Doodle graph from John D . Cook

https://www.johndcook.com/blog/2024/06/27/polygon-circumscribing-constant/
Maximum number of polygons
In[]:=
num=30
Out[]=
30
List of polygon/circle radii
In[]:=
r=RecurrenceTable[{f[n]==f[n-1]/Cos[Pi/n],f[2]==1},f,{n,2,num}];
In[]:=
r=Join[{0},r];
Preliminary drawing commands
In[]:=
drawing={Black,EdgeForm[{Blue,AbsoluteThickness[1]}]}
Out[]=

,EdgeForm
,AbsoluteThickness[1]
Construction drawings with circles
In[]:=
polycircles=Flatten[Table[{Opacity[1],Circle[{0,0},r[[n-1]]],Opacity[0],RegularPolygon[{r[[n]],Pi/2},n]},{n,3,num}]];
In[]:=
construction=Table[Graphics[Join[drawing,Take[polycircles,4*(n-2)]]],{n,3,5}]
Out[]=

,
,

In[]:=
Do[Export[StringJoin[NotebookDirectory[],"/20240628-Circumscribed polygon construction ",ToString[n+2],".png"],construction[[n]],"PNG",ImageSize->350,ImageResolution->100],{n,1,3}];

Circumscribed polygons without circles

In[]:=
polys=Table[RegularPolygon[{r[[n]],Pi/2},n],{n,3,num}];
In[]:=
poly30=Graphics[Join[drawing,{Opacity[0]},polys]]
Out[]=
In[]:=
Export[StringJoin[NotebookDirectory[],"/20240628-Circumscribed polygons.png"],poly30,"PNG",ImageSize->350,ImageResolution->100];