Initialization Code
Initialization Code
Manipulate
Manipulate
The pinwheel tiling divides a 1-2 triangle into five similar right triangles, an example of a substitution tiling. This Demonstration shows a variety of different substitution tilings. Set “number of steps” to 0 to see the underlying system.
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Manipulatelimit=Switch[sys,"AmmannA4",4,"BirdsandBees",4,"Labyrinth",4,"Domino",5,"Drafter4",5,"HalfHex",5,"Limhex",5,"LTetromino",5,"Pinwheel",5,"PPentomino",5,"QuarterHex",5,"Sphinx",5,"Tritan4",5,"Tromino",5,"WaltonQuintic",5,"Binary",6,"Equithirds",6,"Dog",7,"Kite",6,"Drafter3",7,"Harvest",7,"SqrtTwo",7,"AmmannPhiChair",8,"KitesandDarts",8,"PenroseRhomb",8,"RobinsonTriangle",8,"PsiWedge",8,"PsiTrap",8,"TribonacciTrap",8,"QuarticPinwheel",9,"SqrtPhi",10,"TriangleDuo",10,"Tritan3",10,"TwoTriangle",10,"WaltonChair",10,"TribonacciChordQuadrangle",12,"PsiTriple",15,"SqrtPsi",15,"AmmannChair",16,"SqrtRho",17,"PsiChordQuadrangle",19,"PsiQuad",20,"SqrtChi",20,"RhoChordQuadrangle",21,"RhoQuad",24];If[steps>limit,steps=limit-1];root=SubstitutionTilingSystems[sys]["Root"];symbol=Switch,1,x,2,x,3,x,2,x,GoldenRatio,ϕ,Root[-1-#1+&,1],ρ,Root[-1-+&,1],ψ,Root[-1-#1-+&,1],t,Root[-1-#1+&,2],χ,Root[1-#1-2-+&,2],x,Root[-1-+&,1],x;integer=IntegerQ[root];AlgebraicSubstitutionTiling[sys,steps,{"N","ImageSize"{600,410}}],Row[{Control@{{sys,"PsiQuad","tiling system"},{"Pinwheel""pinwheel tiling","Domino""domino tiling","Tromino""tromino tiling","LTetromino""L tetromino tiling","PPentomino""P pentomino tiling","Tritan4""tritan rep-tile tiling","Drafter3""30-60-90 3 tiling","Drafter4""30-60-90 4 tiling","HalfHex""half-hex rep-tile tiling","QuarterHex""quarter-hex rep-tile tiling","Sphinx""sphinx tiling","Labyrinth""labyrinth tiling","Limhex""limhex tiling","Tritan3""semi-detached house tiling","Kite""kite tiling","Equithirds""equithirds tiling","Dog""dog fractal","TriangleDuo""triangle duo tiling","AmmannChair""Ammann chair tiling","AmmannPhiChair""Ammann phi chair tiling","WaltonChair""Walton chair tiling","AmmannA4""Ammann A4 tiling","KitesandDarts""kites and darts tiling","RobinsonTriangle""Robinson triangle tiling","PenroseRhomb""Penrose rhomb tiling","Binary""binary tiling","BirdsandBees""birds and bees","PsiQuad""psi quad tiling","RhoQuad""rho quad tiling","PsiTriple""psi triple tiling","TwoTriangle""two triangle tiling","TribonacciChordQuadrangle""tribonacci chord quadrangle 3-3","PsiChordQuadrangle""psi chord quadrangle 4-3","PsiWedge""psi wedge tiling","PsiTrap""psi trap tiling","TribonacciTrap""tribonacci trap tiling","RhoChordQuadrangle""rho chord quadrangle 5-3","SqrtTwo""sqrt two tiling","SqrtPhi""sqrt phi tiling","SqrtPsi""sqrt psi tiling","SqrtRho""sqrt rho tiling","SqrtChi""sqrt chi tiling","Harvest""harvest substitution tiling","QuarticPinwheel""quartic pinwheel tiling","WaltonQuintic""Walton quintic tiling"}},Spacer[10],Control@{{steps,2,"number of steps"},Range[0,limit],ImageSizeSmall,Appearance"Labeled"}," step limit: ",Dynamic[limit]}],Row[{"found by: ",Dynamic[Row[Riffle[SubstitutionTilingSystems[sys]["Discoverers"],", "]]],Spacer[{20,20}],"root: ",Dynamic[If[integer,root,Row[{N[root],Spacer[15],MinimalPolynomial[root,symbol]," = 0"}]]]}],{limit,4,ControlTypeNone},{root,GoldenRatio,ControlTypeNone},{symbol,ϕ,ControlTypeNone},{integer,False,ControlTypeNone},ContentSize{620,460},AutorunSequencing{{1,20}},SaveDefinitionsTrue,TrackedSymbols{sys,steps,limit,root,symbol,integer}
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