Polyominoes and Polyiamonds as Fundamental Domains of Isohedral Tilings for Crystal Class D 2

Hiroshi Fukuda, Chiaki Kanomata, Nobuaki Mutoh, Gisaku Nakamura, and Doris Schattschneider

Kitasato University

(3 Nov., 2010)

College of Liberal Arts And Sciences, Kitasato University 1 1.1. OurOur ProjectProject 2.2. MethodMethod 3.3. SummarySummary ofof thethe previousprevious resultsresults (tiling(tiling withwith rotationalrotational )symmetry) 4.4. ResultsResults forfor DD 2 5.5. FutureFuture workwork

College of Liberal Arts And Sciences, Kitasato University 2 Our Project

WeWe wantwant toto enumerateenumerate (list(list up)up) allall 2D2D isohedralisohedral tilingtiling forfor givengiven symmetrysymmetry inin whichwhich polyominoespolyominoes (or(or polyiamonds)polyiamonds) areare tilestiles andand fundamentalfundamental domains.domains.

College of Liberal Arts And Sciences, Kitasato University 3 Isohedral tiling

IsohedralIsohedral tilingtiling isis aa tilingtiling byby congruentcongruent tilestiles inin whichwhich anyany twotwo tilestiles areare superposedsuperposed byby anan isometry,isometry, anan symmetry,symmetry, whichwhich keepskeeps wholewhole tilingtiling invariant.invariant.

College of Liberal Arts And Sciences, Kitasato University 4 ForFor example,example, thisthis isohedralisohedral tilingtiling isis invariantinvariant underunder thethe 90°90° rotationrotation atat blackblack andand whitewhite points,points, andand twotwo perpendicularperpendicular translationtranslation vectors,vectors, andand anyany 22 tilestiles areare superposedsuperposed byby thesethese operations.operations. ForFor example,example, greengreen tilestiles areare superposedsuperposed byby thethe rotationrotation andand translation.translation.

College of Liberal Arts And Sciences, Kitasato University 5 Periodic tiling

ItIt isis wellwell knownknown thatthat

“the“the isohedralisohedral tilingtiling isis periodic”,periodic”, wherewhere periodicperiodic meansmeans thatthat thethe patternpattern isis invariantinvariant underunder twotwo independentindependent translations.translations.

TheThe symmetrysymmetry elementselements ofof thethe periodicperiodic patternpattern generategenerate thethe symmetrysymmetry group,group, whichwhich isis calledcalled spacespace groupgroup inin 2D.2D.

College of Liberal Arts And Sciences, Kitasato University 6 17 space group in 2D

ItIt isis provedproved byby CoxeterCoxeter andand MoserMoser inin 19601960 thatthat

“there“there areare 1717 spacespace groupsgroups andand that'sthat's all”.all”.

WhereasWhereas 230230 spacespace groupsgroups inin 3D3D waswas shownshown byby E.E. FedorovFedorov inin 1891.1891. ItIt isis saidsaid thatthat FedorovFedorov alsoalso provedproved 1717 spacespace groupsgroups inin 2D.2D.

College of Liberal Arts And Sciences, Kitasato University 7 Hermann-Mauguin Notation p1,p1, p2,p2, pm,pm, pg,pg, cm,cm, cmm,cmm, pgg,pgg, pmm,pmm, pmg,pmg, p3,p3, p3m1,p3m1, p31m,p31m, p4,p4, p4m,p4m, p4g,p4g, p6,p6, p6mp6m

1717 spacespace groupsgroups areare denoteddenoted byby Hermann-Hermann- MauguinMauguin notation.notation.

College of Liberal Arts And Sciences, Kitasato University 8 Symmetry elements p1,p1, p2,p2, pm,pm, pg,pg, cm,cm, cmm,cmm, pgg,pgg, pmm,pmm, pmg,pmg, p3,p3, p3m1,p3m1, p31m,p31m, p4,p4, p4m,p4m, p4g,p4g, p6,p6, p6mp6m

InIn Hermann-MauguinHermann-Mauguin notation,notation, Numbers,Numbers, n=1,n=1, 2,2, 3,3, 4,4, 66 representsrepresents n-foldn-fold rotationrotation symmetry,symmetry, i.e.,i.e., 2π/n2π/n [rad][rad] rotationrotation symmetry.symmetry. mm isis thethe mirrormirror symmetrysymmetry andand gg isis thethe glideglide reflectionreflection symmetry.symmetry.

College of Liberal Arts And Sciences, Kitasato University 9 Glide

Mirror reflection

Then, translation

This pattern has glide reflection. Mirror reflection along the dashed line and the successive translation is the glide reflection, which does not change the pattern.

College of Liberal Arts And Sciences, Kitasato University 10 p4

ThisThis patternpattern isis anan exampleexample whosewhose symmetrysymmetry isis p4,p4,

College of Liberal Arts And Sciences, Kitasato University 11 p4

ThisThis patternpattern isis anan exampleexample whosewhose symmetrysymmetry isis p4,p4, andand thethe elementselements ofof p4p4 areare

4-fold rotation center

Inequivalent 4-fold rotation center

translation

translation

College of Liberal Arts And Sciences, Kitasato University 12 Fundamental Domain

ForFor eacheach symmetrysymmetry group,group, thethe maximummaximum closedclosed regionregion whichwhich doesdoes notnot includeinclude anyany identicalidentical pointspoints isis calledcalled thethe fundamentalfundamental domain.domain. ForFor givengiven symmetry,symmetry, thethe areaarea isis constant,constant, however,however, thethe shapeshape isis free.free.

College of Liberal Arts And Sciences, Kitasato University 13 ForFor example,example, thethe shadedshaded squaresquare isis aa fundamentalfundamental domaindomain ofof p4,p4,

College of Liberal Arts And Sciences, Kitasato University 14 ForFor example,example, thethe shadedshaded squaresquare isis aa fundamentalfundamental domaindomain ofof p4,p4, andand thisthis isis anotheranother fundamentalfundamental domain.domain.

College of Liberal Arts And Sciences, Kitasato University 15 Therefore,Therefore, forfor thethe isohedralisohedral tiling,tiling, thethe smallestsmallest tiletile isis aa fundamentalfundamental domain,domain, andand InverselyInversely ifif symmetrysymmetry GG isis givengiven first,first, byby takingtaking aa fundamentalfundamental domaindomain asas tiletile wewe obtainobtain anan isohedralisohedral tiling.tiling. IfIf thethe fundamentalfundamental domaindomain TT ofof GG isis notnot pointpoint symmetric,symmetric, thethe fullfull symmetrysymmetry ofof thethe tilingtiling obtainedobtained byby TT isis GG andand TT isis fundamentalfundamental domain.domain.

College of Liberal Arts And Sciences, Kitasato University 16 InIn ourour example,example, thethe fundamentalfundamental domaindomain ofof p4p4 isis notnot pointpoint symmetricsymmetric thusthus thethe fullfull symmetrysymmetry ofof tilingtiling isis p4p4 andand thethe tiletile isis fundamentalfundamental domain.domain.

College of Liberal Arts And Sciences, Kitasato University 17 OnOn thethe otherother hand,hand, ifif fundamentalfundamental domaindomain TT forfor givengiven GG isis pointpoint symmetric,symmetric, thethe fullfull symmetrysymmetry ofof tilingtiling generatedgenerated byby TT andand GG maymay bebe differentdifferent fromfrom G,G, G'G' wherewhere GG isis aa properproper subgroupsubgroup ofof G'.G'.

