Ancient Mathematics: Baking a Pi

Happy Pi Day!


Today, on March 14 (3/14), we’re celebrating one of the most important numbers in all of math: pi! That’s “P-I,” with no “E”! But pie that you eat is a great way to celebrate this circular holiday.

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If you asked mathematicians to say what number is the most important, many would pick 1, or 0, or even infinity, but to a lot of people, the most important number is

3.141592653589723…

(and on and on- the decimals never stop!) This is the number “pi”, represented by the lowercase Greek letter of the same name: π. This number might seem completely random, but it actually holds within it a deep, fundamental truth about our physical universe.


But where does this strange number come from? It’s the ratio of a circle’s circumference (the distance measured all the way around a circle) and its diameter (the distance across the circle’s widest part, measured from one point to the point directly across from it). For any circle, no matter how big or how small, this ratio is always the same, and always equal to π. Since it’s a universal constant, mathematicians gave it a name for easy reference. And since an endless string of numbers is way too long for daily use, π is most often shortened to 3.14. So on 3/14, March 14th, we celebrate the delicious marvel of pi(e)!

On the left a circle denoting its circumference (distance around the outside) and diameter (distance across the inside). On the right a cartoon of a circular pie with the symbol "pi" in the center.
The ratio of a circle's circumference to its diameter is always equal to pi!

Since π represents something so fundamental about geometry, you might hope that it would be a nice, neat number, but instead π is what’s called an irrational number. Irrational numbers can’t be expressed as a fraction of two integers. 3 is a rational number because it can be written as 3/1, or 6/2, or any other equivalent fraction. Likewise, 4 can be written as 4/1, or 8/2. In fact, any number is considered to be rational if it can be expressed in the form p/q, where p and q are integers.

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But π can’t be simplified into a fraction of whole integers, and there’s no way to calculate the exact, terminating value of the number because the decimals never end. Some people find it fun to memorize numbers in the sequence. There’s even a competition for it! The current record-holder is Rajveer Meena, who recited 70,000 digits from memory in 2015. The entire ordeal took around 10 hours!


So, what’s the deal with 3.14…? Why can’t it be written as one of these nice and tidy fractions? Why isn’t π an exact number? These are certainly curiosities worthy of exploration, and we aren’t the first to seek the answers.

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The reason for π’s infinite nature has to do with the way it’s calculated. If you take the length of the diameter of a circle and lay it along the outside of the same circle, you’ll find that it doesn’t reach all the way around. If you take three of the diameter lengths and connect them end-to-end around the circle, you’ll be left with just a tiny bit leftover before you’ve made a complete circle. It turns out that tiny bit is an irrational number: 0.14159265358… So in total, you'd need 3.14159265358…. diameter lengths to make it all the way around the circumference of a circle.


If you’ve already checked out our other posts on ancient mathematics, your mental gears might already be turning as you wonder- at what point in history did people discover π? Like many other kinds of scientific discoveries, multiple civilizations around the world started thinking about the same ideas around the same time independently. It’s fascinating to examine the differences and similarities between similar discoveries made around the world. One of the earliest records we have of mathematicians “discovering” π comes from the Rhind Papyrus, which is an awesome ancient document you can learn more about here.

A very old piece of papyrus with ancient writing on it. The papyrus is frayed and has some holes.
The Rhind Papyrus gives us clues about how ancient mathematicians thought about problems like ratios and the number pi.

Ancient mathematicians used what was immediately accessible to them as tools, which oftentimes meant they used proportions of the human body-- like the length of your fingertips to your elbow (forearm) as compared to the width of your fingers. This meant that they were used to thinking about ratios, such as the number of finger widths per forearm length (about 24 finger widths / 1 forearm length, in case you’re curious!). They also often used physical objects as references to understand areas, like rocks or tiles laid down to fully cover a surface. With just these two ideas, ancient civilizations got really close to understanding the complex, infinite number of π!


