A hexagonal prism is capped with three rhombi at one of its ends. The surface area of , designated , is minimized when , the ratio of the diagonals of the rhombi, equals , which also pertains to the faces of the rhombic dodecahedron. Honeycombs are likewise characterized by this geometry.
P
P
S
d
1
2
≈0.707107In[]:=
Manipulate[Module[{u,h,vA,vB,vC,vD,vE,vF,vG,r1,r3,t1,t2,t6,vCC,sep1,sep2,dia,rd,rdn},u=Sqrt[3]/2;(*histheheightofprism*)h=2;(*MovementofpointCandD:Sqrt[(Sqrt[3]d/2)^2-0.25]*)vA={-0.5,-u,0};vB={-0.5,u,0};vC={0,0,Sqrt[(Sqrt[3]d/2)^2-0.25]};vD={-1,0,-Sqrt[(Sqrt[3]d/2)^2-0.25]};vE={-0.5,u,-h};vF={-0.5,-u,-h};vG={-1,0,-h};r1=Polygon[{vA,vC,vB,vD}];r3=Table[Rotate[r1,i2Pi/3,{0,0,1}],{i,3}];t1=Polygon[{vB,vD,vG,vE}];t2=Polygon[{vA,vD,vG,vF}];t6={RGBColor[0,1,1],Table[Rotate[{t1,t2},i2Pi/3,{0,0,1}],{i,3}]};vCC={1,0,0};sep1={Red,Line[{vA,vB,vCC,vA}]};sep2=Line[Table[{Sin[i2Pi/6+Pi/6],Cos[i2Pi/6+Pi/6],-Sqrt[(Sqrt[3]d/2)^2-0.25]},{i,7}]];dia=Table[Rotate[Line[{vC,vD}],i2Pi/3,{0,0,1}],{i,3}];rd=PolyhedronData["RhombicDodecahedron","Faces"];rdn=Translate[Rotate[Scale[rd,1.0606{1,1,1}],ArcTan[Sqrt[2]],{0,1,0}],{0,0,-0.7071}];(*honeycomb:rdy=Table[Translate[{r3,t6},{0,iSqrt[3],0}],{i,3}];rdx=Table[Translate[{rdy},{i1.5,-iSqrt[3]/2,0}],{i,2}];*)Grid[{{Graphics3D[{{Yellow,If[rdns,rdn]},sep1,{Red,dia},{Opacity[0.5],r3,t6}},BoxedFalse,SphericalRegionTrue,ImageSize400,ViewAngle0.13,ViewPoint{7,-5,3},PlotRange{{-1.2,1.2},{-1,1},{-2,1}}],ParametricPlot[{d,3/2(Sqrt[3]Sqrt[(4(Sqrt[(Sqrt[3]d/2)^2-0.25])^2+1)])+6h-3(Sqrt[(Sqrt[3]d/2)^2-0.25])},{d,0.6,1},AxesLabel{Style["d",Italic],Style["S",Italic]},Epilog{Red,PointSize[Large],Point[{d,3/2(Sqrt[3]Sqrt[(4(Sqrt[(Sqrt[3]d/2)^2-0.25])^2+1)])+6h-3(Sqrt[(Sqrt[3]d/2)^2-0.25])}]}]}}]],{{d,0.7071,Row[{"ratio of diagonals ",Style["d",Italic]}]},0.6,1,.001,Appearance"Labeled"},{{rdns,False,"show rhombic dodecahedron"},{True,False}},TrackedSymbols{rdns,d},ControlPlacementTop]
Out[]=