College of Liberal Arts And Sciences, Kitasato University 18 ForFor example,example, ifif wewe useuse thethe shadedshaded squaresquare fundamentalfundamental domaindomain ofof p4p4 asas aa tile,tile,

College of Liberal Arts And Sciences, Kitasato University 19 ForFor example,example, ifif wewe useuse thethe shadedshaded squaresquare fundamentalfundamental domaindomain ofof p4p4 asas aa tile,tile, thethe fullfull symmetrysymmetry ofof thethe tilingtiling isis p4m.p4m. p4p4 isis aa properproper subgroupsubgroup ofof p4m.p4m.

mirror

College of Liberal Arts And Sciences, Kitasato University 20 1.1. OurOur ProjectProject 2.2. MethodMethod 3.3. SummarySummary ofof thethe previousprevious resultsresults (tiling(tiling withwith rotationalrotational symmetry)symmetry) 4.4. ResultsResults forfor DD 2 5.5. FutureFuture workwork

College of Liberal Arts And Sciences, Kitasato University 21 Method

WeWe willwill enumerateenumerate (list(list up)up) allall 2D2D isohedralisohedral tilingtiling forfor givengiven symmetrysymmetry inin whichwhich polyominoespolyominoes (or(or polyiamonds)polyiamonds) areare tilestiles andand fundamentalfundamental domainsdomains asas follows.follows.

College of Liberal Arts And Sciences, Kitasato University 22 Method

1.1. WeWe firstfirst givegive aa spacespace symmetrysymmetry inin 2D.2D. 2.2. TakeTake aa fundamentalfundamental domaindomain asas polyominopolyomino oror polyiamond.polyiamond. Where A polyomino (or n-omino) is a tile made up of n unit squares that are connected at their edges. Similarly a polyiamond (or n-iamond) is a tile made up of n unit equilateral triangles that are connected at their edges.

College of Liberal Arts And Sciences, Kitasato University 23 3.3. ComputerComputer willwill listlist upup allall polyomioespolyomioes (polyiamonds)(polyiamonds) whichwhich cancan bebe fundamentalfundamental domains.domains. 4.4. UsingUsing thethe subgroupsubgroup relationrelation ship,ship, computercomputer investigatesinvestigates thethe possiblepossible additionaladditional symmetry.symmetry. IfIf additionaladditional symmetrysymmetry isis foundfound thisthis tilingtiling willwill bebe dropped.dropped.

College of Liberal Arts And Sciences, Kitasato University 24 SubgroupSubgroup relationshipsrelationships fromfrom Coxeter'sCoxeter's book.book.

College of Liberal Arts And Sciences, Kitasato University 25 p4

ForFor example,example, allall fundamentalfundamental domainsdomains forfor p4p4

(1-1) (2-1) 4-1 4-2 (4-3) 5-1 5-1-2 (5-2) (5-2-2) 5-2-3 (5-3) 5-4

5-5 5-6 5-7 5-7-2 5-8

College of Liberal Arts And Sciences, Kitasato University 26 p4

ForFor example,example, allall fundamentalfundamental domainsdomains forfor p4p4 andand correspondingcorresponding tilingstilings areare shown.shown. XX isis notnot p4.p4.

(1-1) (2-1) 4-1 4-2 (4-3) 5-1 5-1-2 (5-2) (5-2-2) 5-2-3 (5-3) 5-4

5-5 5-6 5-7 5-7-2 5-8

College of Liberal Arts And Sciences, Kitasato University 27 p4

ForFor example,example, allall fundamentalfundamental domainsdomains forfor p4p4 andand correspondingcorresponding tilingstilings areare shown.shown. XX isis notnot p4.p4.

(1-1) (2-1) 4-1 4-2 (4-3) 5-1 5-1-2 (5-2) (5-2-2) 5-2-3 (5-3) 5-4

5-5 5-6 5-7 5-7-2 5-8

College of Liberal Arts And Sciences, Kitasato University 28 p4

ForFor example,example, allall fundamentalfundamental domainsdomains forfor p4p4 andand correspondingcorresponding tilingstilings areare shown.shown. XX isis notnot p4.p4.

(1-1) (2-1) 4-1 4-2 (4-3) 5-1 5-1-2 (5-2) (5-2-2) 5-2-3 (5-3) 5-4

5-5 5-6 5-7 5-7-2 5-8

College of Liberal Arts And Sciences, Kitasato University 29 p4

ForFor example,example, allall fundamentalfundamental domainsdomains forfor p4p4 andand correspondingcorresponding tilingstilings areare shown.shown. XX isis notnot p4.p4.

(1-1) (2-1) 4-1 4-2 (4-3) 5-1 5-1-2 (5-2) (5-2-2) 5-2-3 (5-3) 5-4

5-5 5-6 5-7 5-7-2 5-8

College of Liberal Arts And Sciences, Kitasato University 30 p4

ForFor example,example, allall fundamentalfundamental domainsdomains forfor p4p4 andand correspondingcorresponding tilingstilings areare shown.shown. XX isis notnot p4.p4.

(1-1) (2-1) 4-1 4-2 (4-3) 5-1 5-1-2 (5-2) (5-2-2) 5-2-3 (5-3) 5-4

5-5 5-6 5-7 5-7-2 5-8

College of Liberal Arts And Sciences, Kitasato University 31 p4

ForFor example,example, allall fundamentalfundamental domainsdomains forfor p4p4 andand correspondingcorresponding tilingstilings areare shown.shown. XX isis notnot p4.p4.

(1-1) (2-1) 4-1 4-2 (4-3) 5-1 5-1-2 (5-2) (5-2-2) 5-2-3 (5-3) 5-4

5-5 5-6 5-7 5-7-2 5-8

College of Liberal Arts And Sciences, Kitasato University 32 p4

ForFor example,example, allall fundamentalfundamental domainsdomains forfor p4p4 andand correspondingcorresponding tilingstilings areare shown.shown. XX isis notnot p4.p4.

(1-1) (2-1) 4-1 4-2 (4-3) 5-1 5-1-2 (5-2) (5-2-2) 5-2-3 (5-3) 5-4

5-5 5-6 5-7 5-7-2 5-8

College of Liberal Arts And Sciences, Kitasato University 33 p4

ForFor example,example, allall fundamentalfundamental domainsdomains forfor p4p4 andand correspondingcorresponding tilingstilings areare shown.shown. XX isis notnot p4.p4.

(1-1) (2-1) 4-1 4-2 (4-3) 5-1 5-1-2 (5-2) (5-2-2) 5-2-3 (5-3) 5-4

5-5 5-6 5-7 5-7-2 5-8

College of Liberal Arts And Sciences, Kitasato University 34 p4

ForFor example,example, allall fundamentalfundamental domainsdomains forfor p4p4 andand correspondingcorresponding tilingstilings areare shown.shown. XX isis notnot p4.p4.

(1-1) (2-1) 4-1 4-2 (4-3) 5-1 5-1-2 (5-2) (5-2-2) 5-2-3 (5-3) 5-4

5-5 5-6 5-7 5-7-2 5-8

College of Liberal Arts And Sciences, Kitasato University 35 p4

ForFor example,example, allall fundamentalfundamental domainsdomains forfor p4p4 andand correspondingcorresponding tilingstilings areare shown.shown. XX isis notnot p4.p4.

