CITE THIS NOTEBOOK: Pentagon spirals with golden ratio by Sándor Kabai. Wolfram Community, SEP 30 2022.
Pentagons are arranged in a circle, then increased in the proportion of the golden ratio to produce spirals. A pentagons can be replaces with any fitting objects, such as circles, stars and polyhedra.
See also: https://demonstrations.wolfram.com/PentagonSpirals/
fi=GoldenRatio;​​r=0.5/Sin[2Pi/10];​​m=0.5/Tan[2Pi/10];​​h=0.5/Tan[2Pi/20];​​frame=Table[{rSin[i2Pi/5],rCos[i2Pi/5],0},{i,6}];​​f1=Polygon[frame];​​circ=Polygon[Table[{Sin[i2Pi/50],Cos[i2Pi/50],0},{i,50}]];​​f2=Translate[f1,{0,h+m,0}];​​f10=Table[Rotate[f2,i2Pi/10,{0,0,1}],{i,10}];​​f5=Table[Rotate[f2,i2Pi/5,{0,0,1}],{i,5}];​​f5a=Rotate[Scale[f5,fi{1,1,1},{0,0,0}],2Pi/10,{0,0,1}];
Ring of ten pentagons
Graphics3D[{f10},BoxedFalse,ImageSize300,SphericalRegionTrue,ViewPoint{0,0,100}]
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5 pentagons selected
Graphics3D[{f10,Cyan,f5},BoxedFalse,ImageSize300,SphericalRegionTrue,ViewPoint{0,0,100}]
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Series of 5 pentagons increasing in size by the golden ratio and rotated in each step by 2Pi/10
f5i=Table[{Hue[0.4^2i],Rotate[Scale[f5a,fi^i{1,1,1},{0,0,0}],i2Pi/10,{0,0,1}]},{i,5}];​​Graphics3D[{f5i},BoxedFalse,ImageSize300,SphericalRegionTrue,ViewPoint{0,0,100}]
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Another approach is to prepare 5 separate spirals
sout=Table[{Hue[0.4^2i],Rotate[Scale[f2,fi^i{1,1,1},{0,0,0}],i2Pi/10,{0,0,1}]},{i,5}];​​Graphics3D[{f10,sout},BoxedFalse,ImageSize300,SphericalRegionTrue,ViewPoint{0,0,100}]
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The same with decreasing size
sin=Table[{Hue[0.4^2i],Rotate[Scale[f2,1/fi^i{1,1,1},{0,0,0}],i2Pi/10,{0,0,1}]},{i,5}];​​Graphics3D[{f10,sin},BoxedFalse,ImageSize300,SphericalRegionTrue,ViewPoint{0,0,100}]
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Five spirals
sout=Table[{Rotate[Scale[f2,fi^i{1,1,1},{0,0,0}],i2Pi/10,{0,0,1}]},{i,4}];​​s5=Table[{Hue[0.7^0.6i],Rotate[sout,i2Pi/5,{0,0,1}]},{i,5}];​​Graphics3D[{f5,s5,f10},BoxedFalse,ImageSize300,SphericalRegionTrue,ViewPoint{0,0,100}]
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Reduce the size to the origonal
s5sm=Scale[s5,1/fi/15.5{1,1,1},{0,0,0}];​​new1=Translate[s5sm,{0,h+m,0}];​​Graphics3D[{f5,new1},BoxedFalse,ImageSize300,SphericalRegionTrue,ViewPoint{0,0,100}]
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Reduce the size to the origonal, another version
Repeat with stars
Repeat with star
Ring of ten pentagons
Series of 5 pentagons increasing in size by the golden ratio and rotated in each step by 2Pi/10
Another approach is to prepare 5 separate spirals
The same with increasing size
Reduce the size to the origonal
Any pentagonal object inserted
Series of 5 pentagons increasing in size by the golden ratio and rotated in each step by 2Pi/10
Another approach is to prepare separate spirals
Five spirals
Same with dodecagedron
Series of 5 pentagons increasing in size by the golden ratio and rotated in each step by 2Pi/10
Another approach is to prepare separate spirals
The same with increasing size
Circles
Series of 5 increasing in size by the golden ratio and rotated in each step by 2Pi/10
Another approach is to prepare separate spirals
Five spirals
Series of 5 increasing in size by the golden ratio and rotated in each step by 2Pi/10
See it from below
Another approach is to prepare separate spirals
Five spirals