To see how, imagine you want to figure out the area of a circle that is 9 units in diameter. You don’t know how to get to a circle’s area from just its diameter, but you do know how to calculate the area of a square - Area = Length x Length. So what’s the length of a square that has the closest area to our circle?

On the left is a square with a height of 8 units. On the right is a circle with a diameter of 9 units.
How do you calculate the area of a circle if you don't know about pi?

You can try to figure this out yourself! First, find 64 round, similarly shaped objects like coins, buttons, cut out paper circles, LEGO blocks or whatever else you have lying around! Make a square that’s 8 objects long and 8 objects wide. Now, try to rearrange the 64 objects into a perfect circle with none leftover. Can you figure out how big or small the circle should be?

On the left is a square with a height of 8 units. It is covered in 64 small pebbles. On the right the 64 pebbles are arranged in a square atop a circle with diameter of 9 units.
You can approximately calculate the area of a circle by comparing it to a square with roughly the same area.

If you stick with it, you’ll find that this ideal circle is 9 objects in diameter. This is the same conclusion the ancient Egyptians came to!

On the left the 64 pebbles form a square with a height of 8 units. On the right they are rearranged to form a circle with a diameter of nine units.
The 64 pebbles can be arranged into a circular that has a diameter of 9 units!

So how close does this simple exercise get us to π? We know now that the area of a circle is given by A = π r^2. If we use our pebble estimation, this gives us 64 = π (9/2)^2 , or π = 3.16. This is amazingly close to 3.14, especially considering how rough our estimate was.

A circle with diameter of 9 is laid atop a square with height 8 to show that the area of both is roughly 64 in our approximation. 64 equals 8 times eight. 64 is approximately equal to pi times the square of nine divided by 2. Then pi is approximately equal to 64 divided by the square of nine divided by two. So pi is approximately 3.16.
Our rough pebble estimation gets us amazingly close to the real value of pi!

Nowadays, we have supercomputers that can calculate the digits of π to insane levels of precision, way more than humans would ever need! We’re currently up to 62 TRILLION digits, and still searching for more. So the next time you enjoy the fruity deliciousness of a P-I-E, take a moment to appreciate the awesome phenomenon of its P-I ratio! Happy Pi Day!

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Credits:

Written by Nicole Naporano

Edited by Caroline Martin

Cartoons by Caroline Martin

Images courtesy of Wikimedia Commons


Primary sources and additional readings:

Math And The Rise Of Civilization. Ep.01 - The Beginning of Numbers

Pi Wikipedia page

 

Explore the irrational beauty of pi with these activities!


Expand (10 minutes): Keep the celebration going by baking a pie with us and learning about the Ancient Cooking: Pi Day and the Science of Baking!


Play (10-15 minutes): Since the decimals of π go on forever, you can find almost any string of numbers inside it. You can use this website to search for particular digits in the number, like your birthday or your lucky number.


Experiment (45-60 minutes): In Pi Day and the Science of Baking we learned that gluten is a protein in flour that makes dough strong and stretchy. How do different amounts of gluten affect a dough? With a grown up's help create some Great Balls of Gluten! to find an answer...


Investigate (20-30 minutes): Another crazy way to calculate the value of π is with probability! If you randomly drop a bunch of toothpicks on a sheet with long, straight parallel lines, the toothpick will land on one of those lines with a probability that depends on π. You can learn about this phenomenon, "Buffon's Needle", in a real life experiment or with a computer simulation. How many toothpicks do you need to get close to the real value of π?


Deepen (15-30 minutes): To learn more about the basics of calculating the area and circumference of circles, and how that relates to π, check out Khan Academy's great introduction.


Experiment (30-45 minutes): The Maillard reaction is a series of complicated chemical reactions that are kickstarted by the heat of cooking. It's what is responsible for the golden brown color and sweet, nutty flavor of a pie crust-- just like we saw in Pi Day and the Science of Baking. You and your grown up can Smell the Maillard Reaction in your own kitchen!