(1-1) (2-1) 4-1 4-2 (4-3) 5-1 5-1-2 (5-2) (5-2-2) 5-2-3 (5-3) 5-4

5-5 5-6 5-7 5-7-2 5-8

College of Liberal Arts And Sciences, Kitasato University 36 p4

ForFor example,example, allall fundamentalfundamental domainsdomains forfor p4p4 andand correspondingcorresponding tilingstilings areare shown.shown. XX isis notnot p4.p4.

(1-1) (2-1) 4-1 4-2 (4-3) 5-1 5-1-2 (5-2) (5-2-2) 5-2-3 (5-3) 5-4

5-5 5-6 5-7 5-7-2 5-8

College of Liberal Arts And Sciences, Kitasato University 37 p4

ForFor example,example, allall fundamentalfundamental domainsdomains forfor p4p4 andand correspondingcorresponding tilingstilings areare shown.shown. XX isis notnot p4.p4.

(1-1) (2-1) 4-1 4-2 (4-3) 5-1 5-1-2 (5-2) (5-2-2) 5-2-3 (5-3) 5-4

5-5 5-6 5-7 5-7-2 5-8

College of Liberal Arts And Sciences, Kitasato University 38 p4

ForFor example,example, allall fundamentalfundamental domainsdomains forfor p4p4 andand correspondingcorresponding tilingstilings areare shown.shown. XX isis notnot p4.p4.

(1-1) (2-1) 4-1 4-2 (4-3) 5-1 5-1-2 (5-2) (5-2-2) 5-2-3 (5-3) 5-4

5-5 5-6 5-7 5-7-2 5-8

College of Liberal Arts And Sciences, Kitasato University 39 p4

ForFor example,example, allall fundamentalfundamental domainsdomains forfor p4p4 andand correspondingcorresponding tilingstilings areare shown.shown. XX isis notnot p4.p4.

(1-1) (2-1) 4-1 4-2 (4-3) 5-1 5-1-2 (5-2) (5-2-2) 5-2-3 (5-3) 5-4

5-5 5-6 5-7 5-7-2 5-8

College of Liberal Arts And Sciences, Kitasato University 40 p4

ForFor example,example, allall fundamentalfundamental domainsdomains forfor p4p4 andand correspondingcorresponding tilingstilings areare shown.shown. XX isis notnot p4.p4.

(1-1) (2-1) 4-1 4-2 (4-3) 5-1 5-1-2 (5-2) (5-2-2) 5-2-3 (5-3) 5-4

5-5 5-6 5-7 5-7-2 5-8

College of Liberal Arts And Sciences, Kitasato University 41 p4

ForFor example,example, allall fundamentalfundamental domainsdomains forfor p4p4 andand correspondingcorresponding tilingstilings areare shown.shown. XX isis notnot p4.p4.

(1-1) (2-1) 4-1 4-2 (4-3) 5-1 5-1-2 (5-2) (5-2-2) 5-2-3 (5-3) 5-4

5-5 5-6 5-7 5-7-2 5-8

College of Liberal Arts And Sciences, Kitasato University 42 p4

ForFor example,example, allall fundamentalfundamental domainsdomains forfor p4p4 andand correspondingcorresponding tilingstilings areare shown.shown. XX isis notnot p4.p4.

(1-1) (2-1) 4-1 4-2 (4-3) 5-1 5-1-2 (5-2) (5-2-2) 5-2-3 (5-3) 5-4

5-5 5-6 5-7 5-7-2 5-8

College of Liberal Arts And Sciences, Kitasato University 43 1.1. OurOur ProjectProject 2.2. MethodMethod 3.3. SummarySummary ofof thethe previousprevious resultsresults (tiling(tiling withwith rotationalrotational symmetry)symmetry) 4.4. ResultsResults forfor DD 2 5.5. FutureFuture workwork

College of Liberal Arts And Sciences, Kitasato University 44 Rotational Symmetry (1)

TheThe symmetrysymmetry groupsgroups whichwhich havehave nn≧≧ 3 3 foldfold rotationalrotational symmetrysymmetry areare thethe followingfollowing 88 groups.groups.

p4,p4, p4g,p4g, p4mp4m p3,p3, p31m,p31m, p3m1p3m1 p6,p6, p6mp6m

WeWe havehave shownshown thethe resultsresults forfor p4.p4. TheThe resultsresults forfor p3,p3, andand p6p6 areare similar.similar.

College of Liberal Arts And Sciences, Kitasato University 45 Rotational Symmetry (2)

TheThe symmetrysymmetry groupsgroups whichwhich havehave nn≧≧ 3 3 foldfold rotationalrotational symmetrysymmetry areare thethe followingfollowing 88 groups.groups.

p4,p4, p4g,p4g, p4mp4m p3,p3, p31m,p31m, p3m1p3m1 p6,p6, p6mp6m

ForFor p4m,p4m, p3m1,p3m1, andand p6m,p6m, therethere areare nono tilingtiling ofof polyominoespolyominoes oror polyiamondspolyiamonds asas fundamentalfundamental domains.domains.

College of Liberal Arts And Sciences, Kitasato University 46 TheThe reasonreason isis thatthat thethe mirrormirror reflectionreflection axesaxes MM can not cross the can not cross the M M  fundamentalfundamental domain.domain. TheThe figurefigure showsshows thethe symmetrysymmetry elementselements ofof p4mp4m.. TheThe lineslines areare MM andand aa fundamentalfundamental domaindomain isis shownshown byby shadedshaded area.area.

College of Liberal Arts And Sciences, Kitasato University 47 ProofProof forfor p4m.p4m. ThusThus thisthis isoscelesisosceles rightright triangletriangle isis thethe onlyonly M M  possiblepossible funamentalfunamental domain,domain, whichwhich isis notnot aa polyominopolyomino nornor aa polyiamond.polyiamond.

College of Liberal Arts And Sciences, Kitasato University 48 Rotational Symmetry (3)

p4,p4, p4gp4g,, p4mp4m p3,p3, p31mp31m,, p3m1p3m1 p6,p6, p6mp6m

Next,Next, forfor p4gp4g fundamentalfundamental domaindomain ofof polyominopolyomino alwaysalways remainremain fundamentalfundamental domaindomain ofof thethe fullfull symmetrysymmetry exceptexcept forfor squaresquare kk2-ominoes.-ominoes. AndAnd p4gp4g fundamentalfundamental domaindomain ofof squaresquare yieldsyields p4mp4m tiling.tiling.

College of Liberal Arts And Sciences, Kitasato University 49 AllAll fundamentalfundamental domaindomain ofof polyominopolyomino forfor p4gp4g

(1-1) 4-1 (4-2) 4-3

College of Liberal Arts And Sciences, Kitasato University 50 AllAll fundamentalfundamental domaindomain ofof polyominopolyomino forfor p4gp4g isis fundamentalfundamental domaindomain ofof fullfull symmetrysymmetry (p4g)(p4g) exceptexcept forfor squaresquare polyomino.polyomino.

(1-1) 4-1 (4-2) 4-3

College of Liberal Arts And Sciences, Kitasato University 51 TheThe allall fundamentalfundamental domaindomain ofof polyominopolyomino forfor givengiven p4gp4g isis fundamentalfundamental domaindomain ofof fullfull symmetrysymmetry (p4g)(p4g) exceptexcept forfor squaresquare polyomino.polyomino.

