Have a helper function
(* helpers *)
Clear[closeL]
closeL[l_List] := Join[l, {First[l]}]
an equilateral triangle is easiest generated from the solution of the cyclotomic equation of third degree
In[24]:= (* equilateral triangle *)
Solve[x^3 == 1, x]
Out[24]= {{x -> 1}, {x -> -(-1)^(1/3)}, {x -> (-1)^(2/3)}}
rotation by 30 Degree is multiplication by Exp[Pi I/6]
, so your first triangle is just:
(* rotation by 30 Degree is multiplication by Exp[Pi I/6], so the first triangle is, keeping the zero in the center for practical reasons:*)
Graphics[Line[closeL[{Re[#], Im[#]} & /@ (Exp[Pi I/2] Last[Last[#]] & /@ Solve[x^3 == 1, x])]]]
to get the solution of the cyclotomic equation upright, i.e. with one horizontal edge.
How to proceed:
- shrink the roots by the right value (9/16)
- rotate the roots by multiplication with another
Exp[Pi I/6]
- continue until you did that 13 times
- plot all solutions collected as shown above