In celebration of Pi Day (3-14), not to be confused with Pi Approximation Day July 22nd (22/7), I thought I'd show you an easy way to build a small model of the Great Pyramid of Giza (Khufu/Cheops) based on the universal constant of Pi (π). In addition, I'll give you a few fun facts about Pi and the pyramid.
Egyptologists and mystics have noticed that the dimensions of the Great Pyramid have the ratio of Pi embedded in them. If you're looking for something to be mystical about, that's fine, but there's a more rational reason (pun intended). The way they might've laid the pyramid out to start building would have Pi already built into it by definition.
So the Pharoah commands you to build the largest structure ever built by mankind. He's a God-King, so you should probably take it pretty seriously. He wants it to be a certain height so that the angle from the palace blocks out half the sky. Knowing the distance from the palace to the site, you calculate the pyramid will have to be about 481 feet tall. Modern day skyscrapers are by definition anything over 40 stories tall, so at an average height of 12 feet per story, your pyramid built circa 2560 BC would qualify as the first skyscraper ever built, and the tallest building in the world for thousands of years until the Eiffel Tower was built in 1887 AD.
Knowing that you can only stack blocks so high based on the size of the base, you need to figure out the size of the base. You're in Egypt surrounded by sand so know that sand (like in an hourglass) will only pile as high as the radius of the cone's circular base. You're making a pyramid from square blocks, so your base will be square. The relationship between the two is the circle's circumference and the square's perimeter.
This leads you to figure out that the height of the pyramid will be the radius of a circle whose circumference is equal to the perimeter of a square base. So, at a height/radius of 481 feet, that means the circle's circumference is 3,024 feet (481 x 2π = 3,024 feet). Divide that by 4 and there's the size of the sides of the square pyramid base (756 feet). You're really proud of your design and how the math worked out so you show it to the Pharoah. He approves and you're off to go try to find 100,000 unemployed folks to help you build it over the next 20 years.
Now, how to lay it out? You could use a long rope, but it would easier would be to make a wheel of a known size and roll it out across the sand. A little more math and you make a wheel that's just over a yard in diameter (36.09634 inches). You can always make it a bit oversized and then sand it down until it's perfect. You have your guys make a giant flat spot in the sand and drive a stake into the ground where you want the first corner. You're also aligning the corners with the largest star in the Orion's Belt constellation, so you know your diagonal. You make a mark on the edge of the wheel and set it on the stake at zero. Now roll your wheel 80 revolutions and drive another stake. This makes your side exactly 756 feet long ((36.09634 x π x 80)/12).
Turn 90° towards your “Orion diagonal” and roll out another 80 revolutions. If you land on your diagonal, great. If not, there are a couple of ways you can square up the corner of your pyramid. One way is to measure 3 wheel revolutions on one side and make a mark, then measure out 4 revolution on the other side. Now take the wheel and roll straight between the two marks. They should be exactly 5 revolutions apart. This will work for any multiples of 3, 4, 5 (e.g. 12, 16 and 20), so you can adjust your legs and stakes accordingly. Pythagoras would become famous for this concept around 520 BC.
Another way to square your sides is to lay out all four sides using the 80 revolutions method and set your four stakes in the corners. Now take your wheel and roll it along the diagonals counting the number of revolutions. If you're base is square, they will be the same. If one diagonal is short, bump your stake out and try again, making sure to maintain the 756 foot distances between stakes.
Now the fun begins and most of the math is over. You'll want the corner edges to be straight, so you'll need to make the sides of each new layer shorter by the same amount all the way up. This depends on the size of the blocks you'll be using for each layer. Let's build a model of the pyramid using sugar cubes to see how that works.
Step 1: Layout
Make your starting "stake" at the "job site" and make an indexing mark on your "wheel". Start with the mark at the stake and roll out the appropriate number of revolutions. In this case, 1. Make your second mark. This corresponds to the 756 foot mark.
With sugar cubes, they're pretty much self-squaring with a little finger pressure, so I didn't have to mess with the whole Pythagorean Theorem thing, but in real life, that would've been super-important.
Step 2: The Build
Now you'll want to use "blocks" that are evenly divisible into your side length. In my case, 8 sugar cubes landed exactly between the marks because of the size of "wheel" I chose.
Lay out the first "floor" of 8 x 8 = 64 "blocks". My "wheel" diameter is 1.35 inches, so the sides of my pyramid are π x D = 4.25 inches. This means that the height of my scale pyramid should be 2.7 inches high (4.25 x 4 = 17/2π). Because my sugar cubes weren't the perfect height (BTW, sugar cubes aren't "cubes", they're a little shorter than they are wide. Sigh...), so the height/aspect ratio of the model is a bit tall.
Whenever you're building something from blocks, you don't want the seams to line up, so if I'd started with a six-sided square and jumped to a four-sided square for the second layer, then the final pyramid would've been too short, but more importantly, the seams would've lined up and it would've been super-rickety. As such, I staggered the seams by half a "block" and the structure is much more stable. This is called "weaving" in the construction world and is what you do when covering something with several sheets of plywood. In fact, the weight of the blocks reinforce each other, even on such a small scale as this.
Continue up, making each level one block smaller until your reach the top. In the real world, the blocks get smaller as you move up the pyramid because they have less weight to bear, so you can use that to reduce the size of each course to nail your perfect height/radius.
Step 3: The Reveal
As you can see, we achieved our straight sides by reducing each course by the same amount (1 block). The pyramid is nice and stable because the seams are woven together/staggered. As previously discussed, our pyramid is a bit tall for it's base, but it's definitely pyramid-shaped. We could've left the top 2-3 layers off and we would've had a pyramid very similar to the ones found in Central and South America from the Aztec empire.
Step 4: Other Fun Facts
Albert Einstein was born on Pi Day (3/14/1879).
How to type Pi:
Hold the Alt key and type 227. Voila! π [Alt] + 227 = π
How far away can a person see the Great Pyramid?
The apparent height of the 481 foot tall (Hobj) pyramid for someone who is 6 feet tall (Heye) is 475 feet (Hobj - Heye). The distance that can be seen until the curvature of the earth gets in the way is: D = √(7 x H)/4 = √(7 x 475)/4 = 28.8 miles!
What's So Special About Pi?
Pi is a non-repeating, non-terminating irrational number, so it goes on forever. It is considered a universal constant like the speed of light. It's been used on space probes to try and convince any aliens that may stumble across them in deep space that we're intelligent beings.
Plato approximated Pi about 2,300 years ago as √2 + √3 = 3.14, which is pretty close for the time.
Archimedes approximated Pi to 99.9% by calculating perimeters of multi-faceted shapes that fit inside and outside a circle. At 96 sides, he determined that Pi is between these two fractions:
223/71 < π < 22/7
Digits of Pi
Akira Haraguchi took more than 16 hours to recite π to 100,000 decimal places, setting what he claims to be a new world record.
Alexander Yee and Shigeru Kondo built an $18,000 computer that has calculated π to 5 Trillion digits! The data is about 6 TB. That's about the same amount of data for 900 hours of Hi-Def video, 5 days worth of music CD's, and as many words as all Americans speak in half an hour!
To Put Things Into Perspective
Thirty-nine decimal places of π:
are accurate enough to compute a circle the size of the known universe with an error less than the radius of a hydrogen atom.