(1-1) 4-1 (4-2) 4-3

College of Liberal Arts And Sciences, Kitasato University 52 TheThe allall fundamentalfundamental domaindomain ofof polyominopolyomino forfor givengiven p4gp4g isis fundamentalfundamental domaindomain ofof fullfull symmetrysymmetry (p4g)(p4g) exceptexcept forfor squaresquare polyomino.polyomino.

(1-1) 4-1 (4-2) 4-3

College of Liberal Arts And Sciences, Kitasato University 53 TheThe allall fundamentalfundamental domaindomain ofof polyominopolyomino forfor givengiven p4gp4g isis fundamentalfundamental domaindomain ofof fullfull symmetrysymmetry (p4g)(p4g) exceptexcept forfor squaresquare polyomino.polyomino.

(1-1) 4-1 (4-2) 4-3

Therefore, for p4g computer can skip investigation of the full symmetry, STEP 4 and excludes simply square polyominoes. College of Liberal Arts And Sciences, Kitasato University 54 Rotational Symmetry (2)

p4,p4, p4gp4g,, p4mp4m p3,p3, p31mp31m,, p3m1p3m1 p6,p6, p6mp6m

Similarly,Similarly, forfor p31mp31m fundamentalfundamental domaindomain ofof polyiamondpolyiamond alwaysalways remainsremains fundamentalfundamental domaindomain ofof thethe fullfull symmetry.symmetry.

College of Liberal Arts And Sciences, Kitasato University 55 AllAll fundamentalfundamental domaindomain ofof polyiamondspolyiamonds forfor p31mp31m

3-1 12-1 12-2 12-3 12-4 12-5 12-6 12-7 12-8 12-9 12-10

12-11 12-12 12-13 12-14 12-15 12-16 12-17 12-18 12-19 12-20

College of Liberal Arts And Sciences, Kitasato University 56 AllAll fundamentalfundamental domaindomain ofof polyiamondspolyiamonds forfor p31mp31m isis fundamentalfundamental domaindomain ofof fullfull symmetrysymmetry (p31m).(p31m).

3-1 12-1 12-2 12-3 12-4 12-5 12-6 12-7 12-8 12-9 12-10

12-11 12-12 12-13 12-14 12-15 12-16 12-17 12-18 12-19 12-20

College of Liberal Arts And Sciences, Kitasato University 57 AllAll fundamentalfundamental domaindomain ofof polyiamondspolyiamonds forfor p31mp31m isis fundamentalfundamental domaindomain ofof fullfull symmetrysymmetry (p31m).(p31m).

3-1 12-1 12-2 12-3 12-4 12-5 12-6 12-7 12-8 12-9 12-10

12-11 12-12 12-13 12-14 12-15 12-16 12-17 12-18 12-19 12-20

College of Liberal Arts And Sciences, Kitasato University 58 AllAll fundamentalfundamental domaindomain ofof polyiamondspolyiamonds forfor p31mp31m isis fundamentalfundamental domaindomain ofof fullfull symmetrysymmetry (p31m).(p31m).

3-1 12-1 12-2 12-3 12-4 12-5 12-6 12-7 12-8 12-9 12-10

12-11 12-12 12-13 12-14 12-15 12-16 12-17 12-18 12-19 12-20

College of Liberal Arts And Sciences, Kitasato University 59 AllAll fundamentalfundamental domaindomain ofof polyiamondspolyiamonds forfor p31mp31m isis fundamentalfundamental domaindomain ofof fullfull symmetrysymmetry (p31m).(p31m).

3-1 12-1 12-2 12-3 12-4 12-5 12-6 12-7 12-8 12-9 12-10

12-11 12-12 12-13 12-14 12-15 12-16 12-17 12-18 12-19 12-20

College of Liberal Arts And Sciences, Kitasato University 60 AllAll fundamentalfundamental domaindomain ofof polyiamondspolyiamonds forfor p31mp31m isis fundamentalfundamental domaindomain ofof fullfull symmetrysymmetry (p31m).(p31m).

3-1 12-1 12-2 12-3 12-4 12-5 12-6 12-7 12-8 12-9 12-10

12-11 12-12 12-13 12-14 12-15 12-16 12-17 12-18 12-19 12-20

College of Liberal Arts And Sciences, Kitasato University 61 AllAll fundamentalfundamental domaindomain ofof polyiamondspolyiamonds forfor p31mp31m isis fundamentalfundamental domaindomain ofof fullfull symmetrysymmetry (p31m).(p31m).

3-1 12-1 12-2 12-3 12-4 12-5 12-6 12-7 12-8 12-9 12-10

12-11 12-12 12-13 12-14 12-15 12-16 12-17 12-18 12-19 12-20

College of Liberal Arts And Sciences, Kitasato University 62 AllAll fundamentalfundamental domaindomain ofof polyiamondspolyiamonds forfor p31mp31m isis fundamentalfundamental domaindomain ofof fullfull symmetrysymmetry (p31m).(p31m).

3-1 12-1 12-2 12-3 12-4 12-5 12-6 12-7 12-8 12-9 12-10

12-11 12-12 12-13 12-14 12-15 12-16 12-17 12-18 12-19 12-20

College of Liberal Arts And Sciences, Kitasato University 63 AllAll fundamentalfundamental domaindomain ofof polyiamondspolyiamonds forfor p31mp31m isis fundamentalfundamental domaindomain ofof fullfull symmetrysymmetry (p31m).(p31m).

3-1 12-1 12-2 12-3 12-4 12-5 12-6 12-7 12-8 12-9 12-10

12-11 12-12 12-13 12-14 12-15 12-16 12-17 12-18 12-19 12-20

College of Liberal Arts And Sciences, Kitasato University 64 AllAll fundamentalfundamental domaindomain ofof polyiamondspolyiamonds forfor p31mp31m isis fundamentalfundamental domaindomain ofof fullfull symmetrysymmetry (p31m).(p31m).

3-1 12-1 12-2 12-3 12-4 12-5 12-6 12-7 12-8 12-9 12-10

12-11 12-12 12-13 12-14 12-15 12-16 12-17 12-18 12-19 12-20

College of Liberal Arts And Sciences, Kitasato University 65 AllAll fundamentalfundamental domaindomain ofof polyiamondspolyiamonds forfor p31mp31m isis fundamentalfundamental domaindomain ofof fullfull symmetrysymmetry (p31m).(p31m).

3-1 12-1 12-2 12-3 12-4 12-5 12-6 12-7 12-8 12-9 12-10

12-11 12-12 12-13 12-14 12-15 12-16 12-17 12-18 12-19 12-20

College of Liberal Arts And Sciences, Kitasato University 66 AllAll fundamentalfundamental domaindomain ofof polyiamondspolyiamonds forfor p31mp31m isis fundamentalfundamental domaindomain ofof fullfull symmetrysymmetry (p31m).(p31m).

3-1 12-1 12-2 12-3 12-4 12-5 12-6 12-7 12-8 12-9 12-10

12-11 12-12 12-13 12-14 12-15 12-16 12-17 12-18 12-19 12-20

College of Liberal Arts And Sciences, Kitasato University 67 AllAll fundamentalfundamental domaindomain ofof polyiamondspolyiamonds forfor p31mp31m isis fundamentalfundamental domaindomain ofof fullfull symmetrysymmetry (p31m).(p31m).

