Expanding structures based on hinged cubes
Expanding structures based on hinged cubes
Sándor Kabai
1. Nine cubes are hinged along the edges to form an expanding structure.
1. Nine cubes are hinged along the edges to form an expanding structure.
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Manipulate[c1=PolyhedronData["Cube","Faces"];c2=Rotate[Translate[c1,{0,1,0}],-α,{0,0,1},{0.5,0.5,0}];c3=Translate[Translate[c1,{-1,1,0}],{1-Cos[α],Sin[α],0}];c8=Table[Rotate[{c2,c3},i2Pi/4,{0,0,1}],{i,4}];Graphics3D[{{Green,c1},c8},BoxedFalse,PlotRange{{-2.5,2.5},{-2.5,2.5},{-0.5,0.5}},SphericalRegionTrue,ViewAngle0.2,ViewPoint->{3,1,5}],{{α,0,"move"},0,Pi/2}]
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2. Combination o f nine cubes.
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Manipulate[c1a=PolyhedronData["Cube","Faces"];c2a=Rotate[Translate[c1a,{0,1,0}],-α,{0,0,1},{0.5,0.5,0}];c3a=Translate[Translate[c1a,{-1,1,0}],{1-Cos[α],Sin[α],0}];c8a=Table[Rotate[{c2a,c3a},i2Pi/4,{0,0,1}],{i,4}];t1=Table[Rotate[Translate[Translate[{c1a,c8a},{-2,2,0}],2{1-Cos[α],Sin[α],0}],i2Pi/4,{0,0,1}],{i,4}];Graphics3D[{{Green,c1},c8,t1},BoxedFalse,SphericalRegionTrue],{{α,0,"move"},0,Pi/2}]
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3. Unit made of four cubes.
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Manipulate[c1=PolyhedronData["Cube","Faces"];c2={RGBColor[0,1,1],Rotate[Translate[c1,{0.5,0.5,0}],α,{0,0,1}]};c3=Rotate[Translate[c1,{0.5,-0.5,0}],-α,{0,0,1}];c4=Translate[Rotate[{{c2},c3},Pi,{0,0,1}],{2Cos[α],0,0}];cx=Table[Translate[{c2,c3,c4},{i2(Cos[α]+Sin[α]),0,0}],{i,nx}];cy=Table[Translate[{{Green,c2},c3,c4,cx},{0,i2(Cos[α]+Sin[α]),0}],{i,ny}];Graphics3D[{c2,c3,c4,cx,cy},BoxedFalse,SphericalRegionTrue],{{α,0,"expand"},0,Pi/4},{{nx,1,"nx"},0,3,1,RadioButton},{{ny,1,"ny"},0,3,1,RadioButton},TrackedSymbolsManipulate]
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4. Planar version of unit made of four cubes
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Manipulate[x=Cos[α];y=Sin[α];vA={2x,0,0};vB={x,y,0};vC={0,0,0};vD={x,-y,0};vE={2x+y,-x,0};vF={2x+2y,0,0};vG={2x+y,x,0};vH={3x+y,x+y,0};vI={2x+y,x+2y,0};vJ={x+y,x+y,0};vK={x,2x+y,0};vL={x-y,x+y,0};t1={Yellow,Polygon[{vA,vB,vC,vD}]};t2=Polygon[{vA,vE,vF,vG}];t3={Yellow,Polygon[{vG,vH,vI,vJ}]};t4=Polygon[{vB,vJ,vK,vL}];tx=Table[Translate[{t1,t2,t3,t4},{i(2x+2y),0,0}],{i,nx}];ty=Table[Translate[{t1,t2,t3,t4,tx},{0,i2(x+y),0}],{i,ny}];Graphics3D[{t1,t2,t3,t4,tx,ty},BoxedFalse,SphericalRegionTrue,ViewPoint{0,0,10},ViewAngle0.1],{{α,0.5,"expand"},0,Pi/2},{{nx,1,"nx"},0,3,1,RadioButton},{{ny,1,"ny"},0,4,1,RadioButton},TrackedSymbolsManipulate]
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5. Planar version with bars along the squares. One telescoping diagonal is enough to operate the whole structure.
6. Two 3D versions of unit with four cubes (or rhombic dodecahedron).