Pentagonal loops in pentagonal tiling
Sándor Kabai
Tiling is created with ring of ten pentagons, then the pentagons are replaced with loops.
Tiling is created with ring of ten pentagons, then the pentagons are replaced with loops.
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Manipulate[p=ParametricPlot3D[{Cos[x]+mCos[(n-1)x],Sin[x]-mSin[(n-1)x],mSin[nx]},{x,0,2Pi},PlotStyle->{Yellow,Tube[0.07]}][[1]];Graphics3D[{p},Boxed->False,SphericalRegion->True,ViewPoint->{0,0,100},ImageSize->400],{{n,5,"m"},1,10,1,RadioButton},{{m,0.5,"m"},0,3,Appearance->"Labeled"},TrackedSymbols:>{n,m}]
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pn=Rotate[Scale[p,0.8{1,1,1},{0,0,0}],Pi/2,{0,0,1}];pent=Polygon[Table[{Sin[i2Pi/5],Cos[i2Pi/5],0},{i,5}]];Graphics3D[{pn,pent},Boxed->False,SphericalRegion->True,ViewPoint->{0,0,100},ImageSize->300]
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(*First"Select all"andevaluateasitis,thenmakepent=pn,andevualteagaintoobtaintheloopedfigures*)fi=GoldenRatio;dx=0.5/Sin[Pi/5];pent=pn;pent=Polygon[Table[{Sin[i2Pi/5],Cos[i2Pi/5],0},{i,5}]];(*Isnertpent=pnhere*)dec0=Table[Rotate[Translate[pent,{0,fi^2,0}],i2Pi/10,{0,0,1}],{i,10}];Graphics3D[{pent,dec0},Boxed->False,SphericalRegion->True,ViewPoint->{0,0,100},ImageSize->300]
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dec10=Table[Rotate[Translate[dec0,{8.05,0,0}],i2Pi/10,{0,0,1}],{i,10}];Graphics3D[{dec10},Boxed->False,SphericalRegion->True,ViewPoint->{0,0,100},ImageSize->300]
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dec5aa=Table[Rotate[Translate[dec0,{0,fi^2,0}],i2Pi/5,{0,0,1}],{i,5}];Graphics3D[{dec5aa},Boxed->False,SphericalRegion->True,ViewPoint->{0,0,100},ImageSize->400]
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dec5aa=Table[Rotate[Translate[dec0,{0,fi^2,0}],i2Pi/5,{0,0,1}],{i,5}];dec0a=Table[Rotate[Translate[dec0,{0,-fi^4,0}],i2Pi/5,{0,0,1}],{i,5}];Graphics3D[{dec0a,Yellow,dec5aa},Boxed->False,SphericalRegion->True,ViewPoint->{0,0,100},ImageSize->400]
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