{"ID":6536113,"CreatedAt":"2026-07-14T01:21:01.169441415Z","UpdatedAt":"2026-07-15T03:28:55.185153975Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2607.10552","arxiv_id":"2607.10552","title":"Which Wallpaper Groups Arise from Tiled Games?","abstract":"Which 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 machine-verified 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 translation-lattice index exactly two, the minimum possible: the tile is the glide's half-step. Two computational tracks accompany the construction. On the graph track, quotienting a straight cover by its translations recovers the tile exactly, $\\beq(M/\\calT)=\\beq(K)$, and swap boundaries add exactly $\\binom m2$, 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.","short_abstract":"Which 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 machine-verified graph automorphism, every r...","url_abs":"https://arxiv.org/abs/2607.10552","url_pdf":"https://arxiv.org/pdf/2607.10552v1","authors":"[\"Matthew Fried\"]","published":"2026-07-12T03:44:20Z","proceeding":"cs.GT","tasks":"[\"cs.GT\",\"math.GR\"]","methods":"[]","has_code":false}