Octahedron Tessellation

ver1={{0,0,0},{1,0,0},{0.5,Sqrt[3]/2,0},{0.5,0,0},{0.25,0.25 Tan[Pi/6],0},{0.75,-0.25 Tan[Pi/6],0},2{0.25,0.25 Tan[Pi/6],0},3{0.25,0.25 Tan[Pi/6],0},{0.5,0,0}+{0.25,0.25 Tan[Pi/6],0},0.5{Cos[Pi/3],Sin[Pi/3],0},Sqrt[3]/6{0,1,0},0.5{Cos[Pi/3],Sin[Pi/3],0}+Sqrt[3]/6{0,1,0},0.5{Cos[Pi/3],Sin[Pi/3],0}+{0.25,0.25 Tan[Pi/6],0},{-1,0,0},{-1,0,0}+{0.5,Sqrt[3]/2,0},{-1,0,0}+2{0.5,Sqrt[3]/2,0},{-1,0,0}-{0.5,Sqrt[3]/2,0}};

al2=Pi-1.9106332362490186`;

hept={GraphicsComplex[ver1,{Red,Polygon[{{3,1,5,4,7,8,13}}]}]};
trr1={GraphicsComplex[ver1,{Red,Polygon[{{10,3,12}}]}]};
trr2={GraphicsComplex[ver1,{Red,Polygon[{{1,10,11}}]}]};
pent={GraphicsComplex[ver1,{Blue,Polygon[{{8,7,4,2,9}}]}]};
trb1={GraphicsComplex[ver1,{Blue,Polygon[{{2,4,6}}]}]};

heptB=Rotate[hept,-Pi/3,{0,0,1},ver1[[3]]];
trr1B=Rotate[{GraphicsComplex[ver1,{Red,Polygon[{{10,3,12}}]}]},-Pi/3,{0,0,1},ver1[[3]]];
trr2B=Rotate[{GraphicsComplex[ver1,{Red,Polygon[{{1,10,11}}]}]},-Pi/3,{0,0,1},ver1[[3]]];
pentB=Rotate[{GraphicsComplex[ver1,{Blue,Polygon[{{8,7,4,2,9}}]}]},-Pi/3,{0,0,1},ver1[[3]]];
trb1B=Rotate[{GraphicsComplex[ver1,{Blue,Polygon[{{2,4,6}}]}]},-Pi/3,{0,0,1},ver1[[3]]];

heptB=Rotate[hept,-Pi/3,{0,0,1},ver1[[3]]];

trr1B=Rotate[{GraphicsComplex[ver1,{Red,Polygon[{{10,3,12}}]}]},-Pi/3,{0,0,1},ver1[[3]]];
trr2B=Rotate[{GraphicsComplex[ver1,{Red,Polygon[{{1,10,11}}]}]},-Pi/3,{0,0,1},ver1[[3]]];
pentB=Rotate[{GraphicsComplex[ver1,{Blue,Polygon[{{8,7,4,2,9}}]}]},-Pi/3,{0,0,1},ver1[[3]]];
trb1B=Rotate[{GraphicsComplex[ver1,{Blue,Polygon[{{2,4,6}}]}]},-Pi/3,{0,0,1},ver1[[3]]];

heptC=Rotate[hept,-2Pi/3,{0,0,1},ver1[[3]]];
trr1C=Rotate[{GraphicsComplex[ver1,{Red,Polygon[{{10,3,12}}]}]},-2Pi/3,{0,0,1},ver1[[3]]];
trr2C=Rotate[{GraphicsComplex[ver1,{Red,Polygon[{{1,10,11}}]}]},-2Pi/3,{0,0,1},ver1[[3]]];
pentC=Rotate[{GraphicsComplex[ver1,{Blue,Polygon[{{8,7,4,2,9}}]}]},-2Pi/3,{0,0,1},ver1[[3]]];
trb1C=Rotate[{GraphicsComplex[ver1,{Blue,Polygon[{{2,4,6}}]}]},-2Pi/3,{0,0,1},ver1[[3]]];

heptD=Rotate[hept,-Pi,{0,0,1},ver1[[3]]];
trr1D=Rotate[{GraphicsComplex[ver1,{Red,Polygon[{{10,3,12}}]}]},-Pi,{0,0,1},ver1[[3]]];
trr2D=Rotate[{GraphicsComplex[ver1,{Red,Polygon[{{1,10,11}}]}]},-Pi,{0,0,1},ver1[[3]]];
pentD=Rotate[{GraphicsComplex[ver1,{Blue,Polygon[{{8,7,4,2,9}}]}]},-Pi,{0,0,1},ver1[[3]]];
trb1D=Rotate[{GraphicsComplex[ver1,{Blue,Polygon[{{2,4,6}}]}]},-Pi,{0,0,1},ver1[[3]]];

{heptR,pentR,trr1R,trr2R,trb1R,heptBR,pentBR,trr1BR,trr2BR,trb1BR,heptCR,pentCR,trr1CR,trr2CR,trb1CR,heptDR,pentDR,trr1DR,trr2DR,trb1DR}=Map[Rotate[#,Pi,{0,0,1},ver1[[15]]/2]&,{hept,pent,trr1,trr2,trb1,heptB,pentB,trr1B,trr2B,trb1B,heptC,pentC,trr1C,trr2C,trb1C,heptD,pentD,trr1D,trr2D,trb1D}];

Manipulate[Graphics3D[{EdgeForm[{Yellow,Thin}],{Rotate[{Rotate[{hept,pent,Rotate[trb1,-r1 al2,{1,0,0},ver1[[4]]]},-r2 al2,ver1[[3]],ver1[[1]]],trr1,trr2,heptB,pentB,trb1BR,Rotate[{trr1B,trr2B,heptC,pentC,Rotate[trb1C,r1 al2,ver1[[16]]-ver1[[15]],ver1[[15]]],Rotate[{trr1C,trr2C,heptD,pentD,Rotate[{trr1D,trr2D},-r1 al2,ver1[[3]],ver1[[3]]],Rotate[trb1D,r1 al2,{1,0,0},ver1[[16]]]},-r2 al2,ver1[[15]],ver1[[3]]]},r2 al2,{1,0,0},ver1[[3]]]},-r2 al2,ver1[[15]],ver1[[1]]]},{Rotate[{heptR,pentR,Rotate[trb1R,r1 al2,{1,0,0},ver1[[15]]]},r2 al2,ver1[[3]],ver1[[14]]],trr1R,trr2R,heptBR,pentBR,trb1B,Rotate[{trr1BR,trr2BR,heptCR,pentCR,Rotate[trb1CR,-r1 al2,ver1[[3]],ver1[[1]]],Rotate[{trr1CR,trr2CR,heptDR,pentDR,Rotate[{trr1DR,trr2DR},r1 al2, ver1[[3]],ver1[[17]]],Rotate[trb1DR,-r1 al2,{1,0,0},ver1[[17]]]},r2 al2,ver1[[15]],ver1[[14]]]},r2 al2,ver1[[14]],ver1[[1]]]},If[sp,Table[Text[Style[i],ver1[[i]]],{i,1,17}],{}]},Boxed->False,ViewPoint->{0,0,19},PlotRange->2,SphericalRegion->True,ImageSize->{480,420}],{{r1,0,"step 1"},0,1,0.01},{{r2,0,"step 2"},0,1,0.01},{{sp,False,"show hint"},{True,False}},SaveDefinitions->True]

The Demonstration shows a tiling of octahedron surface.

G. N. Frederickson, Dissections: Plane & Fancy, New York: Cambridge University Press, 2002 pp. 111-112.

Contributed by: Izidor Hafner