Aperiodic tilings and substitutions

Transcription

1 Aperioi tilings n sustitutions Niols Ollinger LIFO, Université Orléns Journées SDA2, Amiens June 12th, 2013

2 The Domino Prolem (DP) Assume we re given finite set of squre pltes of the sme size with eges olore, eh in ifferent mnner. Suppose further there re infinitely mny opies of eh plte (plte type). We re not permitte to rotte or reflet plte. The question is to fin n effetive proeure y whih we n eie, for eh given finite set of pltes, whether we n over up the whole plne (or, equivlently, n infinite qurnt thereof) with opies of the pltes sujet to the restrition tht joining eges must hve the sme olor. (Wng, 1961) 1/35

3 Wng tiles A tile set τ Σ 4 is tile set with olore eges. The set of τ-tilings X τ τ Z2 is the set of olorings of Z 2 y τ where olors mth long eges. 2/35

4 Perioi Tilings Definition A tiling is perioi with perio p if it is invrint y trnsltion of vetor p. Lemm If tile set mits perioi tiling then it mits iperioi tiling. Lemm Tile sets tiling the plne iperioilly re re (reursively enumerle). 3/35

5 o-tiling Lemm Tile sets tiling the plne re o-re. Sketh of the proof Consier tilings of lrger n lrger squre regions. If the set oes not tile the plne, y ompity, there exists size of squre it nnot over with tiles. 4/35

6 Aperioiity Definition A tiling is perioi if it mits no non-trivil perio. Definition A tile set is perioi if it mits tiling n ll its tilings re perioi. Remrk If there were no perioi tile set, the Domino Prolem woul e eile. 5/35

7 Uneiility of DP Theorem[Berger 1964] DP is uneile. Remrk To prove it one nees perioi tile sets. Seminl self-similrity se proofs (reution from HP): Berger, 1964 (20426 tiles, full PhD thesis) Roinson, 1971 (56 tiles, 17 pges, long se nlysis) Durn et l, 2007 (Kleene s fixpoint existene rgument) Tiling rows seen s trnsuer tre se proof: Kri, 2007 (ffine mps, reution from IP) An others! Mozes, 1990 (non-eterministi sustitutions) Aner n Lewis, 1980 (1-systems n 2-systems) 6/35

8 In this tlk A simple originl onstrution of n perioi tile set se on two-y-two sustitution systems n its pplition to n ol historil onstrution. This work omines tools n ies from: [Berger 64] The Uneiility of the Domino Prolem [Roinson 71] Uneiility n nonperioiity for tilings of the plne [Grünum Shephr 89] Tilings n Ptterns, n introution [Durn Levin Shen 05] Lol rules n glol orer, or perioi tilings 7/35

9 Tiling with fixe tile q q q q q q B q 0 B B No hlting tile. 8/35

10 Finite Tiling q f q q q q q q B q 0 B B 9/35

11 1. Two-y-two Sustitution Systems 2. An Aperioi Tile Set 3. Conlusion

12 Sustitutions Σ = {,,, } s :,,,. 1. Two-y-two Sustitution Systems 10/35

13 Two-y-two sustitutions s : A 2x2 sustitution s : Σ Σ mps letters to squres of letters on the sme finite lphet. S : The sustitution is extene s glol mp S : Σ Z2 Σ Z2 on olorings of the plne: z Z 2, k, S()(2z + k) = s((z))(k) 1. Two-y-two Sustitution Systems 11/35

14 Limit set n history Λ s = y x x,y Z 2 The limit set Λ s Σ Z2 is the mximl ttrtor of S: Λ s = ( S t Σ Z2) σ t N The limit set is the set of olorings mitting n history ( i ) i N where i = σ ui (S( i+1 )). 1. Two-y-two Sustitution Systems 12/35

15 Unmiguous sustitutions A sustitution is perioi if its limit set Λ s is perioi. A sustitution is unmiguous if, for every oloring from its limit set Λ s, there exists unique oloring n unique trnsltion u stisfying = σ u (S( )). Proposition Unmiguity implies perioiity. Sketh of the proof. Consier perioi oloring with miniml perio p, its preimge hs perio p/2. Ie. Construt tile set whose tilings re in the limit set of n unmiguous sustitution system. 1. Two-y-two Sustitution Systems 13/35

16 Coing tile sets into tile sets Definition A tile set τ oes tile set τ, oring to oing rule t : τ τ if t is injetive n X τ = {σ u (t()) X τ, u } : {,,, } 1. Two-y-two Sustitution Systems 14/35

17 Unmiguous self-oing Definition A tile set τ oes sustitution s : τ τ if it oes itself oring to the oing rule s. Proposition A tile set oth mitting tiling n oing n unmiguous sustitution is perioi. Sketh of the proof. X τ Λ s n X τ. Ie. Construt tile set whose tilings re in the limit set of lolly hekle unmiguous sustitution emeing whole history. 1. Two-y-two Sustitution Systems 15/35

