Six-fold rhombus tiling using three rhombuses and a hexagon
Sándor Kabai
Three different kinds of rhombuses are supplemented with a hexagon to produce various tiling.
The three rhombuses are r1, r2, r3, where r3 is a square. When the basic assembly is placed in a square, then as Kürschak theorem can also be demonstrated. https://demonstrations.wolfram.com/KurschaksDodecagon/
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n=12;​​vn=Table[{Sin[i2Pi/n],Cos[i2Pi/n],0},{i,n+1}];​​tuben=Tube[vn];​​polyn=Rotate[Polygon[vn],0Pi/n,{0,0,1}];​​linen=Line[vn];​​pn=Table[Rotate[Translate[tuben,{1,0,0}],i2Pi/n,{0,0,1}],{i,n}];​​ln=Table[Rotate[Translate[linen,{1,0,0}],i2Pi/n,{0,0,1}],{i,n}];​​base=Rotate[{{Cyan,Translate[polyn,{0,1,0}],Yellow,pn}},0Pi/n,{0,0,1}];​​Graphics3D[{base},Boxed->False,ViewPoint->{0,0,100},ImageSize->300]
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n=12;​​vn=Table[{Sin[i2Pi/(n)],Cos[i2Pi/n],0},{i,(n+1)}];​​tuben=Tube[vn];​​polyn=Polygon[vn];​​linen=Line[vn];​​pn=Table[Rotate[Translate[tuben,{0,1,0}],i2Pi/n+Pi/n,{0,0,1}],{i,(n)}];​​ln=Table[Rotate[Translate[linen,{0,1,0}],i2Pi/n,{0,0,1}],{i,n}];​​Graphics3D[{ln},Boxed->False,ViewPoint->{0,0,100},ImageSize->300]
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n=12;​​v12=Table[{Sin[i2Pi/n+Pi/n],Cos[i2Pi/n+Pi/n],0},{i,(n+1)}];​​p12=Polygon[v12];​​p12a=Table[Rotate[Translate[Rotate[p12,Pi/n,{0,0,1}],{0,2.735,0}],i2Pi/12,{0,0,1}],{i,12}];​​Graphics3D[{ln,Cyan,p12,Yellow,p12a},Boxed->False,ViewPoint->{0,0,100},ImageSize->300]​​
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p12i=Table[Rotate[Translate[Rotate[p12,Pi/n,{0,0,1}],{0,3.735,0}],i2Pi/12,{0,0,1}],{i,12}];​​Graphics3D[{ln,p12,p12i},Boxed->False,ViewPoint->{0,0,100},ImageSize->300]
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p12b=Table[Rotate[Translate[Rotate[p12,Pi/n,{0,0,1}],{0,2.735,0}],i2Pi/6+0Pi/12,{0,0,1}],{i,6}];​​Graphics3D[{p12,ln,p12b},Boxed->False,ViewPoint->{0,0,100},ImageSize->300]
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x1=Sin[Pi/12];​​y1=Cos[Pi/12];​​r1=Polygon[{{0,0,0},{x1,y1,0},{0,2y1,0},{-x1,y1,0}}];​​​​x2=Sin[Pi/6];​​y2=Cos[Pi/6];​​r2=Polygon[{{0,0,0},{x2,y2,0},{0,2y2,0},{-x2,y2,0}}];​​​​x3=Sin[Pi/4];​​y3=Cos[Pi/4];​​r3=Polygon[{{0,0,0},{x3,y3,0},{0,2y3,0},{-x3,y3,0}}];​​​​Graphics3D[{Rotate[r3,2Pi/3,{0,0,1}],Pink,Rotate[r2,3Pi/12,{0,0,1}],Cyan,r1},Boxed->False,ViewPoint->{0,0,100},ImageSize->300,SphericalRegion->True]
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r1c=Table[Rotate[r1,i2Pi/12,{0,0,1}],{i,12}];​​r2ring1=Table[Rotate[Translate[r2,{0,1,0}],i2Pi/12+Pi/12,{0,0,1}],{i,12}];​​r3ring=Table[Rotate[Translate[r3,{0,2y1,0}],i2Pi/12,{0,0,1}],{i,12}];​​r2ring2=Table[Rotate[Translate[r2,{1+2y2+x2,-y2,0}],i2Pi/12+Pi/12,{0,0,1}],{i,12}];​​r1ring=Table[Rotate[Translate[r1,{2y1+2y3+x1,-y1,0}],i2Pi/12,{0,0,1}],{i,12}];​​Graphics3D[{Yellow,r1ring,LightGreen,r3ring,Green,r2ring1,Pink,r2ring2,Cyan,r1c},Boxed->False,ViewPoint->{0,0,100},ImageSize->300,SphericalRegion->True]