The un-Earthly thing about pi is that it keeps popping up in weird places and in unexpected ways. Heck, it even shows up in the Bible, defined as exactly three. These appearances combine to make a larger-than-life constant that is almost too spooky not to have extraterrestrial origins.
In honor of this joyous math holiday, an examination of the most startling facts about pi is in order.
Finding a formula for pi took thousands of years
The notion of a constant from which could be derived the area of a circle seems to have originated as far back as 1900 B.C. Around then, both the Egyptians and the Babylonians came up with approximations for pi that were within 1% of the actual value. Being so close so early, you'd think that making a formula for pi would soon follow, in which case you'd be horribly wrong (shame on you).
In fact, despite the fact that the best math cultures in the world were working on a formula pretty much nonstop throughout human history, it took over 3,300 years after the first estimates of pi for humankind to come up with an exact formula for it. Incidentally, the culture to first crack this problem was the Indians, in the 15th century AD.
Due to pi, it's impossible to create a square with the exact same area as a circle
If you're like the average Syfy Wire reader, you have billions of dollars and exactly two children. Let's say one of your kids has had their eye on a circular island you own, so you give it to them. In the interest of not playing favorites, you promise the other child that you'll give them a square piece of your estate that is equal in area to the circular island. Whoops, you've just promised your kid something you can never give them, which brings up stores of issues from broken childhoods (mathematically speaking).
In fact, making a circle and a square with equal areas is an impossible problem that stymied mathematicians for 3,600 years. Alice in Wonderland author Lewis Carroll tried admirably to argue that the problem was impossible, before throwing up his arms in disgust and declaring that people who believed they could make identical areas from the two shapes to be, in so many words, giant idiots. ("The first of these two misguided visionaries filled me with a great ambition to do a feat I have never heard of as accomplished by man, namely to convince a circle squarer of his error! The value my friend selected for Pi was 3.2: the enormous error tempted me with the idea that it could be easily demonstrated to BE an error. More than a score of letters were interchanged before I became sadly convinced that I had no chance.")
The area contained by the standard bell curve is the square root of pi
We are all familiar with the standard bell curve as "that thing that saved us from getting an F in school when we said 'Alexander Graham Bell invented the bell curve.'" The bell curve shows a common way to grade "on a curve," where most people get C's and equal numbers of people do well as do poorly. It's also known as "normal distribution." Although the bell curve is extremely simple and symmetrical, it literally contains the key to the most complicated constant on the planet. It is pretty weird that something as simple as "hey, let's distribute everything normally" involves using one of the most mystifying constants in mathematics. Still, contained within the space underneath a bell curve is an area exactly equal to the square root of pi.
Albert Einstein was born on Pi Day
This bizarre link connects the greatest mathematician and greatest mathematical constant. Pi day is March 14. This is because, when abbreviated 3/14, it matches the first three digits of pi (not unlike February 27, when the world celebrates the delightful 1980s sitcom 227).
The famous set of proofs known as Einstein's equations use pi to support the field equation relating the curvature of space-time to an energy source. The amount of gravity is proportional to the amount of energy and momentum. While I don't purport to fully understand this equation, I do know that it provides evidence for the theory of relativity, so it is super important.
It's also important to note that Einstein's equation contains the variable for gravity. Gravity is another force which has generally eluded specific scientific definition. Because gravity's units involve the meter, which was derived in a method that utilizes pi, gravity involves pi. It's fitting that gravity, perhaps the most baffling scientific property, should be tied to pi, the most mystifying number in the universe.
Pi can compute the size of the universe down to the last atom
So, why does pi go to infinite digits? Is the universe advertising, "Coming soon: even smaller stuff to measure"? You'd think that measuring the largest thing in existence with a margin of error equal to the smallest thing in existence would require a very specific value for pi, with a long line of digits spouting out past the decimal. After all, the Milky Way has one hundred billion stars, and there are an estimated 10 trillion galaxies, which leaves us with 10^23rd stars. And that's stars, not atoms. One could use a value of 10^57 atoms/star to increase the number of atoms being measured to 10^90. And that's not even including the vast empty spaces between stars.