3-1 12-1 12-2 12-3 12-4 12-5 12-6 12-7 12-8 12-9 12-10

12-11 12-12 12-13 12-14 12-15 12-16 12-17 12-18 12-19 12-20

College of Liberal Arts And Sciences, Kitasato University 68 AllAll fundamentalfundamental domaindomain ofof polyiamondspolyiamonds forfor p31mp31m isis fundamentalfundamental domaindomain ofof fullfull symmetrysymmetry (p31m).(p31m).

3-1 12-1 12-2 12-3 12-4 12-5 12-6 12-7 12-8 12-9 12-10

12-11 12-12 12-13 12-14 12-15 12-16 12-17 12-18 12-19 12-20

College of Liberal Arts And Sciences, Kitasato University 69 AllAll fundamentalfundamental domaindomain ofof polyiamondspolyiamonds forfor p31mp31m isis fundamentalfundamental domaindomain ofof fullfull symmetrysymmetry (p31m).(p31m).

3-1 12-1 12-2 12-3 12-4 12-5 12-6 12-7 12-8 12-9 12-10

12-11 12-12 12-13 12-14 12-15 12-16 12-17 12-18 12-19 12-20

College of Liberal Arts And Sciences, Kitasato University 70 AllAll fundamentalfundamental domaindomain ofof polyiamondspolyiamonds forfor p31mp31m isis fundamentalfundamental domaindomain ofof fullfull symmetrysymmetry (p31m).(p31m).

3-1 12-1 12-2 12-3 12-4 12-5 12-6 12-7 12-8 12-9 12-10

12-11 12-12 12-13 12-14 12-15 12-16 12-17 12-18 12-19 12-20

College of Liberal Arts And Sciences, Kitasato University 71 AllAll fundamentalfundamental domaindomain ofof polyiamondspolyiamonds forfor p31mp31m isis fundamentalfundamental domaindomain ofof fullfull symmetrysymmetry (p31m).(p31m).

3-1 12-1 12-2 12-3 12-4 12-5 12-6 12-7 12-8 12-9 12-10

12-11 12-12 12-13 12-14 12-15 12-16 12-17 12-18 12-19 12-20

College of Liberal Arts And Sciences, Kitasato University 72 AllAll fundamentalfundamental domaindomain ofof polyiamondspolyiamonds forfor p31mp31m isis fundamentalfundamental domaindomain ofof fullfull symmetrysymmetry (p31m).(p31m).

3-1 12-1 12-2 12-3 12-4 12-5 12-6 12-7 12-8 12-9 12-10

12-11 12-12 12-13 12-14 12-15 12-16 12-17 12-18 12-19 12-20

College of Liberal Arts And Sciences, Kitasato University 73 AllAll fundamentalfundamental domaindomain ofof polyiamondspolyiamonds forfor p31mp31m isis fundamentalfundamental domaindomain ofof fullfull symmetrysymmetry (p31m).(p31m).

3-1 12-1 12-2 12-3 12-4 12-5 12-6 12-7 12-8 12-9 12-10

12-11 12-12 12-13 12-14 12-15 12-16 12-17 12-18 12-19 12-20

College of Liberal Arts And Sciences, Kitasato University 74 AllAll fundamentalfundamental domaindomain ofof polyiamondspolyiamonds forfor p31mp31m isis fundamentalfundamental domaindomain ofof fullfull symmetrysymmetry (p31m).(p31m).

3-1 12-1 12-2 12-3 12-4 12-5 12-6 12-7 12-8 12-9 12-10

12-11 12-12 12-13 12-14 12-15 12-16 12-17 12-18 12-19 12-20

College of Liberal Arts And Sciences, Kitasato University 75 AllAll fundamentalfundamental domaindomain ofof polyiamondspolyiamonds forfor p31mp31m isis fundamentalfundamental domaindomain ofof fullfull symmetrysymmetry (p31m).(p31m).

3-1 12-1 12-2 12-3 12-4 12-5 12-6 12-7 12-8 12-9 12-10

12-11 12-12 12-13 12-14 12-15 12-16 12-17 12-18 12-19 12-20

College of Liberal Arts And Sciences, Kitasato University 76 AllAll fundamentalfundamental domaindomain ofof polyiamondspolyiamonds forfor p31mp31m isis fundamentalfundamental domaindomain ofof fullfull symmetrysymmetry (p31m).(p31m).

3-1 12-1 12-2 12-3 12-4 12-5 12-6 12-7 12-8 12-9 12-10

12-11 12-12 12-13 12-14 12-15 12-16 12-17 12-18 12-19 12-20

College of Liberal Arts And Sciences, Kitasato University 77 1.1. OurOur ProjectProject 2.2. MethodMethod 3.3. SummarySummary ofof thethe previousprevious resultsresults (tiling(tiling withwith rotationalrotational symmetry)symmetry) 4.4. ResultsResults forfor DD 2 5.5. FutureFuture workwork

College of Liberal Arts And Sciences, Kitasato University 78 D 2 OurOur computercomputer programprogram forfor rotationalrotational symmetrysymmetry asumesasumes rotationalrotational centercenter andand twotwo orthogonalorthogonal translations.translations. TheThe 44 groups,groups, pmm,pmm, pmg,pmg, pgg,pgg, cmmcmm havehave alsoalso suchsuch symmetriessymmetries andand byby slightslight modifications,modifications, ourour computercomputer programprogram worksworks forfor thesethese .symmetries.

College of Liberal Arts And Sciences, Kitasato University 79 HereHere pmm,pmm, pmg,pmg, pgg,pgg, cmmcmm belongsbelongs toto thethe crystalcrystal classclass DD andand nono otherother 2 groupgroup belongsbelongs toto DD .. 2

WhereWhere crystalcrystal classclass isis thethe factorfactor groupgroup ofof spacespace groupgroup G,G, G/T,G/T, wherewhere TT isis translationtranslation subgroupsubgroup ofof G.G. DD ,, isis aa pointpoint groupgroup 2 generatedgenerated byby 2-fold2-fold rotationrotation andand mirrormirror reflection.reflection.College of Liberal Arts And Sciences, Kitasato University 80 pmm pmm,pmm, pmg,pmg, pgg,pgg, cmmcmm

ByBy thethe samesame reasonreason asas p4m,p4m, p3m1p3m1 andand p6m,p6m, therethere areare nono tilingtiling ofof polyominoespolyominoes oror polyiamondspolyiamonds asas fundamentalfundamental domainsdomains forfor pmm.pmm.

College of Liberal Arts And Sciences, Kitasato University 81 cmm pmm,pmm, pmg,pmg, pgg,pgg, cmmcmm

ResultsResults forfor cmmcmm isis almostalmost samesame asas p4g.p4g.

College of Liberal Arts And Sciences, Kitasato University 82 cmm

AllAll fundamentalfundamental domaindomain ofof polyominopolyomino forfor cmmcmm

(1-1) (2-1-1) (2-1-2) (3-1-1) (3-1-2) 3-2 4-1 (4-2-1) (4-2-2) (4-3)

(5-1-1) (5-1-2) 5-2 5-3

College of Liberal Arts And Sciences, Kitasato University 83 cmm

AllAll fundamentalfundamental domaindomain ofof polyominopolyomino forfor cmmcmm areare fundamentalfundamental domaindomain ofof fullfull symmetrysymmetry cmmcmm exceptexcept forfor rectanglerectangle polyominoes.polyominoes.

