Space filling with lattice of rhombic triacontahedra (RT)
Space filling with lattice of rhombic triacontahedra (RT)
by Sándor Kabai
When RT are arranged in a lattice, then the spaces between them correspond to a non-convex polyhedra, which is produces by removing eight obGR from an RT.
When RT are arranged in a lattice, then the spaces between them correspond to a non-convex polyhedra, which is produces by removing eight obGR from an RT.
In[]:=
fi=1.618;alf=ArcTan[fi];bet=ArcTan[1/fi];bet1=ArcTan[1/fi^2];obGR=PolyhedronData["ObtuseGoldenRhombohedron","Faces"];rt=PolyhedronData["RhombicTriacontahedron","Faces"];rtf=Rotate[rt,bet,{0,1,0}];(**halfdiagonalsofgoldenrhombohedron*)y=Sin[alf];x=Cos[alf];(*diagonalofobGR*)diao=2
1
2
3-
;(*diagonalofinsideandoutsidecube*)dia1=2ySqrt[3];dia2=dia1-2diao;(*Insideandoutsidecubes*)u=dia2/(2Sqrt[3]);cub=PolyhedronData["Cube","Faces"];cubin=Scale[cub,2u];cubout=Scale[cub,2y];6
5
In[]:=
(*CoordindatesofobGR*)vO={u,-u,u};vP={y,-y,y};vA={yfi,-y,0};vB={yfi,0,x};vC={y,0,fiy};vD={0,-x,fiy};vE={0,-yfi,y};vF={x,-yfi,0};r1=Polygon[{vB,vC,vD,vO}];r2=Polygon[{vO,vD,vE,vF}];r3=Polygon[{vA,vB,vO,vF}];r4=Polygon[{vP,vC,vD,vE}];r5=Polygon[{vP,vE,vF,vA}];r6=Polygon[{vP,vA,vB,vC}];vOO={-u,-u,u};vPP={-y,-y,y};vAA={-yfi,-y,0};vBB={-yfi,0,x};vCC={-y,0,fiy};vDD={0,-x,fiy};vEE={0,-yfi,y};vFF={-x,-yfi,0};r1m=Polygon[{vEE,vDD,vCC,vPP}];r2m=Polygon[{vFF,vEE,vPP,vAA}];r3m=Polygon[{vPP,vCC,vBB,vAA}];r4m=Polygon[{vOO,vFF,vEE,vDD}];r5m=Polygon[{vOO,vFF,vAA,vBB}];r6m=Polygon[{vOO,vDD,vCC,vBB}];(*EightobGRaroundtheinsidecubeandtheoutsodecube*)Manipulate[full1=Translate[{r1,r2,r3,r4,r5,r6},fly{1,-1,1}];full2=Translate[{r1m,r2m,r3m,r4m,r5m,r6m},fly{-1,-1,1}];full4=Rotate[{full1,full2},Pi,{0,0,1}];full8=Rotate[{full1,full2,full4},Pi,{0,1,0}];full={full1,full2,full4,full8};cubv=PolyhedronData["Cube"][[1,1]];sp=Sphere[{0,0,0},0.06];sp8=Map[Translate[sp,#]&,2ycubv];Graphics3D[{{Red,sp8},{Yellow,full},{Opacity[0.3],cubout},Green,cubin},BoxedFalse,SphericalRegionTrue,PlotRange2.1,ViewAngle0.2,ViewPoint{2,-6,2}],{{fly,0.5,"fly"},0,0.7},TrackedSymbols->fly]
Out[]=
In[]:=
In[]:=
dep1={r1,r2,r3};dep2={r4m,r5m,r6m};dep4=Rotate[{dep1,dep2},Pi,{0,0,1}];dep8=Rotate[{dep1,dep2,dep4},Pi,{0,1,0}];dep={dep1,dep2,dep4,dep8};lap1=Polygon[{vF,vE,vFF,{0,-yfi,-y}}];lap2=Rotate[lap1,Pi,{1,0,0}];lap3=Rotate[Rotate[{lap1,lap2},Pi/2+0.001,{0,0,1}],Pi/2+0.001,{1,0,0}];lap4=Rotate[Rotate[{lap1,lap2},Pi/2+0.001,{1,0,0}],Pi/2+0.001,{0,0,1}];lap={lap1,lap2,lap3,lap4};rtdep={Red,lap,Green,dep};Graphics3D[{rtdep},BoxedFalse,SphericalRegionTrue,ViewPoint{0,-100,0},ImageSize200]
Out[]=
In[]:=
(*Thicconvexchapeisobtainedwhen8obGRareremovedfromRT*)Manipulate[full1=Translate[{r1,r2,r3,r4,r5,r6},fly{1,-1,1}];full2=Translate[{r1m,r2m,r3m,r4m,r5m,r6m},fly{-1,-1,1}];full4=Rotate[{full1,full2},Pi,{0,0,1}];full8=Rotate[{full1,full2,full4},Pi,{0,1,0}];full={full1,full2,full4,full8};Graphics3D[{rtdep,Yellow,full},BoxedFalse,SphericalRegionTrue,PlotRange2.1,ViewAngle0.2,ViewPoint{2,-6,2}],{{fly,0,"fly"},0,0.74},TrackedSymbols->fly]
Out[]=