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28 Is this self-enoing? Iterting the oing rule one otins 56 tiles. oing rule Unfortuntely, this tile set is not self-oing. Ie A synhronizing sustitution s thir lyer. 1. Two-y-two Sustitution Systems 17/35

29 à l Roinson Proposition The ssoite tile set of 104 tiles mits tiling n oes n unmiguous sustitution. 1. Two-y-two Sustitution Systems 18/35

30 à l Roinson Proposition The ssoite tile set of 104 tiles mits tiling n oes n unmiguous sustitution. 1. Two-y-two Sustitution Systems 18/35

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32 Aperioiity: sketh of the proof 1. The tile set mits tiling: Generte vli tiling y iterting the sustitution rule: X τ Λ s. 2. The sustitution is unmiguous: It is injetive n the projetors hve isjoine imges. 3. The tile set oes the sustitution: () eh tiling is n imge of the nonil sustitution Consier ny tiling, level y level, short se nlysis. () the preimge of tiling is tiling Strightforwr y onstrution (preimge remove onstrints). 1. Two-y-two Sustitution Systems 20/35

33 1. Two-y-two Sustitution Systems 2. An Aperioi Tile Set 3. Conlusion

34 Roert Berger (orn 1938) is known for inventing the first perioi tiling using set of 20,426 istint tile shpes. [Roert Berger Wikipei entry]

35 (... ) In 1966 R. Berger isovere the first perioi tile set. It ontins 20,426 Wng tiles, (... ) Berger himself mnge to reue the numer of tiles to 104 n he esrie these in his thesis, though they were omitte from the pulishe version (Berger [1966]). (... ) [GrSh, p.584]

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41 Berger s skeleton sustitution 2. An Aperioi Tile Set 27/35

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43 Berger s forgotten perioi tile set Proposition The ssoite tile set of 103 tiles mits tiling n oes n unmiguous sustitution. Remrk The numer of tiles oes not grow monotonilly in the numer of letters of the synhronizing lyer. 5 letters 104 tiles 11 letters 103 tiles 2. An Aperioi Tile Set 29/35

44 1. Two-y-two Sustitution Systems 2. An Aperioi Tile Set 3. Conlusion

45 To ontinue... Theorem The limit set of 2x2 sustitution is sofi. Ie To enoe Λ s vi lol mthing rules, eorte s into lolly hekle s emeing whole history. Corollry[Berger 1964] DP is uneile. Ie Construt 2x2 sustitution whose limit set ontins everywhere squres of lrger n lrger size, insert Turing omputtion insie those squres. 3. Conlusion 30/35

46 Enforing sustitutions vi tilings Let π mp every tile of τ(s ) to s ()(u) where n u re the letter n the vlue of on lyer 1. Proposition. Let s e ny sustitution system. The tile set τ(s ) enfores s : π ( X τ(s )) = Λs. Ie. Every tiling of τ(s ) oes n history of s n every history of s n e enoe into tiling of τ(s ). = s() = s() ( 1 1 ) ( 0 0 ) 3. Conlusion 31/35

47 Squres everywhere t t h h v v t t h h v v t t t t t t v v t t v v t t v v t t v v h h t t h h t t h h t t h h t t t t v v t t v v t t v v t t v v h h t t h h t t h h t t h h t t t h v 3. Conlusion 32/35