(1-1) (2-1-1) (2-1-2) (3-1-1) (3-1-2) 3-2 4-1 (4-2-1) (4-2-2) (4-3)

(5-1-1) (5-1-2) 5-2 5-3

College of Liberal Arts And Sciences, Kitasato University 84 cmm

AllAll fundamentalfundamental domaindomain ofof polyominopolyomino forfor cmmcmm areare fundamentalfundamental domaindomain ofof fullfull symmetrysymmetry cmmcmm exceptexcept forfor rectanglerectangle polyominoes.polyominoes.

(1-1) (2-1-1) (2-1-2) (3-1-1) (3-1-2) 3-2 4-1 (4-2-1) (4-2-2) (4-3)

(5-1-1) (5-1-2) 5-2 5-3

College of Liberal Arts And Sciences, Kitasato University 85 cmm

AllAll fundamentalfundamental domaindomain ofof polyominopolyomino forfor cmmcmm areare fundamentalfundamental domaindomain ofof fullfull symmetrysymmetry cmmcmm exceptexcept forfor rectanglerectangle polyominoes.polyominoes.

(1-1) (2-1-1) (2-1-2) (3-1-1) (3-1-2) 3-2 4-1 (4-2-1) (4-2-2) (4-3)

(5-1-1) (5-1-2) 5-2 5-3

College of Liberal Arts And Sciences, Kitasato University 86 cmm

AllAll fundamentalfundamental domaindomain ofof polyominopolyomino forfor cmmcmm areare fundamentalfundamental domaindomain ofof fullfull symmetrysymmetry cmmcmm exceptexcept forfor rectanglerectangle polyominoes.polyominoes.

(1-1) (2-1-1) (2-1-2) (3-1-1) (3-1-2) 3-2 4-1 (4-2-1) (4-2-2) (4-3)

(5-1-1) (5-1-2) 5-2 5-3

College of Liberal Arts And Sciences, Kitasato University 87 cmm

AllAll fundamentalfundamental domaindomain ofof polyominopolyomino forfor cmmcmm areare fundamentalfundamental domaindomain ofof fullfull symmetrysymmetry cmmcmm exceptexcept forfor rectanglerectangle polyominoes.polyominoes.

(1-1) (2-1-1) (2-1-2) (3-1-1) (3-1-2) 3-2 4-1 (4-2-1) (4-2-2) (4-3)

(5-1-1) (5-1-2) 5-2 5-3

College of Liberal Arts And Sciences, Kitasato University 88 cmm

AllAll fundamentalfundamental domaindomain ofof polyominopolyomino forfor cmmcmm areare fundamentalfundamental domaindomain ofof fullfull symmetrysymmetry cmmcmm exceptexcept forfor rectanglerectangle polyominoes.polyominoes.

(1-1) (2-1-1) (2-1-2) (3-1-1) (3-1-2) 3-2 4-1 (4-2-1) (4-2-2) (4-3)

(5-1-1) (5-1-2) 5-2 5-3

College of Liberal Arts And Sciences, Kitasato University 89 cmm

AllAll fundamentalfundamental domaindomain ofof polyominopolyomino forfor cmmcmm areare fundamentalfundamental domaindomain ofof fullfull symmetrysymmetry cmmcmm exceptexcept forfor rectanglerectangle polyominoes.polyominoes.

(1-1) (2-1-1) (2-1-2) (3-1-1) (3-1-2) 3-2 4-1 (4-2-1) (4-2-2) (4-3)

(5-1-1) (5-1-2) 5-2 5-3

College of Liberal Arts And Sciences, Kitasato University 90 cmm

AllAll fundamentalfundamental domaindomain ofof polyominopolyomino forfor cmmcmm areare fundamentalfundamental domaindomain ofof fullfull symmetrysymmetry cmmcmm exceptexcept forfor rectanglerectangle polyominoes.polyominoes.

(1-1) (2-1-1) (2-1-2) (3-1-1) (3-1-2) 3-2 4-1 (4-2-1) (4-2-2) (4-3)

(5-1-1) (5-1-2) 5-2 5-3

College of Liberal Arts And Sciences, Kitasato University 91 cmm

AllAll fundamentalfundamental domaindomain ofof polyominopolyomino forfor cmmcmm areare fundamentalfundamental domaindomain ofof fullfull symmetrysymmetry cmmcmm exceptexcept forfor rectanglerectangle polyominoes.polyominoes.

(1-1) (2-1-1) (2-1-2) (3-1-1) (3-1-2) 3-2 4-1 (4-2-1) (4-2-2) (4-3)

(5-1-1) (5-1-2) 5-2 5-3

College of Liberal Arts And Sciences, Kitasato University 92 cmm

AllAll fundamentalfundamental domaindomain ofof polyominopolyomino forfor cmmcmm areare fundamentalfundamental domaindomain ofof fullfull symmetrysymmetry cmmcmm exceptexcept forfor rectanglerectangle polyominoes.polyominoes.

(1-1) (2-1-1) (2-1-2) (3-1-1) (3-1-2) 3-2 4-1 (4-2-1) (4-2-2) (4-3)

(5-1-1) (5-1-2) 5-2 5-3

College of Liberal Arts And Sciences, Kitasato University 93 cmm

AllAll fundamentalfundamental domaindomain ofof polyominopolyomino forfor cmmcmm areare fundamentalfundamental domaindomain ofof fullfull symmetrysymmetry cmmcmm exceptexcept forfor rectanglerectangle polyominoes.polyominoes.

(1-1) (2-1-1) (2-1-2) (3-1-1) (3-1-2) 3-2 4-1 (4-2-1) (4-2-2) (4-3)

(5-1-1) (5-1-2) 5-2 5-3

College of Liberal Arts And Sciences, Kitasato University 94 cmm

AllAll fundamentalfundamental domaindomain ofof polyominopolyomino forfor cmmcmm areare fundamentalfundamental domaindomain ofof fullfull symmetrysymmetry cmmcmm exceptexcept forfor rectanglerectangle polyominoes.polyominoes.

(1-1) (2-1-1) (2-1-2) (3-1-1) (3-1-2) 3-2 4-1 (4-2-1) (4-2-2) (4-3)

(5-1-1) (5-1-2) 5-2 5-3

College of Liberal Arts And Sciences, Kitasato University 95 cmm

AllAll fundamentalfundamental domaindomain ofof polyominopolyomino forfor cmmcmm areare fundamentalfundamental domaindomain ofof fullfull symmetrysymmetry cmmcmm exceptexcept forfor rectanglerectangle polyominoes.polyominoes.

(1-1) (2-1-1) (2-1-2) (3-1-1) (3-1-2) 3-2 4-1 (4-2-1) (4-2-2) (4-3)

(5-1-1) (5-1-2) 5-2 5-3

College of Liberal Arts And Sciences, Kitasato University 96 cmm

AllAll fundamentalfundamental domaindomain ofof polyominopolyomino forfor cmmcmm areare fundamentalfundamental domaindomain ofof fullfull symmetrysymmetry cmmcmm exceptexcept forfor rectanglerectangle polyominoes.polyominoes.

(1-1) (2-1-1) (2-1-2) (3-1-1) (3-1-2) 3-2 4-1 (4-2-1) (4-2-2) (4-3)

(5-1-1) (5-1-2) 5-2 5-3

College of Liberal Arts And Sciences, Kitasato University 97 pmg pmm,pmm, pmgpmg,, pgg,pgg, cmmcmm pmgpmg hashas twotwo newnew features.features. 1)1) BothBoth polyominopolyomino andand polyiamondpolyiamond cancan bebe fundamentalfundamental domains.domains.

