In Carl Sagan’s Contact, the extraterrestrials embedded a message in the irrational number pi (the circumference of a circle divided by its radius). But some other numbers are critical to the structure of our universe too — and why they are critical does not make obvious sense.

➤ Perhaps the most fundamental and mysterious one is the fine structure constant of the universe:

A seemingly harmless, random number with no units or dimensions has cropped up in so many places in physics and seems to control one of the most fundamental interactions in the universe.

Its name is the fine-structure constant, and it’s a measure of the strength of the interaction between charged particles and the electromagnetic force. The current estimate of the fine-structure constant is 0.007 297 352 5693, with an uncertainty of 11 on the last two digits. The number is easier to remember by its inverse, approximately 1/137.

If it had any other value, life as we know it would be impossible. And yet we have no idea where it comes from.

Paul Sutter, “Life as we know it would not exist without this highly unusual number” at Space.com (March 24, 2022)

Many famous scientists have reflected on 1/137:

The brilliant physicist Richard Feynman (1918-1988) famously thought so, saying there is a number that all theoretical physicists of worth should “worry about”. He called it “one of the greatest damn mysteries of physics: a magic number that comes to us with no understanding by man”…

What’s special about alpha is that it’s regarded as the best example of a pure number, one that doesn’t need units. It actually combines three of nature’s fundamental constants – the speed of light, the electric charge carried by one electron, and the Planck’s constant, as explains physicist and astrobiologist Paul Davies to Cosmos magazine. Appearing at the intersection of such key areas of physics as relativity, electromagnetism and quantum mechanics is what gives 1/137 its allure.

Paul Ratner, “Why the number 137 is one of the greatest mysteries in physics” at Big Think (October 31, 2018)

Nobelist Wolfgang Pauli (1945) is said to have remarked, “When I die, my first question to the devil will be: What is the meaning of the fine structure constant?” At any rate, he thought about it a great deal during his life.

University of Nottingham physics professor Laurence Eaves thinks the number 1/137 would be good for starting communication with intelligent aliens as they would be likely to know about it and to realize they were dealing with other intelligent entities.

➤ Here’s another thought-provoking number. Consider the irrational number known as phi (ϕ) or the Golden Ratio. Jordan Ellenberg author of Shape: The Hidden Geometry of Information, Biology, Strategy, Democracy, and Everything Else (2021):

Among the mysteries of the irrationals, one number holds a special place: the so-called golden ratio. The golden ratio’s value is about 1.618 (but not exactly 1.618, since then it would be the ratio 1,618/1,000, and therefore not irrational) and it’s also referred to by the Greek letter φ, which is pronounced “fee” if you’re a mathematician and “fie” if you are in a fraternity. If you want an exact description, the golden ratio can be expressed as (1/2)(1+√5.)

Jordan Ellenberg, “The Most Irrational Number” at Slate (June 8, 2021)

We find this number everywhere too:

The golden ratio is sometimes called the “divine proportion,” because of its frequency in the natural world. The number of petals on a flower, for instance, will often be a Fibonacci number. The seeds of sunflowers and pine cones twist in opposing spirals of Fibonacci numbers. Even the sides of an unpeeled banana will usually be a Fibonacci number—and the number of ridges on a peeled banana will usually be a larger Fibonacci number.

Resource Library, “The Golden Ratio” at National Geographic Society

➤ Then there is pi (π), which (outside of Carl Sagan’s novel and film) burbles on forever without forming a pattern, yet it is fundamental in nature too:

The number crops up in the natural world, too. It appears everywhere there’s a circle, of course, such as the disk of the sun, the spiral of the DNA double helix, the pupil of the eye, the concentric rings that travel outward from splashes in ponds. Pi also appears in the physics that describes waves, such as ripples of light and sound. It even enters into the equation that defines how precisely we can know the state of the universe, known as Heisenberg’s uncertainty principle.

Finally, pi emerges in the shapes of rivers. A river’s windiness is determined by its “meandering ratio,” or the ratio of the river’s actual length to the distance from its source to its mouth as the crow flies. Rivers that flow straight from source to mouth have small meandering ratios, while ones that lollygag along the way have high ones. Turns out, the average meandering ratio of rivers approaches — you guessed it — pi.

Natalie Wolchover, “What Makes Pi So Special?” at LiveScience (August 9, 2012)

But the world of numbers is a strange place anyway.

➤ Numbers like the square root of minus 1 don’t appear to make sense but our computers and the entire modern world depend on them. And then there are the hyper real (infinite) numbers. As Jonathan Bartlett says, “Thinking about infinities is somewhat mind-bending, but it turns out that actually manipulating infinities with the hyperreal system is incredibly easy if you are familiar with basic algebra.”

➤ And there’s Chaitin’s unknowable number which, as Robert J. Marks says, is critical to computer function: “The number exists. If you write programs in C++, Python, or Matlab, your computer language has a Chaitin number. It’s a feature of your computer programming language. But we can prove that even though Chaitin’s number exists, we can also prove it is unknowable.”

Will we ever figure out why these numbers are what they are? Dr. Sutter isn’t sure:

Today, we have no explanation for the origins of this constant. Indeed, we have no theoretical explanation for its existence at all. We simply measure it in experiments and then plug the measured value into our equations to make other predictions.

Someday, a theory of everything — a complete and unified theory of physics — might explain the existence of the fine-structure constant and other constants like it. Unfortunately, we don’t have a theory of everything, so we’re stuck shrugging our shoulders.

Paul Sutter, “Life as we know it would not exist without this highly unusual number” at Space.com (March 24, 2022)

One thing is for sure: If we live in a mysterious universe, so do any other intelligent civilizations.

You may also wish to read: Why the unknowable number exists but is uncomputable. Sensing that a computer program is “elegant” requires discernment. Proving mathematically that it is elegant is, Chaitin shows, impossible. Gregory Chaitin walks readers through his proof of unknowability, which is based on the Law of Non-contradiction.