△ Abd is an equilateral triangle,

BO=DO= 1，

AO=√ (Addo? )=3

Arc length of the first rotation =(60π×√3)/ 180=(√3)π/3.

∫ Arc length of the first and second coils = (60 π×√ 3)/180+(60 π×√ 3)/180 = (2×√ 3) π/3.

The arc length of the third turn is: (60 π×1)180 = π/3.

If it goes through 3n times (n is a positive integer)

∫3n÷3 = n，

Therefore, after 3n(n is a positive integer) times, the total length of the path that the diamond center O passes through is:

n×[(2√3+ 1)π/3+π/3]=πn(2√3+ 1)/3

3n = 36

So substitute (8√3+4)π.