc_module_name":46,"category_name":133,"show_sort_weight":132,"slug":134},10,"Lifestyle","lifestyle",{"id":136,"doc_module":4,"doc_module_name":46,"category_name":137,"show_sort_weight":106,"slug":138},19,"General","general"]