Just like it does every year, March 14th is now upon us. While there are many reasons to celebrate the day, mathematically inclined residents of any country that writes the date in (month/day) fashion should immediately be excited by the prospect of seeing the numbers “3” and “14” next to one another, as 3.14 is famously a good approximation for one of the most well-known numbers that can’t neatly be written down as just a simple set of digits: π. Pronounced “pi” and celebrated worldwide by baking enthusiasts as “Pi day,” it’s also a great opportunity to share some facts about π with the world.

While the first two facts that you’ll read here about π are generally very well-known, I seriously doubt anyone, even an actual mathematician, will get to the end of the list and know all 11 of these facts. Follow along and see how well you do!

The transcendental number, π, dates back to antiquity, and has as its definition that it’s the ratio of a circle’s circumference to its diameter. The fact that it’s approximately 3.14 as a decimal, or 22/7 as a fraction, has led to the made-up holiday known as “Pi day.”

1.) Pi, or π as we’re going to call it from now on, is the ratio of a perfect circle’s circumference to its diameter. One of the very first lessons I ever gave when I began teaching was to have my students bring in any “circle” from home. It could’ve been a pie tin, a paper plate, a mug with a circular bottom or top, or any other object that had a circle somewhere on it, with only one catch: I would give you a flexible tape measure, and you’d have to measure both the circumference and the diameter of your circle.

With more than 100 students between all of my classes, each student took their measured circumference and divided it by their measured diameter, which should’ve given an approximation for π. As it turned out, whenever I run this experiment and average all of the students’ data together, the average always comes out to somewhere between 3.13 and 3.15: often landing right on 3.14, which is the best 3-digit approximation of π of all. Approximating π, although there are many methods that are better than this crude one I used, is unfortunately the best that you can do.

Although it’s tempting to attempt to represent the quantity π as a fraction, with common estimates like 22/7 doing an alright job, it turns out that there is no exact representation of this number, π, in fractional form.

2.) π cannot be calculated exactly, because it is impossible to represent as a fraction of exact (integer) numbers. If you can represent a number as a fraction (or ratio) between two integers, i.e., two whole numbers of either positive or negative values, then that’s a number whose value you can know exactly. This is true for numbers whose fractions don’t repeat, like 2/5 (or 0.4), and it’s true for numbers whose fractions do repeat, like 2/3 (or 0.666666…).

But π, like all irrational numbers, cannot be represented this way and cannot be calculated exactly as a result. All we can do is approximate π, and while we’ve been doing that extremely well with our modern mathematical techniques and calculational tools, we’ve been doing quite a good job of this historically as well, even going back thousands of years.

One of the ways to approximate the area within a circle, which enables an approximation for π for any known diameter, is to either inscribe or circumscribe a regular polygon that touches a circle in N location, where “N” is the number of sides in your regular polygon. This is shown for a pentagon, hexagon, and octagon, respectively. Archimedes used up to a 96-sided polygon to achieve his best approximations to π .

3.) The “method of Archimedes” has been used to approximate π for more than 2000 years. Calculating the area of a circle is hard, particularly if you don’t already know what “π” is. But calculating the area of a regular polygon is easy, especially if you know the formula for the area of a triangle, and realize that any regular polygon can be broken up into a series of isosceles triangles. You have two ways to go:

• you can inscribe a regular polygon inside of a circle, and know that the “true” area of your circle must be bigger than that,
• or you can circumscribe a regular polygon about the outside of a circle, and know that the “true” area of your circle must be less than that.

The more sides you make to your regular polygon, in general, the closer you’ll get to the value of π. In the 3rd century BC, Archimedes took the equivalent of a 96-sided polygon to approximate π, and found that it must lie between the two fractions 220/70 (or 22/7, which is why π day in Europe is the 22nd of July) and 223/71. The decimal equivalents for those two approximations are 3.142857… and 3.140845…, which is pretty impressive for some 2000+ years ago!

This statue shows 5th century Chinese mathematician Zu Chongzhi, and is found in Tinglin Park in Kunshan. Zu Chongzhi found the greatest fractional approximation of π with a denominator less than 10,000: 355/113. It was the best approximation for π in the world until approximately the late 14th century.

4.) The approximation for π known as Milü, discovered by Chinese mathematician Zu Chongzhi, was the best fractional approximation of π for about 900 years: the longest “best approximation” in recorded history. In the 5th Century, mathematician Zu Chongzhi discovered the remarkable fractional approximation of π: 355/113. For those of you who like the decimal approximation of π, this works out to 3.14159292035… which gets the first seven digits of π correct, and is only off from the true value by about 0.0000002667, or 0.00000849% of the true value.

