The mathematical constant π (≈ 3.1415…) is defined as the ratio between the circumference of a circle and its diameter. In mathematical terms, this becomes C = 2πr, where C is the circumference and r is the radius of the circle (the diameter is twice the radius). Most people are probably aware of this basic relationship for a circle from school math. Fewer people, however, might be aware that the mathematical constant π shows up in a large range of different areas of science and places out in nature, from the brightness of supernovas and music to electrical engineering and rainbows. It even occurs in probability problems involving needle dropping and statistical distributions. For the naive observer, these seem to have very little to do with circles.

Many proponents of new age woo think this means that there is something deeply mysterious and supernatural going on that signals the mystical nature of reality. They shiver at the thought of these occurrences of π being well-understood from a scientific and mathematical standpoint. This mirrors the complains made by the poet Keats against Newton when the latter explained how the rainbow worked (an honor that actually goes to Theodoric of Freiberg).

Keats thought Newton robbed people of the awe and wonder of nature. In reality, science is the poetry of reality, and knowing the scientific facts about how rainbows work adds to the awe and wonder of nature. It does not, and cannot, subtract.

### So why does π show up in physics, math and biology?

So why does π show up so frequently in different areas of math, physics and biology that superficially seem completely unrelated to the diameter and circumference of a circle? The reality behind this apparent conundrum is more awe-inspiring than any new age fairy tale. The fact is that virtually all instances of π in math and science traces back to the geometry of circles and spheres in some way. In many cases, this involves an understanding of complicated mathematics and scientific, which is why it seems so mysterious to many anti-science activists whose grasp of these areas are somewhat limited.

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Many appearances of π in physics has to do with the spread of different forces in three spatial dimensions. This relates to calculations that involve calculations the surface area or volume of a sphere that can be calculated using π. Others involve different kinds of waves that are modeled by trigonometric functions whose period can be understood with reference to the unit circle. This explains why π shows up in music, light and quantum mechanics. It shows up in the famous Euler’s identity, which comes from a circle in the complex plane and the fact that a rotation of π radians takes you from 1 to -1.

Many other instances in mathematics derive from this identity by substituting any of the other elements of Euler’s identity (1, 0, e and i) or from other trigonometric considerations. In probability theory, π shows up in the Buffon’s needle problem because of the angle (think segments of a circle and trigonometry) between the needle and the lines of the background. The reason that π shows up in the normal distribution traces back to the fact that in order for the distribution to have the properties it does, the calculation of the area under the curve will involve π. This traces back to the fact that a certain exponent turns out to be the square of the radius of a circle.

What about biology? Well, there are circles and spirals in biology as well, from pupils to the DNA double helix. There are also many biological processes that are periodic or oscillatory, from circadian rhythms to pattern developments. Things that are periodic or oscillatory can be modeled using different trigonometric functions and these involve π for geometric reasons.

Let us look at a few case studies.

### Why does π show up in the brightness of supernovas?

The apparent brightness of a light source (such as a supernova) can be calculated by taking the luminosity of the light source (amount of light radiated per second) and divide it by 4πd^{2}, where d is the distance:

Light propagates outward in a sphere and the surface area of a sphere is 4πr^{2}. This explains the geometric reason for why formulas for the apparent brightness of a supernova involve π, even though it might superficially seem to have nothing to do with circles. No need for woo.

### Why does π show up in needle drops?

Draw several parallel and equally spaced, horizontal lines across a paper. Take a needle that is as long as the distance between lines. Drop it randomly on the paper and record if it crosses a line or not. Repeat this experiment (called Buffon’s needle problem) over and over and record how many of these needle drops land over a line. The probability that this will happen can be shown to be 2 / π. How come π shows up in a needle drop experiment? How can this have anything to do with circles? It turns out that the way you calculate this probability involves an angle between the needle and the lines. This produces a segment of a circle and ties back to trigonometry and ultimately the unit circle. Again, nothing mysterious.

### Why does π show up in pattern development in animals?

To simplify a complicated biological research field, the patterns you see on many organism, from spots and strikes to limbs and cell division, are due (in part) to oscillations related to negative feedback loops with a delay. During development, many cases of gene expressions occurs in an oscillatory manner: it increases and decreases in predictable patterns. Because many other cellular processes are influenced by this, they too become periodic or oscillatory. Because periodic and oscillatory systems can be modeled with trigonometry, any occurrence of π can be easily traced back to circles.

### Why does π show up in the normal distribution?

Many observable things in nature follow a normal distribution. Or more precisely, many things in nature can be approximated to a natural distribution. If you have a large enough sample size, the central limit theorem tells us that you can go ahead and use statistical methods that assume normal distribution regardless of whatever distribution your random variable actually has out there in the real world.

In the probability density function of the normal distribution, π shows up. How come π shows up in statistics that, on the surface, seems to have very little, if nothing, to do with circles? It turns out that in order to for the probability density function to give a probability, the area under the curve (also called integral) must be equal to 1 (as the sum of all possibilities must be 100%). For this to work out, it must be the case that a certain integral (called Gaussian integral) is equal to the square root of π. How does π come in? Because it is the area under:

The exponent part x^{2}+y^{2} is the square of the radius of a circle, which brings us back to circles.

### Conclusion

There is nothing mysterious about finding π in statistics, physics, biology or any other field of science and math. They all traces back to spheres, circles, periods and oscillations that have well-understood and non-mysterious mathematical connection to π. This does not ruin the intriguing role of π in different fields of science. Quite the opposite, the knowledge only adds to the awe, wonder and excitement of nature.

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It’s nice to see you touch upon some of the woo and hub-bub that some people try to inject into mathematics. Is there a chance that you could cover these topics further in the future? Like looking at cranks that try to debunk Cantor’s work on infinite sets, golden ratio woo, or other psuedomathematics topics?

It would be interesting to do a similar post on phi.