Tiling with two rhombuses and a pentagon
Sándor Kabai
Various units are formed from two rhombuses (72°and 36°) for the purposes of covering (tiling) the plane. Supplementary pentagonal units are also used.
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fi=GoldenRatio;​​alf=ArcTan[fi];​​bet=ArcTan[1/fi];​​bet1=ArcTan[1/fi^2];​​(*Diagnalsofrhombuses*)​​y1=Cos[Pi/5];​​x1=Sin[Pi/5];​​vA={x1,0,0};​​vB={0,y1,0};​​vC={-x1,0,0};​​vD={0,-y1,0};​​t1=Polygon[{vA,vB,vC,vD}];​​u1=Table[Rotate[Translate[{t1},{0,y1,0}],i2Pi/5+0Pi/2,{0,0,1}],{i,5}];​​y2=2y1Sin[Pi/5];​​x2=Sqrt[1-y2^2];​​vE={x2,0,0};​​vF={0,y2,0};​​vG={-x2,0,0};​​vH={0,-y2,0};​​t2=Polygon[{vE,vF,vG,vH}];​​u2=Table[Rotate[Translate[t2,{0,y2,0}],i2Pi/10,{0,0,1}],{i,10}];​​Row[Graphics3D[#,Boxed->False,ImageSize->{100,100},ViewPoint->{0,0,100}]&/@{t1,t2,u1,u2}]​​
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u1o=Table[Rotate[Translate[t2,{1+x2,0,0}],i2Pi/5-Pi/2,{0,0,1}],{i,5}];​​Row[Graphics3D[#,Boxed->False,ImageSize->140,ViewPoint->{0,0,100}]&/@{u1,u1o,{Cyan,u1,Yellow,u1o}}]
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u2a=Table[Rotate[Translate[t1,{0,y1+1,0}],i2Pi/10+Pi/10,{0,0,1}],{i,10}];​​u2b=Table[Rotate[Translate[t1,{2y2+x1,0,0}],i2Pi/10+Pi/10,{0,0,1}],{i,10}];​​u2c=Table[Rotate[Translate[t2,{1+2y1+x2,0,0}],i2Pi/10,{0,0,1}],{i,10}];​​Graphics3D[{u2,Cyan,u2a,Pink,u2b,Yellow,u2c},Boxed->False,ViewPoint->{0,0,100},ImageSize->200]
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z2=Table[Rotate[Translate[t1,{0,y1+1,0}],i2Pi/10+Pi/10,{0,0,1}],{i,2}];​​z3=Table[Rotate[Translate[t1,{0,y1+1,0}],i2Pi/10+Pi/10,{0,0,1}],{i,3}];​​z4=Table[Rotate[Translate[t1,{0,y1+1,0}],i2Pi/10+Pi/10,{0,0,1}],{i,4}];​​z5=Table[Rotate[Translate[t1,{0,y1+1,0}],i2Pi/10+Pi/10,{0,0,1}],{i,5}];​​z6=Table[Rotate[Translate[t1,{0,y1+1,0}],i2Pi/10+Pi/10,{0,0,1}],{i,6}];​​z7=Table[Rotate[Translate[t1,{0,y1+1,0}],i2Pi/10+Pi/10,{0,0,1}],{i,7}];​​z8=Table[Rotate[Translate[t1,{0,y1+1,0}],i2Pi/10+Pi/10,{0,0,1}],{i,8}];​​z9=Table[Rotate[Translate[t1,{0,y1+1,0}],i2Pi/10+Pi/10,{0,0,1}],{i,9}];​​Row[Graphics3D[#,Boxed->False,ImageSize->120,ViewPoint->{0,0,100}]&/@{z2,z3,z4,z5,z6,z7,z8,z9}]​​
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v1={u2,Green,u2a,Cyan,u2b,Pink,u2c};​​v2={u2,Pink,u2a,Cyan,u2b};​​v3={Cyan,u2b,Yellow,u2c};​​v4={Yellow,u2,Pink,u2a};​​Row[Graphics3D[#,Boxed->False,ImageSize->120,ViewPoint->{0,0,100}]&/@{v1,v2,v3,v4}]
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Row[Graphics3D[#,Boxed->False,ImageSize->120,ViewPoint->{0,0,100}]&/@{u2,u2a,u2b,u2c}]
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u12={Table[Rotate[Translate[{t1},{0,y1,0}],i2Pi/5,{0,0,1}],{i,2}]};​​u13={Table[Rotate[Translate[{t1},{0,y1,0}],i2Pi/5,{0,0,1}],{i,3}]};​​u14={Table[Rotate[Translate[{t1},{0,y1,0}],i2Pi/5,{0,0,1}],{i,4}]};​​Row[Graphics3D[#,Boxed->False,ImageSize->120,ViewPoint->{0,0,100}]&/@{u1,u14,u13,u12}]
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u22=Table[Rotate[Translate[t2,{0,y2,0}],i2Pi/10,{0,0,1}],{i,2}];​​u23=Table[Rotate[Translate[t2,{0,y2,0}],i2Pi/10,{0,0,1}],{i,3}];​​u24=Table[Rotate[Translate[t2,{0,y2,0}],i2Pi/10,{0,0,1}],{i,4}];​​u25=Table[Rotate[Translate[t2,{0,y2,0}],i2Pi/10,{0,0,1}],{i,5}];​​u26=Table[Rotate[Translate[t2,{0,y2,0}],i2Pi/10,{0,0,1}],{i,6}];​​u27=Table[Rotate[Translate[t2,{0,y2,0}],i2Pi/10,{0,0,1}],{i,7}];​​u28=Table[Rotate[Translate[t2,{0,y2,0}],i2Pi/10,{0,0,1}],{i,8}];​​u29=Table[Rotate[Translate[t2,{0,y2,0}],i2Pi/10,{0,0,1}],{i,9}];​​Row[Graphics3D[#,Boxed->False,ImageSize->120,ViewPoint->{0,0,100}]&/@{u29,u28,u27,u26,u25,u24,u23,u22}]
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