In fact, if you calculate the best fractional approximations of π as a function of increasing denominator:

Beginning with the fraction “3/1” and raising either the numerator or denominator allows one to compute increasingly superior fractional approximations for π, with 355/113 making the best approximation one can find with a diameter below 10,000.

you won’t find a superior one until you hit upon the fraction 52163/16604, which is just barely better. Whereas 355/113 differed from the true value of π by 0.00000849%, 52163/16604 differs from the true value of π by 0.00000847%.

This remarkable fraction, 355/113, was the best approximation of π that existed until the late 14th/early 15th century, when the Indian mathematician Madhava of Sangamagrama came up with a superior method for approximating π: one based on the summation of infinite series.

All the real numbers can be divided into groups: natural numbers are always zero or positive, integers are always in whole number increments, rationals are all ratios of integers, and then irrationals can either be expressible as being derived from a polynomial equation (real algebraic) or not (transcendental). Transcendentals are always real, but there are complex algebraic solutions to polynomial equations that extend into the imaginary plane. The jump from “rational numbers” to “real algebraic” numbers is a leap from countably infinite numbers to uncountably infinite numbers: a different type of infinity.

5.) π is not only an irrational number, but it’s also a transcendental number, which has a special meaning. In order to be a rational number, you need to be able to express your number as a fraction with integers for their numerator and a denominator. By that account, π is irrational, but so is a number like the square root of a positive integer, such as √3. However, there’s a big distinction between a number like √3, which is known as a “real algebraic” number, and π, which is not just irrational but also transcendental.

The difference?

If you can write down a polynomial equation with integer exponents and factors, and only use sums, differences, multiplication, division, and exponents, all of the real solutions to that equation are real algebraic numbers. For example, √3 is a solution to the polynomial equation, x² – 3 = 0, with -√3 as its other solution. But no such equations exist for any transcendental numbers, including π, e, and γ.

It was long considered a “holy grail” of mathematics to be able to square the circle: to construct a square with an area of π, given a circle of circumference π, using only a compass and straightedge. If π is transcendental, which it is, this cannot be done, although this was not proven until 1882.

In fact, one of history’s most famous unsolved math puzzles is to create a square with the same area as a circle using only a compass and straightedge. In fact, the difference between the two types of irrational numbers, real algebraic and transcendental ones, can be used to prove that constructing a square whose length has a side of “√π” is impossible given a circle of area “π” and a compass and a straightedge alone.

Of course, this wasn’t proven until 1882, showing just how complicated it is to rigorously prove something that seems obvious (upon exhausting yourself) in mathematics!

If you threw darts completely at random, some of them would land within the circle while others would land within the square but not within the circle. The ratio of “total darts within the circle” to “total darts within the square, including darts within the circle” is π/4, which allows one to approximate π simply by throwing darts!

6.) You can very simply approximate π by throwing darts. Want to approximate π, but don’t want to do any mathematics more advanced than simply “counting” to get there?

No problem, simply take a perfect circle, draw a square around it, where one side of the square is exactly equal to the diameter of the circle, and start throwing darts. You’ll immediately find that:

• some of the darts land inside the circle (option 1),
• some of the darts land outside the circle but inside the square (option 2),
• and some darts land outside both the square and circle (option 3).

As long as your darts are truly landing in a random location, you’ll find that the ratio of “the darts that land inside the circle (option 1)” to “the darts that land inside of the square (options 1 and 2 combined)” is precisely π/4. This method of approximating π is an example of a simulation technique very commonly used in particle physics: the Monte Carlo method. In fact, if you write a computer program to simulate this type of dartboard, then congratulations, you’ve just written your first Monte Carlo simulation!

Although π can be approximated with a simple fraction, there are sequences of fractions known as “continued fractions” that, were one to truly take an infinite number of terms, could calculate π to any arbitrary precision.

7.) You can very excellently, and relatively quickly, approximate π by using a continued fraction. Although you can’t represent π as a simple fraction, just as you can’t represent it as a finite or repeating decimal, you can represent it as something known as a continued fraction, or a fraction where you calculate an increasing number of terms in its denominator to arrive at an increasingly superior (and accurate) approximation.

There are many examples of formulae that one can calculate, repetitively, to arrive at a good approximation for π, but the advantage of the three shown above is that they’re simple, straightforward, and provide an excellent approximation with only a relatively small number of terms. For example, using only the first 10 terms of the final series shown gives the first 8 digits of π correctly, with only a small error in the 9th digit. More terms means a better approximation, so feel free to plug in as many numbers as you like and see how satisfying it can be!

This color-coded depiction of the first 1000+ digits of pi shows sequences of repeating digits in various colors, with “double digits” in yellow, “triple digits” in cyan, and the one “sextuple digit” sequence of 9s, the Feynman point, shown in red.

8.) After 762 digits of π, you arrive at a string of six 9s in a row: known as the Feynman Point. Now, we head into territory that requires some pretty deep calculations. Some have wondered, “What sort of patterns are there to find embedded within the number π?” If you write out the first 1,000 digits, you can find some interesting patterns.

