CITE THIS NOTEBOOK: Space filling based on golden pair of cubes by Sándor Kabai. Wolfram Community OCT 3 2022.
CITE THIS NOTEBOOK: Space filling based on golden pair of cubes by Sándor Kabai. Wolfram Community OCT 3 2022.
Space filling with cube, dodecahedron and bilunabirotunda
Space filling with cube, dodecahedron and bilunabirotunda
When dodecahedra are arranged in a lattice with edge to edge connection, then the space between the dodecahedra can be filled with cubes and bilunabirotunda.
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fi=GoldenRatio;rt=GraphicsComplex[PolyhedronData["RhombicTriacontahedron","VertexCoordinates"],Polygon[PolyhedronData["RhombicTriacontahedron","FaceIndices"]]];cub=GraphicsComplex[PolyhedronData["Cube","VertexCoordinates"],Polygon[PolyhedronData["Cube","FaceIndices"]]];dod=GraphicsComplex[PolyhedronData["Dodecahedron","VertexCoordinates"],Polygon[PolyhedronData["Dodecahedron","FaceIndices"]]];bb=GraphicsComplex[PolyhedronData["Bilunabirotunda","VertexCoordinates"],Polygon[PolyhedronData["Bilunabirotunda","FaceIndices"]]];c1=Translate[Scale[cub,1/fi],fi{0.5,0.5,0.5}];dodn=Rotate[Scale[dod,0.618{1,1,1},{0,0,0}],-ArcTan[0.618],{0,1,0}];bbn=Scale[bb,1/fi{1,1,1},{0,0,0}];bb1=Translate[Rotate[bbn,Pi/2+0.001,{0,0,1}],{fi/2,fi/2,0}];bb2=Translate[Rotate[bbn,Pi/2+0.001,{1,0,0}],{fi/2,0,fi/2}];bb3=Translate[Rotate[bbn,Pi/2+0.001,{0,1,0}],{fi,fi/2,fi/2}];cbase={Cyan,dodn,LightRed,,bb1,bb2,bb3,Yellow,c1};n=4;cx=Table[Translate[{cbase},{ifi,0,0}],{i,0,n}];cy=Table[Translate[{cx},{0,ifi,0}],{i,0,n}];cz=Table[Translate[{cx,cy},{0,0,ifi}],{i,0,n}];ax=Cylinder[{{-1.3,0,0},{2.6,0,0}},0.04];ay=Cylinder[{{0,-1.6,0},{0,3,0}},0.04];az=Cylinder[{{0,0,-1.6},{0,0,2}},0.04];Graphics3D[{{Red,ax,ay,az},cbase},BoxedFalse,SphericalRegionTrue,ViewAngle0.25]
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Take a pair of cubes
Graphics3D[{{Red,ax,ay,az},cub,Green,c1},BoxedFalse,SphericalRegionTrue,ViewAngle0.25]
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Replace large cube with dod
Graphics3D[{{Red,ax,ay,az},Cyan,dodn,Green,c1},BoxedFalse,SphericalRegionTrue,ViewAngle0.25]
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Add bb3
Graphics3D[{{Red,ax,ay,az},LightRed,bb3,Cyan,dodn,Green,c1},BoxedFalse,SphericalRegionTrue]
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Add bb1
Graphics3D[{{Red,ax,ay,az},LightRed,bb3,bb1,Cyan,dodn,Green,c1},BoxedFalse,SphericalRegionTrue,ViewAngle0.25]
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Add bb2, and this is the unit that fills space
Graphics3D[{{Red,ax,ay,az},LightRed,bb3,bb2,bb1,Cyan,dodn,Green,c1},BoxedFalse,SphericalRegionTrue,ViewAngle0.25]
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