Archimedes: Doing more math in 200BC than any of us can today

Most of you probably have heard of Archimedes of Syracuse. He was a Greek mathematician and (possibly mad) scientist alive in the 200B.C.'s. I like to call him the Corn Syrup Mathematician™ because, like corn syrup, you would be astounded to believe how many things people use today that contain something that Archimedes worked with.

Fun fact. Much of what we know about Archimedes was recorded by Vitruvius. I wonder if
we could get Morgan Freeman to narrate some of Archimedes work...

One of the really cool things about Archimedes is that a lot of the topics he worked on make sense. Look at other famous mathematicians and scientists. Does anyone think that what Einstein talked about truly makes sense? Does anyone look at the work of Heisenberg or Schrodinger and think "Oh, now that you put it like that, I totally get it"? In no way am I saying those scientists are incorrect, but the scientific breakthroughs they made are hard to conceptualize. Archimedes on the other hand is a great example to show to people that you can make great insights simply by using common sense and prior knowledge.

I honestly cannot tell if this is the most insightful thing I've heard today or useless.

Archimedes discovered many principles dealing with fluid displacement, buoyancy, and density. He built an Archimedes Screw, something that is still used today. He designed many different kind of siege warfare machines, and supposedly burned a fleet of ships using mirrors.

He also worked in mathematics. He gave an approximation of the square root of 3 (which was accurate to the hundred thousandths decimal). He calculated the area contained by a parabola and a straight line. He calculated the areas of triangles and parallelograms. He approximated π.

Coincidence? I think not.

The work he was most proud of looked at a sphere inside a cylinder where the height of the cylinder and the diameter of the end of the cylinder was equal to the diameter of the sphere. (I want to note here that at this time in history, calculus did not exist. Archimedes was able to accomplish all of this without integrals, infinite series, or limits. Or if he did use those concepts, he had to discover those concepts himself.) Archimedes actually requested that after he died, instead of a gravestone, he wanted a carving of the following figure placed over him.

Yeah, instead of an epitaph, could you just carve my 9th grade
essay on "Haroun and the Sea of Stories" onto a marble slate?

He first calculated the area of a circle by looking at a circle made up of 6 equilateral triangles. He realized that if you put the triangles next to each other, the the length is slightly less than the circumference and the height of reach triangle is slightly less than r. As you increase the number of triangles, the height of the triangles gets closer and closer to r while the length gets closer and closer to the circumference.

Area_Circle = (Area of 1 Triangle) * (Number of Triangles)

A_Circle = (1/2 * Circumference/n * r) * n

A_Circle = 1/2 * C * r

Since C = 2πr, we know that

A_Circle = 1/2 * 2πr * r

A_Circle = πr²

Now Archimedes thought about the surface area of the sphere. He decided to first start by looking at the surface area of a section of the sphere, defined by a section of the sphere cut off by a plane slicing through the sphere.

Archimedes claimed that the surface area of the section of the sphere that was sliced off is equal to the area of a circle with radius equal to the distance from the perimeter of the intersection to the top of the center of the circular area.

Since we are looking at the surface area, let's imagine that the sphere is empty. It is just a shell that outlines the sphere. For reasons that will be clear very soon, let's think of the sphere as the empty casing around a spherical roast beef at Arby's.

[Insert Product Placement Here]

I chose a hunk of roast beef so we can imagine slicing it with a meat slicer. If we were to keep slicing the meat thinner and thinner, we could imagine slicing a 2-Dimensional slice of meat. But remember that our special roast beef is just a hollow shell, so these 2-Dimensional slices are just circles.

"Isosceles. You know, I love the name Isosceles. If I had a kid,
I would name him Isosceles. Isosceles Kramer." - Cosmo Kramer

If we were to take all of these circles, cut them, and lay them out straight so that their ends were level, we would see something like this (this is very similar to the situation we looked at above).

As the slices get thinner, the number of slices increases and the picture above starts to look like a triangle.

Area_{Cross Section} = 1/2 * Circumference * Radius

A_CS = 1/2 * C * r

Since C = 2πr, we know that

A_CS = 1/2 * 2πr * r

A_CS = πr²

Therefore, we know that the surface area of the portion of the sphere that is cut off by a plane slicing through the sphere is equal to the area of the circle defined by the intersection of the surface of the sphere and the plane. Now if we imagine that the plane cutting the sphere goes through the center point of the sphere, the radius of the intersection circle is equal to the distance from point (0, 0, r) to 

(r, 0, 0). This distance is equal to √2r. Since this only calculates the surface area of the top half of the sphere, the surface area of the sphere is twice the area of the circle with radius √2r.

Surface Area_Sphere = 2*π*Radius²

SA_S = 2*π*(√2r)²

SA_S = 4πr²

Now at this point, Archimedes kind of goes off the deep end. He started talking about approximating the volume of a sphere using pyramids. I looked at how he calculated the volume of a sphere and had the following reaction.

I don't quite understand it myself yet. There is a strong possibility that I write another blog post about more of Archimedes work, but for now