Sándor Kabai
Zomeball geometry
The nodes (Zomeballs) used in the Zometool construction system have a shape of a truncated triacontahedron (RTT). This shape is close to that of small rhombicosidodecahedron (SRD), the difference being that the squares are replaced with golden rectangles.
Zomeball geometry
The nodes (Zomeballs) used in the Zometool construction system have a shape of a truncated triacontahedron (RTT). This shape is close to that of small rhombicosidodecahedron (SRD), the difference being that the squares are replaced with golden rectangles.
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Needs["PolyhedronOperations`"];rt=PolyhedronData["RhombicTriacontahedron","Faces"];rtn=Rotate[rt,Pi,{0,0,1}];rtt=Truncate[rt,0.5];Graphics3D[{rtt},BoxedFalse,SphericalRegionTrue,ImageSize200]
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This animation shows the process of truncation.
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Manipulate[rtn=Rotate[rt,Pi,{0,0,1}];rtman=Truncate[rt,trunc];Graphics3D[{rtman},BoxedFalse,SphericalRegionTrue,ImageSize250,PlotRange->1.6,ViewPoint{100,100,100},ViewAngle0.006],{{trunc,0,"truncate"},0,0.5},TrackedSymbolstrunc]
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The rectangular faces correspond to the those of RT.
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Graphics3D[{rtt,Yellow,Scale[rt,0.999]},BoxedFalse,SphericalRegionTrue,ImageSize200]
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The triangular faces correspond to the those of Icosahedron .
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ico=PolyhedronData["Icosahedron","Faces"];icon=Rotate[Scale[ico,1.823],Pi,{0,0,1}];Graphics3D[{rtt,Green,icon},BoxedFalse,SphericalRegionTrue,ImageSize200]
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The pentagonal faces correspond to the those of dodecahedron .
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dod=PolyhedronData["Dodecahedron","Faces"];dodn=Rotate[Scale[dod,1.25],Pi,{0,0,1}];Graphics3D[{rtt,Cyan,dodn},BoxedFalse,SphericalRegionTrue,ImageSize200]
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srd=PolyhedronData["SmallRhombicosidodecahedron","Faces"];srdn={Rotate[Rotate[srd,bet,{0,1,0}],Pi,{0,0,1}]};Graphics3D[{srdn},BoxedFalse,SphericalRegionTrue,ImageSize200]
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p1=0.2;rttp=Normal[rtt]/.p:Polygon[__](p/.pts:{{_?NumericQ,_,_},__}Module[{cent,innervtx},cent=Mean[pts];innervtx={1-p1,p1}.{#,cent}&/@pts;Join[pts,pts[[{1}]],innervtx,innervtx[[{1}]]]])//#&;Graphics3D[{rttp},BoxedFalse,SphericalRegionTrue,ImageSize200]
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The above method of perforation leaves lines at the vertices. To eliminate these lines it is possible to construct the assembly from triangles, squares and pentagon as shown below. In the first line of the code (perf) the extent of preformation can be adjusted.
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perf=0.5;x=0.5257;fi=1.618;alf=ArcTan[1.618];bet=ArcTan[0.618];bet1=ArcTan[1/1.618^2];(*zome-as3*)u=0.4253;vP={2uSqrt[3]/3,0,0};vQ={-uSqrt[3]/3,u,0};vR={-uSqrt[3]/3,-u,0};i1={vP,vQ,vR,vP};i2=perf{vP,vQ,vR,vP};ic1=Polygon[Join[i1->i2]];ic2=Rotate[Translate[ic1,{0,0,1.38}],bet1,{0,1,0}];ic3=Rotate[ic2,Pi,{0,0,1}];ic4=Table[Rotate[Rotate[{ic2,ic3},alf,{0,1,0}],i2Pi/5,{0,0,1}],{i,5}];ic5=Rotate[ic4,Pi,{0,1,0}];as3={ic4,ic5};(*zome-as4*)vA={1/fi/2,-1/2,0};vB={1/fi/2,1/2,0};vC={-1/fi/2,1/2,0};vD={-1/fi/2,-1/2,0};rA={1/fi,0,0};rB={0,1,0};rC={-1/fi,0,0};rD={0,-1,0};r0=Polygon[{rA,rB,rC,rD}];r1=Polygon[{vA,vB,vC,vD}];v1={vA,vB,vC,vD,vA};v2=perf{vA,vB,vC,vD,vA};r1=Polygon[Join[v1->v2]];r2=Translate[r1,{0,0,fi}];r3=Translate[r1,{0,0,-fi}];r4=Rotate[Rotate[{r2,r3},Pi/2,{1,0,0}],Pi/2,{0,1,0}];r5=Rotate[Rotate[{r2,r3},Pi/2,{1,0,0}],Pi/2,{0,0,1}];as4={Scale[Table[Rotate[Rotate[{r2,r3,r4,r5},bet,{1,0,0}],i2Pi/5-Pi/2,{0,0,1}],{i,5}],1/1.1755]};(*zome-as5*)m=2.787;n=5;dodperf=0.35;dx=0.52570.5/Sin[Pi/5];frame=Table[{dxSin[i2Pi/5],dxCos[i2Pi/5],m/2},{i,6}];hole=Table[{0.5dxSin[i2Pi/5],0.5dxCos[i2Pi/5],m/2},{i,6}];f1=Polygon[Join[frame->hole]];f2=Table[Rotate[Rotate[Rotate[{f1},Pi,{0,0,1}],2bet,{1,0,0}],i2Pi/5,{0,0,1}],{i,5}];f3=Rotate[{f1,f2},Pi,{1,0,0}];as5={Rotate[{f1,f2,f3},Pi/2+Pi,{0,0,1}]};zome=Scale[{as3,as4,as5},0.2,{0,0,0}];zome1=Rotate[zome,Pi,{0,0,1}];Graphics3D[{zome},BoxedFalse,SphericalRegionTrue,ImageSize200]
If SRDs are places at the vertices of a larger SRD, then we get a cluster of 60 SRDs, which are connected with their square faces. The envelop of this cluster is a SRD.
If RTs are placed at the vertices of SRD, then the envelope corresponds to RTT.
The envelope of a cluster of 60 RTTs is also an RTT.
The use of Zomaball can be well illustrated with assembling a rhombic hexecontahedron (the logo of Wolfram Alpha). https://demonstrations.wolfram.com/ConstructingARhombicHexecontahedronWithZometool/