Escher Degree of Non-periodic L-tilings by 2 Prototiles

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1 Origil Pper Form, 27, 37 43, 2012 Esher Degree o No-perioi L-tiligs y 2 Prototiles Kzushi Ahr, Mmi Murt Ao Ojiri Deprtmet o Mthemtis, Meiji Uiversity, Higshi-Mit, Tm-ku, Kwski, Kgw , Jp E-mil ress: hr@mth.meiji..jp (Reeive Ferury 15, 2012; Aepte April 27, 2012) For give tilig o the eulie ple E 2,well the egree o reeom o perture eges o prototiles esher egree. I this pper we osier o-perioi L-tiligs y 2 prototiles oti the esher egree o them. Key wors: L-tilig, No-perioi hierrhil tilig, Esheriztio 1. Itroutio A o-perioi L-tilig is limit o the sequee o pthes s show i Fig. 1. (Ote this is lle hir tilig.) It is well kow tht this is tilig o the eulie ple E 2 tht it hs o perioiity o prllel trsltio. I [1], Sugihr itroues esheriztio o ple tilig. Let T e tilig o E 2.Iwehve iite set S = { 1, 2,, l } o oete regios, eh tile o T is (oriettio preservig) ogruet to oe o 1, 2,, l, the we ll S the protoset 1, 2,, l prototiles. I we pertur some o eges o prototiles get other tilig o the ple, we ll the proess o perturtio esheriztio o the tilig T. This is mous tehique i rtworks o M. C. Esher. For exmple, see Fig. 2. The let igure is tilig y oe prllelogrm. We pertur the horizotl eges slte eges iepeetly s i the right igure. I this pper, we etermie the esher egree o L-tiligs, tht is, the egree o reeom o perture eges o prototiles o L-tiligs. I the protoset o L-tilig osists o oe prototile, the the esher egree is oe. This is show i Theorem A. I the protoset o L-tilig osists o two prototiles, the we show tht there re 6 types o o-trivil tiligs. (A o-trivil tilig is tilig whose esher egree is more th 1.) This is show i Theorem B Theorem C. I [2], Goom-Struss gives perioi protoset erive rom the L-tilig. We hve ew kis o protosets lso rom the L-tilig i these theorems, ut oe o them re perioi. This is show i Appeix B. From the viewpoit o tilig rules, the esher egree my give ew metho to oservig equivlet lsses o tiligs with tilig rules mthig rules. Aout mthig rules sustitutio rules, see [3]. This pper is orgize s ollows. I Setio 2 we prepre some ottios si lemms. I Setio 3 we osier L-tilig y oe prototile. I Setio 4, 5 we osier L-tiligs y two prototiles. I Appeix, we show igures o tiligs. Copyright Soiety or Siee o Form, Jp. 2. Prelimiry I this setio, irst we itroue o-perioi hierrhil tilig. Let S = { 1, 2,, l } e set o oete regios λ > 1ostt. Let i (i = 1, 2,,l)eλ sle-up opy o i. Suppose tht eh i (i = 1, 2,,l) e tile y prototiles 1,, l. Tht is, eh i e ivie ito some o opies o 1,, l. Let i e λ sle-up opy o i. The i the sme wy i e tile y 1,, l. Sustitutig 1,, l ito 1,, l,wehvetilig o i y S. The tilig rules o i sys re lle sustitutio rules. A tilig o E 2 otie y sustitutio rules is lle hierrhil tilig, i it oes t hve o perioiity o prllel trsltio, it is lle o-perioi. A L-tilig is exmple o o-perioi hierrhil tilig. Let = S ={}. Let λ = 2 =, the it gives sustitutio rule. We ll this tilig L-tilig this rule L-sustitutio. DEFINITION 2.1 (s-spread) We ll,pth o oe L-sustitutio rom, 1-spre. We ll (s),pth o s times L-sustitutio rom, s-spre. Next, we eie ege perture ege. DEFINITION 2.2 (EDGE) Let ege e pir o segmet oe-sie eighorhoo. See Fig. 3. Forege, we regr segmet s ege o prototile, oe-sie eighorhoo s isie o prototile. I the sequel, we o ot istiguish ege o prototile rom ege i this eiitio. Next, we eie perture ege. DEFINITION 2.3 (PERTURBED EDGE) For ege, ixig the oth es o the ege perturig it little, we get perture ege. See Fig. 4. For two perture eges,, = ithey re ogruet. I orer to pertur eges o prototiles, there exists restritio o wy o perturig, euse eh prototile must e oete. I our otext, we oly oer egree o reeom o perture prototiles, so we osier oly per- 37

