What is the relationship between pi and Feigenbaum's constant? I am student who is new to bifurcation theory, and am interested in better understanding how Feigenbaum's constant relates to other constants. 
This morning I divided pi (using math.pi in Python) by Feigenbaum's constant (to 30 decimal places; δ = 4.669201609102990671853203821578), which yielded 0.6728329416029073. 
I found this number (rounded up to 6 decimal places) in a chart of elliptic integrals, but do not have enough understanding of bifurcation to draw a meaningful connection to the Heuman lambda function. 
Is there any literature exploring period doubling and pi?
 A: 
What is the relationship between π and Feigenbaum's constant ? My inquiry was born  out of looking at this graphic.

$\quad$ They are both related to shapes determined by quadratic equations. Thus, the value of  the former is given by $~\dfrac\pi8=\displaystyle\int_0^1\sqrt{x(1-x)}~dx,~$ and that of the latter by the logistic map $x_{n+1}=r~x_n(1-x_n).~$ Notice any similarities there ? ;-$)$ Also, looking at the graphic, one  cannot help but notice that the periods in question resemble a parabola, which is hardly  surprising, since the recursive expression of $\{x\}_{n\in\mathbb N}$ is parabolic, and, as we all know, circles  and ellipses, as well as parabolas, are ultimately conic sections. But this should not by any  means be misconstrued by thinking that their numerical values are supposedly connected.  Thus, Euler's famous constant e is also related to the other conic section, the hyperbola,  since $~\displaystyle\int_1^e\dfrac1x~dx=1,~$ but no one actually expects them to be numerically related. $($Hope  this helps$).$
