Embedded pentagons and bowtie tiling
Sándor Kabai
A series of pentagons are embedded into one another. The size is determined in the function of rotation. Spheres are place at the vertices of polygons. The resulting image can be inserted into pentagon tiling, e.g, into bowtie tiling.
A series of pentagons are embedded into one another. The size is determined in the function of rotation. Spheres are place at the vertices of polygons. The resulting image can be inserted into pentagon tiling, e.g, into bowtie tiling.
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Manipulate[poly1=Line[Table[{Sin[2Pii/n],Cos[2Pii/n],0},{i,n+1}]];sp={Sphere[{0,0,0},size]};sp1=Table[{Hue[0.34^2i],Translate[sp,{Sin[i2Pi/n],Cos[i2Pi/n],0}]},{i,n}];α=rot2Pi/n;a=Cos[Pi/n]/Cos[Pi/n-α];polygons=Table[Scale[Rotate[poly1,i*α,{0,0,1}],a^i{1,1,1},{0,0,0}],{i,fr}];spheres=Table[Scale[Rotate[{sp1},i*α,{0,0,1}],a^i{1,1,1},{0,0,0}],{i,fr}];Graphics3D[{poly1,polygons,spheres,sp1},ImageSize400,ViewPoint{0,0,100}],{{rot,0.2,"slide vertices"},1,0},{{fr,25,"number of polygons"},1,60},{{n,5,"sides of polygons"},5,10,1,RadioButton},{{size,0.1,"point size"},0.01,0.14},TrackedSymbols{rot,fr,n,size}]
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(*Thelatestimagecanbeusedinbowtietiling,mandncanbevaried*)m=2;n=1;u=2.3967;p=5;p0=Polygon[Table[{Sin[2Pix/p],Cos[2Pix/p],0},{x,0,p}]];(*p0canbevaried*)p0={poly1,spheres,sp1};p0={poly1,sp1,spheres};p1=Translate[p0,{0,0.809,0}];p2=Rotate[p1,Pi,{0,0,1}];p3=Translate[Rotate[{p1,p2},Pi/2,{0,0,1}],{0,u,0}];p4=Translate[Rotate[p3,Pi/2,{0,0,1}],{0,u,0}];p5=Translate[Rotate[{p1,p2},-Pi/2,{0,0,1}],{-u,0,0}];ux=Table[Translate[{p1,p2,p4,p3,p5},{i2u,0,0}],{i,m}];uy=Table[Translate[ux,{0,i2u,0}],{i,n}];Graphics3D[{ux,uy},ImageSize380,Boxed->False,ViewPoint->{0,0,100}]
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