2 38 K. Ahr et l Fig. 1. No-perioi L-tilig. Fig. 6. Deiitio o, 1. Fig. 7. Eges o the prllelogrm. Fig. 2. Perturtio o eges o prllelogrm. Fig. 3. A ege. Fig. 8. Mthig o the tilig (P1). Fig. 4. Exmples o perture eges. Fig. 5. A exmple o prout o perture eges. = ture ege whih is little perture to voi the restritio. Next, we eie prout o perture eges. DEFINITION 2.4 (PRODUCT OF EDGES) Let 1, 2,, k e perture eges. I they re ple o stright lie rom right to let orm row, the we ll it prout o 1, 2,, k we eote this prout y 1 2 k. See Fig. 5. For perture ege, weeie two opertios, 1. For perture ege, is symmetry (right-sielet) imge o. I the sme wy, 1 is upsie-ow imge o. See Fig. 6. It is esy to show the ollowig lemm. LEMMA 2.5 (1) () =,( 1 ) 1 = (2) =,() 1 = 1 1 (3) ( 1 ) = () 1 I tilig, i tile with perture ege other tile with perture ege re eighors t, we hve = 1.Weeote this reltio y.weote sy tht mthes. The ollowig lemm is trivil. LEMMA 2.6 (1) i oly i (2) I the = (3) i oly i Let T e tilig with respet to protoset S. Suppose tht ll prototiles re polygos. Here we ssume tht there is o vertex o tile lyig o ege o other tile. DEFINITION 2.7 (ESCHERIZATION, ESCHER DEGREE) (1) Let T S e s ove. I we pertur eges o prototiles suh tht the perture prototiles give other tilig, we ll this proess esheriztio. (2) I the set o esheriztio o T is prmetrize y some perture eges, the esher egree is the umer o the prmeters. Exmple. Let e prllelogrm (P1) tilig o E 2 s i Fig. 2. Let,,, e eges o s i Fig. 7. From the mthig o the tilig, we hve. Tht is, i we pertur, the the ege hges suh tht = 1, we pertur iepeetly o. The the ege hges suh tht = 1.See Fig. 8. We ll reltios otie rom the tilig property egemthigs.

3 Esher Degree o No-perioi L-tiligs y 2 Prototiles 39 e g h Fig. 9. Prototile. e h g Fig. 11. Perture eges o. e g h h e h g g e h g e e h g l m k j i o p Fig. 10. Mthig i 1-spre o. Fig. 12. Two prototiles. Hee ll esheriztio o the tilig (P1) is prmetrize y eges. Sothe esher egree is Esher Degree o L-tilig y Oe Prototile I this setio we show tht the esher egree o L-tilig y oe prototile is oe. We ssume tht L-igure prototile hs 8 perture eges,,,, h s i Fig. 9. THEOREM 3.1 (THEOREM A) (1) I o-perioi L- tilig y oe prototile,wehve = = e = g = = = h. (2) The esher egree o this tilig is oe. See Fig. 18. REMARK 3.2 = = e = g mes = = e = g, = = = h = = = h, Proo: (1) Cosierig 1-spre, we iretly hve,, h,,, h, g, h (see Fig. 10). This ollows tht = = e = g = = = h. (2) The proo o (1) implies the ollowig lemm. LEMMA 3.3 There exists 1-spre o i oly i stisies = = e = g = = = h. Let e 1-spre,,, h its eges (see Fig. 11). The wehve = h, =, = e = g = e, = = h = g (see Fig. 10). I = = e = g the we = = = h = = e = g esily show tht. (For exmple, h = = = = h = e implies = h = e =, h implies h.) This ollows tht 1-spre o exists, tht is, 2-spre o exists. I the sme wy, i we hve (s 1)-spre (s 1) o with eges (s 1), (s 1),, h (s 1), there exists s-spre, the it stisies tht (s 1) = (s 1) = e (s 1) = g (s 1). I we set (s 1) = (s 1) = (s 1) = h (s 1) (s) = h (s 1) (s 1), (s) = (s 1) (s 1), (s) = e (s) = g (s) = (s 1) e (s 1), (s) = (s) = h (s) = (s 1) g (s 1) iutively, the they stisy (s) = (s) = e (s) = g (s). (s) = (s) = (s) = h (s) it ollows tht 1-spre o (s) exists, tht is, we hve (s + 1)-spre o. I stisies = = e = g = = = h the s-spre (s) exists or y s. Hee the esheriztio is prmetrize y the esher egree is oe. This ompletes the proo. 4. L-tilig y Two Prototiles (1) I this setio, we osier the ses where two prototiles, (Fig. 12) mke oe 1-spre. From Theorem A, i stisies = = e = g = = = h the hs s-spre or s = 1, 2, 3,.Weoserve 5 ptters o i Fig. 13. We ll them o.8, o.4, o.2, o.3, o.5 respetively. (The umerig orer is ot seig or eseig. These umerigs re etermie y the orer o.) A we hve the ollowig theorem. THEOREM 4.1 (THEOREM B) (1) For o.2, o.3, o.4, the esher egree is 1. (2) For o.5, we hve = e = m = =, = g = i = k = o = h = j = l = p the esher egree is 2 (see Fig. 19). (3) For o.8, we hve

