Seven-fold rhombus tiling examples
Sándor Kabai
In a former submission the basis for seven-fold rhombus tiling was explored, where the relationship with polar zonohedra was also described. Now a couple of examples are given for constructing tiling from the three different tiles.
Two basic tiling patterns can be illustrated by depicting seven-sided an fourteen-sided polygons aligned around a point of rotation seven and fourteen times with and angular spacing Pi/7 and Pi/14. The patters contain only three different rhombuses. (see the toy: https://www.fractiles.com/)
In a former submission the basis for seven-fold rhombus tiling was explored, where the relationship with polar zonohedra was also described. Now a couple of examples are given for constructing tiling from the three different tiles.
Two basic tiling patterns can be illustrated by depicting seven-sided an fourteen-sided polygons aligned around a point of rotation seven and fourteen times with and angular spacing Pi/7 and Pi/14. The patters contain only three different rhombuses. (see the toy: https://www.fractiles.com/)
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v7=Table[{Sin[i2Pi/7],Cos[i2Pi/7],0},{i,8}];tube7=Tube[v7];p7=Table[Rotate[Translate[tube7,{0,-1,0}],i2Pi/14,{0,0,1}],{i,14}];poly7=Translate[Polygon[v7],{0,-1,0}];seven={poly7,Green,p7};Graphics3D[{tube7},Boxed->False,ViewPoint->{0,0,100},ImageSize->200]
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v7=Table[{Sin[i2Pi/7],Cos[i2Pi/7],0},{i,8}];tube7=Tube[v7];p7=Table[Rotate[Translate[tube7,{0,-1,0}],i2Pi/14,{0,0,1}],{i,14}];poly7=Translate[Polygon[v7],{0,-1,0}];seven={poly7,Green,p7};
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v14=Table[{Sin[i2Pi/14],Cos[i2Pi/14],0},{i,14}];tube14=Tube[v14];poly14=Polygon[v14];line14=Line[v14];p14=Table[Rotate[Translate[tube14,{0,-1,0}],i2Pi/14,{0,0,1}],{i,14}];l14=Table[Rotate[Translate[line14,{0,-1,0}],i2Pi/14,{0,0,1}],{i,14}];fourteen={Cyan,Translate[poly14,{0,1,0}],Yellow,p14};Row[Graphics3D[#,Boxed->False,ImageSize->200,ViewPoint->{0,0,100}]&/@{seven,fourteen}]
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n=14;Do[zon[i,j]={2Sin[πj/n]Cos[πi/n],2Sin[πj/n]Sin[πi/n],0.2(n/2-j)},{i,0,3n},{j,0,n}];pz14=Table[Polygon[{zon[i+j+1,j],zon[i+j+2,j+1],zon[i+j+1,j+2],zon[i+j,j+1]}],{i,0,2n,2},{j,0,n-2}];Graphics3D[{pz14},Boxed->False,ViewPoint->{0,0,100},ImageSize->200,SphericalRegion->True];
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Row[Graphics3D[#,Boxed->False,ImageSize->200,ViewPoint->{0,0,100}]&/@{{Yellow,p14},pz14}]
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Creating the tiles: t1, t2, t3. The envelope is a heptagon. The side is 1, the x and y values are the half diagonals of rhombuses; used for determining the distances of rings. The angles are Pi/7, 2Pi/7 and Pi/14.
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(*TileNo.1:*)y1=Cos[Pi/7];x1=Sin[Pi/7];vA={x1,0,0};vB={0,y1,0};vC={-x1,0,0};vD={0,-y1,0};t1={Cyan,Polygon[{vA,vB,vC,vD,vA}]};t17=Table[Rotate[Translate[{t1},{0,y1,0}],i2Pi/7+Pi/2,{0,0,1}],{i,7}];
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(*TileNo.2:t2*)y2=2y1Sin[Pi/7];x2=Sqrt[1-y2^2];vE={x2,0,0};vF={0,y2,0};vG={-x2,0,0};vH={0,-y2,0};t2={LightPurple,Polygon[{vE,vF,vG,vH,vE}]};t27=Table[Rotate[Translate[t2,{1+x2,0,0}],i2Pi/7,{0,0,1}],{i,7}];t127={t17,t27};
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(*TileNo.3:t3*)y3=(1+2x2)Sin[Pi/7];x3=Sqrt[(1+2x2)^2-y3^2]-2y1;vI={x3,0,0};vJ={0,y3,0};vK={-x3,0,0};vL={0,-y3,0};t3={Yellow,Polygon[{vI,vJ,vK,vL,vI}]};t37=Table[Rotate[Translate[t3,{2y1+x3,0,0}],i2Pi/7+Pi/7,{0,0,1}],{i,7}];t1237={t17,t27,t37};sp=Sphere[{0,0,0},0.06];
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t1cent=Translate[t1,{0,y1,0}];t2cent=Rotate[Translate[t2,{0,y2,0}],-5Pi/14,{0,0,1}];t3cent=Rotate[Translate[t3,{0,y3,0}],3Pi/14,{0,0,1}];Graphics3D[{sp,t1cent,t2cent,t3cent},Boxed->False,ViewPoint->{0,0,100},ImageSize->200]
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(*Thethreerhombusesformrings,witht1atthecentre*)Row[Graphics3D[#,Boxed->False,ImageSize->160,ViewPoint->{0,0,100}]&/@{t17,t27,t37}]
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Row[Graphics3D[#,Boxed->False,ImageSize->160,ViewPoint->{0,0,100}]&/@{t17,t27,t1237}]
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If rhombuses t3 is placed at the centre (14 of them), then 6 rings can be formed.
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(*Herearetherings*)t114=Table[Rotate[Translate[Rotate[t1,Pi/2,{0,0,1}],{y1+1,0,0}],i2Pi/14,{0,0,1}],{i,14}];t214={Table[Rotate[Translate[t2,{0,-y2-2y3,0}],i2Pi/14,{0,0,1}],{i,14}]};t314={Table[Rotate[Translate[t3,{0,-y3,0}],i2Pi/14,{0,0,1}],{i,14}]};y4=(2y3+2y2)Sin[Pi/14];x4=(y4/Tan[Pi/14])-(1+2y1);t414={Table[Rotate[Translate[t2,{1+2y1+x4,0,0}],i2Pi/14,{0,0,1}],{i,14}]};t514=Table[Rotate[Translate[t1,{2y3+2y2+x1,0,0}],i2Pi/14+Pi/14,{0,0,1}],{i,14}];t614={Table[Rotate[Translate[t3,{1+2y1+2x2+x3,0,0}],i2Pi/14,{0,0,1}],{i,14}]};v14=Table[(1+2y1+2x4+2x3){Sin[i2Pi/14+Pi/14],Cos[i2Pi/14+Pi/14],0},{i,15}];p14=Tube[v14];Graphics3D[{p14,t114,t214,t314,t414,t514,t614},Boxed->False,ViewPoint->{0,0,100},ImageSize->300]
Forming various units.