48 x3,1 R5,1 (3, 1) x2,1 R5,1 (2, 1) x1,1 R5,1 (1, 1) q3,1 R4,1 (3, 1) q2,1 R4,1 (2, 1) q1,1 R4,1 (1, 1) j2,1 R3,1 (2, 1) j1,1 R3,1 (1, 1) 3,1 R2,1 (3, 1) 2,1 R2,1 (2, 1) 1,1 R2,1 (1, 1) v2,1 R1,1 (2, 1) v1,1 R1,1 (1, 1) 2,1 S2,1 1,1 S2,1 3,1 S1,1 2,1 S1,1 1,1 S1,1 x3,2 R5,1 (3, 2) x2,2 R5,1 (2, 2) x1,2 R5,1 (1, 2) q3,2 R4,1 (3, 2) q2,2 R4,1 (2, 2) q1,2 R4,1 (1, 2) j2,2 R3,1 (2, 2) j1,2 R3,1 (1, 2) 3,2 R2,1 (3, 2) 2,2 R2,1 (2, 2) 1,2 R2,1 (1, 2) v2,2 R1,1 (2, 2) v1,2 R1,1 (1, 2) x3,3 R5,1 (3, 3) x2,3 R5,1 (2, 3) x1,3 R5,1 (1, 3) q3,3 R4,1 (3, 3) q2,3 R4,1 (2, 3) q1,3 R4,1 (1, 3) j2,3 R3,1 (2, 3) j1,3 R3,1 (1, 3) 3,3 R2,1 (3, 3) 2,3 R2,1 (2, 3) 1,3 R2,1 (1, 3) v2,3 R1,1 (2, 3) v1,3 R1,1 (1, 3) 2,1 1,1 g3,1 R5,2 (3, 1) g2,1 R5,2 (2, 1) g1,1 R5,2 (1, 1) z3,1 R4,2 (3, 1) z2,1 R4,2 (2, 1) z1,1 R4,2 (1, 1) s2,1 R3,2 (2, 1) s1,1 R3,2 (1, 1) l3,1 R2,2 (3, 1) l2,1 R2,2 (2, 1) l1,1 R2,2 (1, 1) e2,1 R1,2 (2, 1) e1,1 R1,2 (1, 1) 2,2 S2,1 1,2 S2,1 3,2 S1,1 2,2 S1,1 1,2 S1,1 g3,2 R5,2 (3, 2) g2,2 R5,2 (2, 2) g1,2 R5,2 (1, 2) z3,2 R4,2 (3, 2) z2,2 R4,2 (2, 2) z1,2 R4,2 (1, 2) s2,2 R3,2 (2, 2) s1,2 R3,2 (1, 2) l3,2 R2,2 (3, 2) l2,2 R2,2 (2, 2) l1,2 R2,2 (1, 2) e2,2 R1,2 (2, 2) e1,2 R1,2 (1, 2) p3,1 R5,3 (3, 1) p2,1 R5,3 (2, 1) p1,1 R5,3 (1, 1) i3,1 R4,3 (3, 1) i2,1 R4,3 (2, 1) i1,1 R4,3 (1, 1) 2,1 R3,3 (2, 1) 1,1 R3,3 (1, 1) u3,1 R2,3 (3, 1) u2,1 R2,3 (2, 1) u1,1 R2,3 (1, 1) n2,1 R1,3 (2, 1) n1,1 R1,3 (1, 1) 2,1 S2,2 1,1 S2,2 3,1 S1,2 2,1 S1,2 1,1 S1,2 p3,2 R5,3 (3, 2) p2,2 R5,3 (2, 2) p1,2 R5,3 (1, 2) i3,2 R4,3 (3, 2) i2,2 R4,3 (2, 2) i1,2 R4,3 (1, 2) 2,2 R3,3 (2, 2) 1,2 R3,3 (1, 2) u3,2 R2,3 (3, 2) u2,2 R2,3 (2, 2) u1,2 R2,3 (1, 2) n2,2 R1,3 (2, 2) n1,2 R1,3 (1, 2) 2,2 1,2 y3,1 R5,4 (3, 1) y2,1 R5,4 (2, 1) y1,1 R5,4 (1, 1) r3,1 R4,4 (3, 1) r2,1 R4,4 (2, 1) r1,1 R4,4 (1, 1) k2,1 R3,4 (2, 1) k1,1 R3,4 (1, 1) 3,1 R2,4 (3, 1) 2,1 R2,4 (2, 1) 1,1 R2,4 (1, 1) w2,1 R1,4 (2, 1) w1,1 R1,4 (1, 1) 2,2 S2,2 1,2 S2,2 3,2 S1,2 2,2 S1,2 1,2 S1,2 y3,2 R5,4 (3, 2) y2,2 R5,4 (2, 2) y1,2 R5,4 (1, 2) r3,2 R4,4 (3, 2) r2,2 R4,4 (2, 2) r1,2 R4,4 (1, 2) k2,2 R3,4 (2, 2) k1,2 R3,4 (1, 2) 3,2 R2,4 (3, 2) 2,2 R2,4 (2, 2) 1,2 R2,4 (1, 2) w2,2 R1,4 (2, 2) w1,2 R1,4 (1, 2) y3,3 R5,4 (3, 3) y2,3 R5,4 (2, 3) y1,3 R5,4 (1, 3) r3,3 R4,4 (3, 3) r2,3 R4,4 (2, 3) r1,3 R4,4 (1, 3) k2,3 R3,4 (2, 3) k1,3 R3,4 (1, 3) 3,3 R2,4 (3, 3) 2,3 R2,4 (2, 3) 1,3 R2,4 (1, 3) w2,3 R1,4 (2, 3) w1,3 R1,4 (1, 3) Mozes 1990 or (2, 1) T (2, 1) (2, 2) (2, 1) T (2, 2) (2, 2) or (1, 1) (1, 2) (1, 1) U (2, 6) (1, 2) (3, 1) (3, 2) (3, 1) (3, 2) (2, 1) (2, 2) (2, 1) (2, 2) T (1, 1) T (1, 2) (1, 1) (1, 2) (1, 1) (1, 2) Theorem[Mozes 1990] The limit set of non-eterministi retngulr sustitution (+ some hypothesis) is sofi. 3. Conlusion 33/35

49 Goomn-Struss 1998 Theorem[Goomn-Struss 1998] The limit set of homotheti sustitution (+ some hypothesis) is sofi. 3. Conlusion 34/35

50 Fernique-O 2010 Theorem[Fernique-O 2010] The limit set of omintoril sustitution (+ some hypothesis) is sofi. 3. Conlusion 35/35

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