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 .


Is it just coincidence or some actual reason?

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    $\begingroup$ related on math.se: Does the number pi have any significance besides being the ratio of a circle's diameter to its circumference?. I don't think there is much more to be said about it here $\endgroup$
    – glS
    Commented Jan 24, 2015 at 11:03
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    $\begingroup$ I think many $\pi$'s appear, because of the norm of the Fourier transform, which is used almost everywhere in physics. $\endgroup$
    – Tim
    Commented Jan 24, 2015 at 11:41
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    $\begingroup$ At root, it's due to $e^{i\pi}+1 = 0$ $\endgroup$ Commented Jan 24, 2015 at 12:09
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    $\begingroup$ $\pi$ is a dangerous invasive item. All hail $\tau$ :-) You can always make $\pi$ go away by embedding in inside other constants, as suggested in Christoph's answer. A more interesting question might be: why do we only use $\pi$ and $e$ , of all transcendental numbers, in physics? $\endgroup$ Commented Jan 24, 2015 at 13:42
  • $\begingroup$ Mystified as to how can a constant that describes a ratio in Euclidean Geometry be used in field equation that I though changed the geometry of space as the energy density increases. Could be wrong here. $\endgroup$
    – user86411
    Commented Jul 16, 2016 at 21:54

2 Answers 2


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...)


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


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