Significance of $\pi$ in physics We all know this magical mathematical constant. 
My question being , how and why pi just shows up in every other physics derivation or formula or even statistics for that matter .
http://en.wikipedia.org/wiki/List_of_formulas_involving_%CF%80
Is it just coincidence or some actual reason? 
 A: Because circles are everywhere, sometimes in the guise of periodicity and rotational symmetry.
In the particular case of gravity, the appearance of $\pi$ in Einstein's field equations is due to an (arguably) unfortunate choice of $G$ and it would go away if we switched to a rationalized unit system. (But note that if we did so, $\pi$ would re-appear in Newton's law of gravitation...)
A: You wanted to know about Pi:
Pi also shows up in the Rocky Planet Density Equation ( gravitational compression )
Density of Rocky Planets:
Den = ( 1 + Pi ) x 10^-9 * R ^3  + ( 1 + sqrt 2 ) x 10^-1 * R  + 2900 kg/m^3.
The first term has Pi in it and One. It is the " radial - triaxial " coefficient
of radial compression of rocky planet materials associated with the gravitational
compression caused by increasing mass due to increasing radius R ( in Kilometers ).
The second term has One and the square root of 2, but has a gradient of 0.1, and the 
third term is a constant related to the near surface materials. 2900 is a blend of
Basalts and granites ( 2941.05, and 2657.05 blended by volume ).
Anyway, notice that the 10^-9 takes a lot of Radius R to create radial compression
in three axis.  Remember that Pi is used in the circumference, it is used in the 
surface area, it is used in the Volume, and now it is used in the Material
Density of semi-spherical planets. ( spinning oblate spheroids ).
Also, if the only thing you know about a planet is its Radius ( use an array of 
Radiuses ), then you can get the Planets, Radius ( given ), Surface Area ( 4 pi R^2),
Volume, ( 4/3 Pi R^3 ), Density ( density equation above ), Mass ( Volume x Density ),
Gravity ( G M / R^2 ), fraction of gravity ( fraction of density) X ( fraction of Radius ),
and surface composition ( variation from 2900 ) .  
Venus is 2658 ( Mostly granitic ), while Mars is 2954 ( mostly basaltic ).
So Pi and a few other constants are very useful, as are 1 + these constants.
Pi, and 1+Pi, PHI and 1+PHI, and 2+PHI, and PHI-1, and e, and 1+e, and 
1 and 1 + square root of 2. 
There is also an exponential growth curve that passes through the points
(0, 0.581976707 ), and ( 1, 1.581976707 ) that has an area under the curve
between these two points of 1.00000000....... That same curve has an area
under the curve from ( - infinity, 0 ) to ( 0, 0.581976707 ) = e.
You can get the points at each integer by raising 0.581976707 to the n by using
the Y^x function on a calculator.  eg:  ( 0.581976707 ) ^2 = 0.338696887.
The ratio of 0.338696887 / 0.581976707 = 0.581976707.
Every time you step one number to the left, the height is 58.1976707 % of the
prior height, or if you step one number to the right, the height is 158.1976707 %
of the number before it.
The Numbers Pi, e , and PHI control much of the Universe. as does ratios
of the integrals, and derivatives of these numbers.
Look at the numbers 1 / ( 1+e ) and 1 minus 1/ ( 1+e ) and compare it to
Baryonic and Non- Baryonic Material fractions ( percentages ) in the Universe.
28.9 % and 73.1 %
Mike Clark
Golden, Colorado, USA