College of Liberal Arts And Sciences, Kitasato University 98 pmg

2)2) ThereThere areare uncountablyuncountably manymany familiesfamilies If pmg

College of Liberal Arts And Sciences, Kitasato University 99 pmg

2)2) ThereThere areare uncountablyuncountably manymany familiesfamilies If pmg has a line made up of boundaries and 2-fold centers,

College of Liberal Arts And Sciences, Kitasato University 100 pmg

2)2) ThereThere areare uncountablyuncountably manymany familiesfamilies If pmg has a line made up of boundaries and 2-fold centers, we will ontain pmg by sliding belt-like region.

College of Liberal Arts And Sciences, Kitasato University 101 pmg

2)2) ThereThere areare uncountablyuncountably manymany familiesfamilies If pmg has a line made up of boundaries and 2-fold centers, we will ontain pmg by sliding belt-like region.

College of Liberal Arts And Sciences, Kitasato University 102 pmg

2)2) ThereThere areare uncountablyuncountably manymany familiesfamilies If pmg has a line made up of boundaries and 2-fold centers, we will ontain pmg by sliding belt-like region.

College of Liberal Arts And Sciences, Kitasato University 103 pmg

2)2) ThereThere areare uncountablyuncountably manymany familiesfamilies If pmg has a line made up of boundaries and 2-fold centers, we will ontain pmg by sliding belt-like region.

College of Liberal Arts And Sciences, Kitasato University 104 pmg

2)2) ThereThere areare uncountablyuncountably manymany familiesfamilies If pmg has a line made up of boundaries and 2-fold centers, we will ontain pmg by sliding belt-like region. We will call this line slide line.

College of Liberal Arts And Sciences, Kitasato University 105 WeWe cancan proveprove thatthat forfor pmgpmg 1)1) AtAt leastleast oneone tilingtiling inin thisthis familyfamily hashas nono additionaladditional symmetrysymmetry otherother thanthan pmg,pmg, 2)2) DisplacedDisplaced tilings,tilings, inin whichwhich componentcomponent unitunit squaressquares ofof polyominoespolyominoes (polyiamonds)(polyiamonds) areare notnot inin thethe squaresquare (triangular)(triangular) lattice,lattice, isis possiblepossible ifif andand onlyonly ifif pmgpmg hashas slideslide line.line. 3)3) TheThe familyfamily hashas non-displacednon-displaced tiling.tiling.

College of Liberal Arts And Sciences, Kitasato University 106 pmg

AllAll fundamentalfundamental domaindomain ofof polyominopolyomino forfor pmgpmg

1-1-1 (1-1-2) 2-1-1 2-1-2 (2-1-3) (2-1-4) 3-1-1 (3-1-2) 3-1-3 (3-1-4) 3-2-1 3-2-2 4-1-1 (4-1-2)

4-1-3 (4-1-4) 4-2-1 (4-2-2) 4-3-1 (4-3-2) 4-4-1 4-4-2 (4-5)

College of Liberal Arts And Sciences, Kitasato University 107 pmg

AllAll fundamentalfundamental domaindomain ofof polyominopolyomino forfor pmgpmg andand correspondingcorresponding tilingstilings areare shown.shown. XX isis notnot pmg.pmg.

1-1-1 (1-1-2) 2-1-1 2-1-2 (2-1-3) (2-1-4) 3-1-1 (3-1-2) 3-1-3 (3-1-4) 3-2-1 3-2-2 4-1-1 (4-1-2)

4-1-3 (4-1-4) 4-2-1 (4-2-2) 4-3-1 (4-3-2) 4-4-1 4-4-2 (4-5)

Slide line

College of Liberal Arts And Sciences, Kitasato University 108 pmg

AllAll fundamentalfundamental domaindomain ofof polyominopolyomino forfor pmgpmg andand correspondingcorresponding tilingstilings areare shown.shown. XX isis notnot pmg.pmg.

1-1-1 (1-1-2) 2-1-1 2-1-2 (2-1-3) (2-1-4) 3-1-1 (3-1-2) 3-1-3 (3-1-4) 3-2-1 3-2-2 4-1-1 (4-1-2)

4-1-3 (4-1-4) 4-2-1 (4-2-2) 4-3-1 (4-3-2) 4-4-1 4-4-2 (4-5)

College of Liberal Arts And Sciences, Kitasato University 109 pmg

AllAll fundamentalfundamental domaindomain ofof polyominopolyomino forfor pmgpmg andand correspondingcorresponding tilingstilings areare shown.shown. XX isis notnot pmg.pmg.

1-1-1 (1-1-2) 2-1-1 2-1-2 (2-1-3) (2-1-4) 3-1-1 (3-1-2) 3-1-3 (3-1-4) 3-2-1 3-2-2 4-1-1 (4-1-2)

4-1-3 (4-1-4) 4-2-1 (4-2-2) 4-3-1 (4-3-2) 4-4-1 4-4-2 (4-5)

Slide line

College of Liberal Arts And Sciences, Kitasato University 110 pmg

AllAll fundamentalfundamental domaindomain ofof polyominopolyomino forfor pmgpmg andand correspondingcorresponding tilingstilings areare shown.shown. XX isis notnot pmg.pmg.

1-1-1 (1-1-2) 2-1-1 2-1-2 (2-1-3) (2-1-4) 3-1-1 (3-1-2) 3-1-3 (3-1-4) 3-2-1 3-2-2 4-1-1 (4-1-2)

4-1-3 (4-1-4) 4-2-1 (4-2-2) 4-3-1 (4-3-2) 4-4-1 4-4-2 (4-5)

Slide line

College of Liberal Arts And Sciences, Kitasato University 111 pmg

AllAll fundamentalfundamental domaindomain ofof polyominopolyomino forfor pmgpmg andand correspondingcorresponding tilingstilings areare shown.shown. XX isis notnot pmg.pmg.

1-1-1 (1-1-2) 2-1-1 2-1-2 (2-1-3) (2-1-4) 3-1-1 (3-1-2) 3-1-3 (3-1-4) 3-2-1 3-2-2 4-1-1 (4-1-2)

4-1-3 (4-1-4) 4-2-1 (4-2-2) 4-3-1 (4-3-2) 4-4-1 4-4-2 (4-5)

College of Liberal Arts And Sciences, Kitasato University 112 pmg

AllAll fundamentalfundamental domaindomain ofof polyominopolyomino forfor pmgpmg andand correspondingcorresponding tilingstilings areare shown.shown. XX isis notnot pmg.pmg.

1-1-1 (1-1-2) 2-1-1 2-1-2 (2-1-3) (2-1-4) 3-1-1 (3-1-2) 3-1-3 (3-1-4) 3-2-1 3-2-2 4-1-1 (4-1-2)

4-1-3 (4-1-4) 4-2-1 (4-2-2) 4-3-1 (4-3-2) 4-4-1 4-4-2 (4-5)

College of Liberal Arts And Sciences, Kitasato University 113 pmg

AllAll fundamentalfundamental domaindomain ofof polyominopolyomino forfor pmgpmg andand correspondingcorresponding tilingstilings areare shown.shown. XX isis notnot pmg.pmg.