• The 33rd digit of π, a “0,” is how far you have to go to get all 10 of the digits, 0-through-9, to appear in your expression for π.
• There are a few instances of “triply repeating” numbers in-a-row in the first 1,000 digits, including “000” (two times), “111” (two times), “555” (two times), and “999” (two times).
• But those two instances of “999” repeating are next to each other; after the 762nd digit of π, you actually get six 9s in a row.

Why is this so noteworthy? Because physicist Richard Feynman noted that if he could memorize π to “the Feynman Point,” he could recite the first 762 digits of π and then say, “nine-nine-nine-nine-nine-nine and so on…” and that would be extremely satisfying. It turns out that, although all consecutive combinations of digits can be proven to appear somewhere in π, you won’t find a string of 7 identical digits in a row until you’ve written out nearly 2 million digits of π!

If you take the natural log (base “e”) of the number 262,537,412,640,768,744, and divide it by the square root of (163), you get an approximation for π that’s successful for the first 31 digits. The reason why has been known since the work of Charles Hermite in 1859.

9.) You can outstandingly approximate π, accurate to 31 digits, by dividing two mundane-appearing irrational numbers. One of the most bizarre properties of π is that it shows up in some really unexpected places. Although the formula eiπ = -1 is arguably the most famous one, perhaps a better and even more bizarre fact is this: if you take the natural logarithm of a particular 18 digit integer, 262,537,412,640,768,744, and you then divide that number by the square root of the number 163, you get a number that’s identical to π for the first 31 digits.

Why is this so, and how did we get such a good approximation for π?

It turns out that in 1859, mathematician Charles Hermite discovered that the combination of three irrational (and two transcendental) numbers e, π, and √163 make what’s known as an “approximate integer” by combining them in the following fashion: e^π√163 is almost exactly an integer. The integer that it almost is? 262,537,412,640,768,744; in fact it “equals” 262,537,412,640,768,743.99999999999925…, so rearranging that formula is how you get this incredibly good approximation for π.

The following famous four space/astronomy/physics heroes all have a birthday on March 14th: Pi day. Can you tell who each of them are? (Spoilers in the text below!)

10.) Four famous physics/astronomy and space heroes from history have their birthday on π day. Look at the image above, and you’ll see a collage of four faces, showing people of various levels of fame in physics/astronomy/space circles. Who are they?

• First up is Albert Einstein, born on March 14, 1879. Known for his contributions to relativity, quantum mechanics, statistical mechanics, and energy-mass equivalence, Einstein is also the most famous person out there with a π-day birthday.
• Next is Frank Borman, born on March 14, 1928, who turns 95 years old on this day in 2023. He commanded Gemini 7 and was NASA liaison at the White House during the Apollo 11 moon landing, but he is best known for commanding the Apollo 8 mission, which was the first mission to bring astronauts to the Moon, to fly around the Moon, and to photograph the site of Earth “rising” over the Moon’s horizon.
• The third image is perhaps the least known today, but is of Giovanni Schiaparelli, born March 14, 1835. His work during the 19th century gave us the greatest maps, of their time, of the other rocky planets within our Solar System: Mercury, Venus, and most famously, Mars.
• And the final image is of Gene Cernan, born March 14, 1934, who is (at present) the final and most-recent human to set foot on the Moon, as he re-entered the Apollo 17 lunar module after crewmate Harrison Schmitt. Cernan died on January 16, 2017 at the age of 82.

Although the open star cluster Messier 38 goes by many names, a color view of the stars within it clearly shows a different pattern than its most common name of “the starfish cluster” would indicate. Here, with a little bit of artificial highlighting, I’ve selected a particular shape that, with help, you should be able to pick out and recognize on your own.

11.) And there’s a famous star cluster that truly looks like a “π” in the sky! Look at the image above; can you see it? This “pi”cturesque view is of the open star cluster Messier 38, which you can find by locating the bright star Capella, the third-brightest star in the northern celestial hemisphere behind Arcturus and Rigel, and then moving about a third-of-the-way back toward Betelgeuse. Right in that location, before you reach the star Alnath, you’ll find the location of the star cluster Messier 38, where a red-green-blue color composite clearly reveals a familiar shape.

Unlike the newest, youngest star clusters out there, none of the remaining stars in Messier 38 will ever go supernova; the survivors are all too low in mass for that. The most massive stars within the cluster have already died, and now, some 220 million years after these stars have formed, it’s only the A-class, F-class, G-class (Sun-like) and cooler stars which remain. And remarkably, the brightest, bluest survivors make an approximate π-shape in the sky. Even though there are four other star clusters that are relatively nearby, none of them are related to Messier 38, which is 4,200 light-years away and contains hundreds, perhaps even thousands of stars. For a real-life look at π-in-the-sky, simply find this star cluster and the sights are yours to behold!

Happy π day to one and all, and may you celebrate it in a sweet and fitting fashion!