Why does pi show up in formulas with no circles?

Pi turns up outside geometry because many problems contain a circle or sphere that is not obvious at first. The sources point to two main reasons: symmetry and oscillations. A third, smaller factor is the way physical constants are defined, which can move a factor of pi from one equation to another. [1]

The normal distribution, or bell curve, is used as a simple model for complex phenomena, and scientists generally assume that experimental error follows it. Its formula contains pi, and the area under the basic curve e to the power of minus x squared equals the square root of pi. This is so even though the curve has no obvious link to circles. [2] [3]

The explanation offered by The Conversation is that the two-dimensional function f(x)f(y) stays the same if the coordinate axes are rotated. Round things relate to circles, and circles involve pi. A MathOverflow answer describes the same idea: because the distribution is orthogonally invariant, one can integrate over a sphere and then over the radius, which brings in a constant tied to the sphere's size. The WIRED article gives the same trick, expanding the integral from one dimension to two to get a surface with circular symmetry. [4] [5] [1]

Anything that spreads equally in all directions forms a sphere, and a sphere's surface area is 4πR squared. WIRED uses sunlight as an example: as light travels outward, the power is spread over a growing sphere, so intensity falls with distance in a formula containing pi. The magnetic field of a moving charge has circular symmetry around it, so pi appears in that equation too. [1]

The magnetic constant is defined as 4π times 10 to the power of minus 7 in the units given. WIRED notes that it is possible to define constants so that pi does not appear in the magnetic field expression, but the magnetic constant also enters the speed of light, so pi would then appear there instead. In its words, one way or another pi is going to show up. [1]

A mass on a spring, a pendulum, a vibrating diatomic molecule and the current in a radio circuit can all be described with sine or cosine functions. Those functions can be pictured as a triangle whose angle keeps increasing, which is a point moving around a circle, so pi appears. WIRED states that it can be found in any oscillation modeled with sines or cosines. [1]

In quantum mechanics, pi appears so often that physicists combined it with the Planck constant to make h-bar, which is h divided by 2π. It features in the uncertainty principle. WIRED traces this partly to the integrals of wave functions, which have the same form as the bell-curve integral and so produce pi by the same two-dimensional trick. [1]