1-1-1 (1-1-2) 2-1-1 2-1-2 (2-1-3) (2-1-4) 3-1-1 (3-1-2) 3-1-3 (3-1-4) 3-2-1 3-2-2 4-1-1 (4-1-2)

4-1-3 (4-1-4) 4-2-1 (4-2-2) 4-3-1 (4-3-2) 4-4-1 4-4-2 (4-5)

Slide line

College of Liberal Arts And Sciences, Kitasato University 114 pmg

AllAll fundamentalfundamental domaindomain ofof polyominopolyomino forfor pmgpmg andand correspondingcorresponding tilingstilings areare shown.shown. XX isis notnot pmg.pmg.

1-1-1 (1-1-2) 2-1-1 2-1-2 (2-1-3) (2-1-4) 3-1-1 (3-1-2) 3-1-3 (3-1-4) 3-2-1 3-2-2 4-1-1 (4-1-2)

4-1-3 (4-1-4) 4-2-1 (4-2-2) 4-3-1 (4-3-2) 4-4-1 4-4-2 (4-5)

College of Liberal Arts And Sciences, Kitasato University 115 pmg

AllAll fundamentalfundamental domaindomain ofof polyominopolyomino forfor pmgpmg andand correspondingcorresponding tilingstilings areare shown.shown. XX isis notnot pmg.pmg.

1-1-1 (1-1-2) 2-1-1 2-1-2 (2-1-3) (2-1-4) 3-1-1 (3-1-2) 3-1-3 (3-1-4) 3-2-1 3-2-2 4-1-1 (4-1-2)

4-1-3 (4-1-4) 4-2-1 (4-2-2) 4-3-1 (4-3-2) 4-4-1 4-4-2 (4-5)

Slide line

College of Liberal Arts And Sciences, Kitasato University 116 pmg

AllAll fundamentalfundamental domaindomain ofof polyominopolyomino forfor pmgpmg andand correspondingcorresponding tilingstilings areare shown.shown. XX isis notnot pmg.pmg.

1-1-1 (1-1-2) 2-1-1 2-1-2 (2-1-3) (2-1-4) 3-1-1 (3-1-2) 3-1-3 (3-1-4) 3-2-1 3-2-2 4-1-1 (4-1-2)

4-1-3 (4-1-4) 4-2-1 (4-2-2) 4-3-1 (4-3-2) 4-4-1 4-4-2 (4-5)

College of Liberal Arts And Sciences, Kitasato University 117 pmg

AllAll fundamentalfundamental domaindomain ofof polyominopolyomino forfor pmgpmg andand correspondingcorresponding tilingstilings areare shown.shown. XX isis notnot pmg.pmg.

1-1-1 (1-1-2) 2-1-1 2-1-2 (2-1-3) (2-1-4) 3-1-1 (3-1-2) 3-1-3 (3-1-4) 3-2-1 3-2-2 4-1-1 (4-1-2)

4-1-3 (4-1-4) 4-2-1 (4-2-2) 4-3-1 (4-3-2) 4-4-1 4-4-2 (4-5)

College of Liberal Arts And Sciences, Kitasato University 118 pmg

AllAll fundamentalfundamental domaindomain ofof polyominopolyomino forfor pmgpmg andand correspondingcorresponding tilingstilings areare shown.shown. XX isis notnot pmg.pmg.

1-1-1 (1-1-2) 2-1-1 2-1-2 (2-1-3) (2-1-4) 3-1-1 (3-1-2) 3-1-3 (3-1-4) 3-2-1 3-2-2 4-1-1 (4-1-2)

4-1-3 (4-1-4) 4-2-1 (4-2-2) 4-3-1 (4-3-2) 4-4-1 4-4-2 (4-5)

College of Liberal Arts And Sciences, Kitasato University 119 pmg

AllAll fundamentalfundamental domaindomain ofof polyiamondpolyiamond forfor pmgpmg andand correspondingcorresponding tilingstilings areare shown.shown. XX isis notnot pmg.pmg.

2-1-1 (2-2-2) 4-1-1 4-1-2 (4-1-3) (4-1-4) 4-2 (4-3)

Slide line

College of Liberal Arts And Sciences, Kitasato University 120 pmg

AllAll fundamentalfundamental domaindomain ofof polyiamondpolyiamond forfor pmgpmg andand correspondingcorresponding tilingstilings areare shown.shown. XX isis notnot pmg.pmg.

2-1-1 (2-2-2) 4-1-1 4-1-2 (4-1-3) (4-1-4) 4-2 (4-3)

College of Liberal Arts And Sciences, Kitasato University 121 pmg

AllAll fundamentalfundamental domaindomain ofof polyiamondpolyiamond forfor pmgpmg andand correspondingcorresponding tilingstilings areare shown.shown. XX isis notnot pmg.pmg.

2-1-1 (2-2-2) 4-1-1 4-1-2 (4-1-3) (4-1-4) 4-2 (4-3)

Slide line

College of Liberal Arts And Sciences, Kitasato University 122 pmg

AllAll fundamentalfundamental domaindomain ofof polyiamondpolyiamond forfor pmgpmg andand correspondingcorresponding tilingstilings areare shown.shown. XX isis notnot pmg.pmg.

2-1-1 (2-2-2) 4-1-1 4-1-2 (4-1-3) (4-1-4) 4-2 (4-3)

Slide line

College of Liberal Arts And Sciences, Kitasato University 123 pmg

AllAll fundamentalfundamental domaindomain ofof polyiamondpolyiamond forfor pmgpmg andand correspondingcorresponding tilingstilings areare shown.shown. XX isis notnot pmg.pmg.

2-1-1 (2-2-2) 4-1-1 4-1-2 (4-1-3) (4-1-4) 4-2 (4-3)

College of Liberal Arts And Sciences, Kitasato University 124 pmg

AllAll fundamentalfundamental domaindomain ofof polyiamondpolyiamond forfor pmgpmg andand correspondingcorresponding tilingstilings areare shown.shown. XX isis notnot pmg.pmg.

2-1-1 (2-2-2) 4-1-1 4-1-2 (4-1-3) (4-1-4) 4-2 (4-3)

College of Liberal Arts And Sciences, Kitasato University 125 pmg

AllAll fundamentalfundamental domaindomain ofof polyiamondpolyiamond forfor pmgpmg andand correspondingcorresponding tilingstilings areare shown.shown. XX isis notnot pmg.pmg.

2-1-1 (2-2-2) 4-1-1 4-1-2 (4-1-3) (4-1-4) 4-2 (4-3)

Slide line

College of Liberal Arts And Sciences, Kitasato University 126 pmg

AllAll fundamentalfundamental domaindomain ofof polyiamondpolyiamond forfor pmgpmg andand correspondingcorresponding tilingstilings areare shown.shown. XX isis notnot pmg.pmg.

2-1-1 (2-2-2) 4-1-1 4-1-2 (4-1-3) (4-1-4) 4-2 (4-3)

College of Liberal Arts And Sciences, Kitasato University 127 pgg pmm,pmm, pmgpmg,, pggpgg,, cmmcmm

FinallyFinally pggpgg hashas samesame featuresfeatures asas pmg.pmg. 1)1) BothBoth polyominopolyomino andand polyiamondpolyiamond cancan bebe fundamentalfundamental domains.domains.

College of Liberal Arts And Sciences, Kitasato University 128 2)2) ThereThere areare familyfamily ofof uncountablyuncountably manymany tilingtiling originatedoriginated byby slide-lines.slide-lines. 3)3) DisplacementDisplacement pointspoints whichwhich

College of Liberal Arts And Sciences, Kitasato University 129