4 40 K. Ahr et l. o.8: o.4: o.2: o.3: o.5: Fig. 13. Five ptters o 1-spre. l = = p, k = m = o, = g = h, e = i the esher = j egree is 4 (see Fig. 20). Proo: I stisies = = e = g = = = h,the hs -spre or y. So, it is suiiet to solve the egemthig i = = e = g = = = h. For exmple, or o.5, ege-mthig is give s j i Fig. 14 we hve, i, p, e,, h k, g j, h i. = = e = g p = lm = e = lm implies = = = h k = o = g = o. We solve the system o equtio we hve = e = m = =, = g = i = k = o. Iversely i = h = j = l = p = e = m = =, = g = i = k = o = h = j = l = p the = = e = g = = = h is stisie. For other tiligs, we solve the system o reltios i similr wy. REMARK 4.2 I we hve = = e = g = i = k = m = o, it ollows tht =. = = = h = j = l = = p So we retur the se o oe prototile the esher egree is 1. This mes tht there re o solutio o two istit prototiles. I the two prototiles se, i the esher egree is more th 1, well the tilig o-trivil. There re two types (o.5 o.8) o o-trivil tiligs y two prototiles oe 1-spre. 5. L-tilig y Two Prototiles (2) I this setio, we osier two prototiles, (Fig. 12) mke two 1-spres,. There re 16 possiilities or so sor. We etermie umerig or s i Fig. 15, we represet tilig y pir o two umers or. For exmple, (5, 10) mes tht give i Fig. 16. We remove the se = two trivil ses (0, 15) (15, 0), there remis 238 omitios. I we exhge the role o, wekow tht (i, j) (15 j, 15 i) re equivlet. From the ollowig lemm, we olue tht the umer o remiig omitios is 119. LEMMA 5.1 (i, 15 i) (15 i, i) re equivlet. Proo: I we eote (s) (i, j) (resp. (s) (i, j) ) y the s- spre o (resp. ) otilig (i, j), itisesily show tht (2k 1) (i,15 i) = (2k 1) (15 i,i),(2k 1) (i,15 i) = (2k 1) (15 i,i),(2k) (i,15 i) = (2k) (15 i,i),(2k) (i,15 i) = (2k) (15 i,i) m l j k o i p p e h i o g j e h k g l m Fig. 14. or o.5 tilig. ompletes the proo. Here we hve the thir theorem., or y k = 1, 2,. This THEOREM 5.2 (THEOREM C) (1) I the esher egree o (i, j) tilig is more th 1, the (i, j) = (5, 10), (10, 5), (0, 2), (13, 15), (0, 8), (7, 15), (0, 10), (5, 15). (2) For the tilig (5, 10) (equivletly (10, 5)), = = g = m = j = l = p, e = i = k = o the esher egree is 2 (see Fig. 21). = = h = (3) For the tilig (0, 2) (equivletly (13, 15)), = = e = g = i = k the esher egree is 5 = = = h = j = p (see Fig. 22). (4) For the tilig (0, 8) (equivletly (7, 15)), = = e = g = k = m = o the esher egree = = = h = l = = p is 3 (see Fig. 23). (5) For the tilig (0, 10) (equivletly (5, 15)), = = e = g = k = o = = = h = l = p, m j, i the esher egree is 3 (see Fig. 24). For eh (i, j), wesolve system o equtios o egemthigs o s-spre (s = 1, 2, ). I some ses, oly i (resp. oly j) etermies the result. For exmple, the ollowig lemm hols. LEMMA 5.3 The esher egree o (1, j) is 1 or y j. Proo: Assume tht is o.1 (see Fig. 17). From the ege-mthig o,weget = e = k = i, = h =, g j. Eges o is give y = p, =, = e = e, = = g, g = lm, h = o

