Sarrus linkage
The sides of a square prism are folded at half height until the top and bottom squares meet. The movement corresponds to that of the Sarrus linkage, which converts a limited circular motion to a linear motion. Stack such prisms to produce a retractable structure., e.g. a foldable shelf system.
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Manipulate[n=2;t=4;m=0.3;vA={1,-1,0};vB={1,1,0};vC={-1,1,0};vD={-1,-1,0};Ex=-1-mSin[α];Ey=-1;Ez=mCos[α];vE={Ex,Ey,Ez};Fx=-1-mSin[α];Fy=1;Fz=mCos[α];vF={Fx,Fy,Fz};Gx=-1;Gy=-1;Gz=m2Cos[α];vG={Gx,Gy,Gz};Jx=-1;Jy=1;Jz=m2Cos[α];vJ={Jx,Jy,Jz};bottom=Polygon[{vA,vB,vC,vD}];p1=Polygon[{vC,vD,vE,vF}];p2=Polygon[{vJ,vG,vE,vF}];Hx=1;Hy=-1;Hz=m2Cos[α];vH={Hx,Hy,Hz};Ix=1;Iy=1;Iz=m2Cos[α];vI={Ix,Iy,Iz};top=Polygon[{vG,vH,vI,vJ}];side=Table[Rotate[{Green,p1,Cyan,p2},iPi/2,{0,0,1}],{i,n}];tower=Table[{Translate[{bottom,side,top},{0,0,i2mCos[α]}]},{i,t}];Graphics3D[{tower},SphericalRegionTrue,BoxedFalse,ViewPoint{-2,4,1},ImageSize{350,350}],{{α,0.6,"move"},0.2,1.4,ImageSizeTiny,ControlPlacementLeft},TrackedSymbols{α}]
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