So, how many digits of pi would be needed to calculate the size of the universe down to the very last atom? Just 39. According to this YouTube video, that's how specific you need to get with pi to calculate the size of, well, anything. Why is it, then, that people are hard at work making super computers that crank out pi to one trillion places? Partly, just to show that they can. Also, deriving such a complicated constant as pi is a great test of a computer's power.
Of course, you may be wondering, "Why would we need to measure the size of the universe?" Physics is based on the notion that the Big Bang started our universe in motion. Having an exact size of the universe to compare to the estimated speeds at which everything in the universe is flying away from each other would be a great confirmation/refutation of the Big Bang Theory.
Unfortunately, we can't currently use pi to specifically identify the size of the universe down to the last particle; we just don't have enough hard data. I mean, at this point the best way we have of measuring the distance to extremely remote stars is to study the brightness of the light that reaches us. While based on brilliant theory, this method is still not extremely accurate, only estimating the size of the universe to the nearest 100 billion light years. Nothing ruins a trip to the edge of the universe like finding you still need to travel 100 billion light years when you thought you were just an atom away.
You can get a pretty good estimate of pi just by dropping a needle
Warning: This paragraph contains math. Skip to the next paragraph if you find math triggering. If a needle of length ℓ is dropped n times on a surface on which parallel lines are drawn t units apart, and if x of those times it comes to rest crossing a line (x > 0), then one may approximate pi based on the results.
Due to all the effort that has been put into developing supercomputers that can determine pi to one trillion digits, it might be surprising to learn that you can get a pretty good simulation of pi by dropping sticks onto a chart. If you drop a stick onto a grid of parallel lines, the probability that the stick will land on a line is directly related to pi, specifically:
Some math genius came up with this unexpected solution for pi over 300 years ago, back when dropping needles onto graphs was probably the height of entertainment. However, it wasn't until computers came around that we could run this experiment the extreme number of times needed to get good results. There's even a needle-dropping computer program you can use right now.
The probability of two numbers having no shared factors is related to pi
Pi's direct attachment to circles isn't the only eerie thing about the constant. In fact, it's shocking that pi keeps coming up in situations where one wouldn't normally see any connection. Take prime numbers, which as far as I can see should have nothing to do with geometric patterns unless you're, like, at a Phish concert. However, if you take any two prime numbers at random, the odds that they have no factors in common except 1 is a factor of pi (specifically 6/π^2).
The reason that this seemingly unrelated math principle deals ultimately with pi is less shocking as it is pretty math-y. The probability of two random numbers having no shared factors can be rewritten as a series of equally diminishing terms, something that looks a little like this:
Pi comes into play because, ultimately, solving for pi involves solving for an infinite number of diminishing rectangles inside a circle. Here's the full proof. WARNING: Footnote 2 of the proof has the most gregarious misuse of the word "silly" that I have ever seen.
Rivers could be directly related to pi
Well, perfectly normal rivers, anyway. Under ideally average conditions (uniform gentle slope on an homogeneously erodible substrate), the sinuosity of a meandering river approaches pi (according to Cambridge professor Hans-Henrik Stolum.) The sinuosity is the ratio between the actual length and the straight-line distance from mouth to source.
The entire solution can be found here, and it contains the exciting phrase "The meandering process oscillates in space and time between a state in which the river platform is ordered and one in which it is chaotic," which is the awesomest way anyone has ever described what amounts to a river that is exactly average.
Of course, this has been met with by blowback from some of the field of ... I'm not even sure ... aquatic mathematics? They assert that it is probably impossible to gather sufficient real-world data to suggest a correlation between average sinuosity of a 'normal' river and the exact value of pi. That's okay, I'm still going to describe every river I write about as, "curved with a sinuosity exactly equal to pi," and be the next Ken Kesey.