Towers of RH

Sándor Kabai
Twelve towers of rhombic hexecontahedra (RH) are placed around a central RH. The size of each subsequent RH is reduced by the golden ratio. This assembly can be used to illustrate infinite series.

In[]:=

ManipulateModule{rh,ϕ,β,t1,t2,t3},rh=GraphicsComplex[PolyhedronData["RhombicHexecontahedron","VertexCoordinates"],Polygon[PolyhedronData["RhombicHexecontahedron","FaceIndices"]]];ϕ=1.618;β=ArcTan[0.618];t0=TableSpecularity[0.8],Hue[

3

0.85

i],TranslateScale{rh},

{1,1,1}

i

ϕ

,{0,0,0},0,0,

i

∑

n

1

n-3

ϕ

-

sink

n-1

ϕ

,{i,fr};t1=Translate[t0,{0,0,cross}];t2=TableRotateRotate[t1,2β,{0,1,0}],

i2π

5

+

2π

10

,{0,0,1},{i,5};t3=Rotate[{t1,t2},π,{0,1,0}];Graphics3D[{{Brown,If[rhs,rh,{}]},If[t1s,t1,{}],If[t2s,t2,{}],If[t3s,t3,{}]},BoxedFalse,SphericalRegionTrue,ViewPoint{0,0,-100},ImageSize400],{{fr,4,"tower height"},1,5,1,ControlTypeRadioButton},{{cross,0,"cross"},1,-10},{{sink,0,"sink"},0,1},{{rhs,False,"central figure"},{True,False},ImageSizeTiny,ControlPlacementLeft},{{t1s,False,"top tower"},{True,False},ImageSizeTiny,ControlPlacementLeft},{{t2s,True,"upper 5 towers"},{True,False},ImageSizeTiny,ControlPlacementLeft},{{t3s,True,"lower 6 towers"},{True,False},ImageSizeTiny,ControlPlacementLeft},TrackedSymbols{sink,rhs,fr,t1s,t2s,t3s}