5 Esher Degree o No-perioi L-tiligs y 2 Prototiles 41 o.0: o.1: o.2: o.3: o.8: o.9 o.10: o.11: o.4: o.5: o.6: o.7: o.12: o.13: o.14: o.15: Fig. 15. Numerig or. we hve = e = h, =, i the ege-mthig o, we get itiol oitios, =, = o, p g I the ege- hee = = e = k = i = o = = = h =, g j = p. mthig i,weoti other oitio, hee g, e, we hve = = e = g = k = i = o = = = h = j = = p.ithe ege-mthig i,wehve g, hee m g, we l hve = = e = g = i = k = m = o. This implies tht = = = h = j = l = = p withi t most 4-spre ll eges,,, p re relte =. This ompletes the proo. A i similr wy, we show tht the esher egree o y tilig other th (i, j) = (5, 10), (10, 5), (0, 2), (13, 15), (0, 8), (7, 15), (0, 10), (5, 15) is 1. (2) I (i, j) = (5, 10) se, the ege-mthig o is j, i, p, e,, h k, g j, h i (see the let o Fig. 16). The ege-mthig o is k, l, h k, j m, i, p, o, p (see the right o Fig. 16). The ege-mthigs o, re equivlet to = = g = m = j = l = p, e = i = k = o = = h =. Let,,, o, p e eges o, s i Fig. 11, we hve = p, = k, = lm, = o, e = e, = g, g = lm, h = o i = hi, j = j, k = e, l = g, m = lm, = o, o = e, p = g Sie the ormul () hols or,,wehve = = g = m = j = l = p, e = i = k = o = = h =. For exmple, p = l = m implies = p = lm =, so o. These re ege-mthig o 2-spre,. Iutively, we oserve s ollows. Suppose tht we hve (s), (s). Let (s), (s),, o (s), p (s) e eges o (s), (s). = m l j k o i p p e h i o g j e h k g l m = Fig. 16. Tilig (5, 10). e g k l j h k i h l j m p g o i m p o e e g h = p =? e h i o g j e h k g l m (1,j ) se Fig. 17. (1, j) tilig. The reltio etwee (s 1) s (s) s re give y (s) = p (s 1) (s 1), (s) = (s 1) k (s 1), (s) = l (s 1) m (s 1), (s) = (s 1) o (s 1), e (s) = (s 1) e (s 1), (s) = (s 1) g (s 1), g (s) = l (s 1) m (s 1), h (s) = (s 1) o (s 1) i (s) = h (s 1) i (s 1), j (s) = j (s 1) (s 1), k (s) = (s 1) e (s 1), l (s) = (s 1) g (s 1), m (s) = l (s 1) m (s 1), (s) = (s 1) o (s 1), o (s) = (s 1) e (s 1), p (s) = (s 1) g (s 1), or y s = 0, 1, 2,. Usig simple lultios, we hve the ollowig lemm. LEMMA 5.4 (s 1) = (s 1) = g (s 1) = m (s 1) I (s 1) = j (s 1) = l (s 1) = p, (s 1)

6 42 K. Ahr et l. Fig. A.1. Tilig o oe prototile. Fig. A.4. Tilig o two prototiles (5,10). Fig. A.2. Tilig o two prototiles, o.5. Fig. A.5. Tilig o two prototiles (0,2). Fig. A.3. Tilig o two prototiles, o.8. Fig. A.6. Tilig o two prototiles (0,8). e (s 1) = i (s 1) = k (s 1) = o (s 1), the (s 1) = (s 1) = h (s 1) = (s 1) (s) = (s) = g (s) = m (s) (s) = j (s) = l (s) = p, e(s) = i (s) = k (s) = o (s) (s) (s) = (s) = h (s) =. (s) Proo: For exmple, p (s 1) = l (s 1) (s 1) = m (s 1) implies (s) = p (s 1) (s 1) = l (s 1) m (s 1) = (s). Other reltios re show i similr wy. From this lemm, (s + 1)-spres (s+1), (s+1) exist or y s. (3), (4), (5) re show i similr wy s (2). REMARK 5.5 I Appeix, we show igures o these tiligs. I (0, 2), (0, 8) tiligs, (s) otis oly oe tile. This mes tht these tiligs re equivlet to tilig o s tiligs. Akowlegmets. The uthos woul like to thk Pro Koukihi Sugihr or his givig them suh iterestig topis. The uthors lso thk the reree or his ki useul vie. Fig. A.7. Tilig o two prototiles (0,10). Appeix A. From Figs. A.1 to A.7 re pitures o tiligs pperig i Theorems A, B, C. Appeix B. For protoset S, i y tilig y S hs o perioiity

7 Esher Degree o No-perioi L-tiligs y 2 Prototiles 43 Fig. B.1. Perioi tilig o oe prototile. Fig. B.2. Perioi tilig o two prototiles o.8. Fig. B.3. Perioi tilig o two prototiles o.5. Fig. B.4. Perioi tilig o two prototiles (5,10). o prllel trsltio the we ll S perioi protoset. Ay tiligs we oti i this pper re ot perioi protosets. From Figs. B.1 to B.4 re igures o perioi tiligs. [2] Goom-Struss, C. (1999) A smll perioi set o tiles, Eur. J. Comitoris, 20, [3] Goom-Struss, C. (1998) Mthig rules sustitutio tiligs, A. Mth., 147, Reerees [1] Sugihr, K. (2011) Esher Mgi, Uiversity o Tokyo Press (i Jpese).