CITE THIS NOTEBOOK: Pentagonal art template and cluster of 12 rhombic triacontahedra by Sándor Kabai. Wolfram Community OCT 27 2022.
Pentagonal Art Template - Wolfram Demonstrations Project
Eight parallel lines are drawn with spacings, 1, , 1, , , 1, , where is the golden ratio. Then more lines are added with the same spacings. These 16 lines plus a central line are repeated 10 times after rotating them by 36 degrees. The intersecting lines produce a network which can be used as a template for establishing various patterns having pentagonal symmetry (e.g., a Penrose tiling). The set of intersection points is similar to the X-ray diffraction pattern of certain quasicrystals.
Eight parallel lines are drawn with spacings
ϕ
ϕ
ϕ
ϕ
ϕ
ϕ
In[]:=
ϕ=GoldenRatio;u=24;w1={Thickness[0.0005],Line[{{{0,-u,0},{0,u,0}},{{ϕ,-u,0},{ϕ,u,0}},{{ϕ+1,-u,0},{ϕ+1,u,0}},{{2ϕ+1,-u,0},{2ϕ+1,u,0}},{{2ϕ+2,-u,0},{2ϕ+2,u,0}},{{3ϕ+2,-u,0},{3ϕ+2,u,0}},{{4ϕ+2,-u,0},{4ϕ+2,u,0}},{{4ϕ+3,-u,0},{4ϕ+3,u,0}},{{5ϕ+3,-u,0},{5ϕ+3,u,0}}}]};w2=Translate[w1,{5ϕ+3,0,0}];w={w1,w2};w5=Table[Rotate[w,i2Pi/10,{0,0,1}],{i,10}];x=5ϕ+3;y=xTan[2Pi/20];vA={x,y,0};d=Sqrt[x^2+y^2];vB={x,y+d,0};vC={0,d,0};vD={0,0,0};vE={0,d+2y,0};vF={-x,y+d,0};p1={Thickness[0.005],Line[{vD,vA,vB,vC}]};star=Table[Rotate[p1,i2Pi/5,{0,0,1}],{i,5}];pe1={Thickness[0.005],Line[{vB,vE,vF}]};dec=Table[Rotate[pe1,i2Pi/5,{0,0,1}],{i,5}];(*Firstsetoflines*)Graphics3D[{w1,dec,star},ImageSize300,ImageSizeTiny,BoxedFalse,SphericalRegionTrue,ViewPoint{0,0,100}]
Out[]=
First and second set of lines
Graphics3D[{w1,w2,dec,star},ImageSize300,ImageSizeTiny,BoxedFalse,SphericalRegionTrue,ViewPoint{0,0,100}]
Out[]=
Ten times the 1st and 2nd sets of lines
Graphics3D[{w5},ImageSize500,ImageSizeTiny,BoxedFalse,SphericalRegionTrue,ViewPoint{0,0,100}]
Out[]=
Assembling perforated RT from faces
(*Halfdiametersofgoldenrhombus*)fi=GoldenRatio;(*Anglesofgoldenrhombus*)alf=ArcTan[fi];bet=ArcTan[1/fi];bet1=ArcTan[1/fi^2];rtx=Cos[alf];rty=Sin[alf];m=2rtyfi;icov=PolyhedronData["Icosahedron","VertexCoordinates"];rtvA={rtx,0,0};rtvB={0,rty,0};rtvC={-rtx,0,0};rtvD={0,-rty,0};rts1={rtvA,rtvB,rtvC,rtvD,rtvA};(*setperforatehere*)perf=0.9;rts2=perf{rtvA,rtvB,rtvC,rtvD,rtvA};r1=Polygon[Join[rts1->rts2]];r2=Translate[r1,{0,0,firty}];r3=Rotate[r2,Pi,{1,0,0}];r4=Rotate[Rotate[{r2,r3},Pi/2+0.001,{1,0,0}],Pi/2+0.001,{0,1,0}];r5=Rotate[Rotate[{r2,r3},Pi/2+0.001,{0,1,0}],Pi/2+0.001,{1,0,0}];rtp={FaceForm[Yellow,Blue],Table[Rotate[Rotate[{r2,r3,r4,r5},bet,{1,0,0}],i2Pi/5-Pi/2,{0,0,1}],{i,5}]};rtpn=Rotate[rtp,Pi,{0,0,1}];rt=PolyhedronData["RhombicTriacontahedron","Faces"];rtn=Rotate[rt,Pi,{0,0,1}];rt12=Map[Translate[rtn,#]&,m*icov];rt12p=Map[Translate[rtp,#]&,m*icov];Graphics3D[{rtp},ViewPoint{0,0,100},ImageSize200,ViewAngle0.01,BoxedFalse,SphericalRegionTrue]
Out[]=
In[]:=
rt12p1=Rotate[Scale[rt12p,6.14{1,1,1},{0,0,0}],2Pi/20,{0,0,1}];Graphics3D[{w5,Cyan,rt12p1},ViewPoint{0,0,100},ImageSize500,ViewAngle0.01,BoxedFalse,SphericalRegionTrue]
Out[]=
