Assignment 1 Reflection

     We did our presentation on the ancient Egyptian area of a circle. I really enjoyed the initial process of figuring out what was happening. They had this formula A = (8/9d)^2 and at this time we had no idea why or how they arrived here. We followed along the instructions of cutting off corners of a circle inscribed in a square and that went smooth enough. When it came time to find how they derived the formula from this method, we had a bit more difficulty. The supposed proof came quick enough, it was a lot of fun to bounce ideas off each other to figure it out. David brought up the ancient Babylonian approximation of a square root and from there, the proof fell together and we were happy.

    Then the next class came and we realized we knew nothing! The more we worked, the more we were mystified by the math that they had done. We had so many questions such as "did ancient Egyptians have a similar method of approximating square roots like the ancient Babylonians did?" David also brought up that the ancient Babylonian formula for approximating square roots was a linear approximation. Now, we were wondering if the ancient Babylonians knew about linear approximation, and if they had one for cube roots as well, or if they stumbled onto this formula somehow. The more I reflect, the more questions I am left with. My main take away was how its truly amazing how much math the ancient civilizations had figured out before the concept of mathematics as we know it now existed. Then, Duncan found an interesting conjecture of how they could've arrived at the area of a circle in a different way with ropes and that was also really interesting. When we think of circles in a modern way, we think about a radius sweeping 360 degrees to make a circle. That's why our formula uses radius instead of diameter. The rope method had a different take on area that I hadn't considered before. And the ancient Egyptians' had a formula using diameter, which makes me wonder how they viewed circles. Was it from inside to out like the spiraling of a rope, or was it outside in starting with the circumference and 'filling in' the middle, or was it something else entirely?

    Finally, we expanded our problem into 3D. This was a lot of fun as well, we had a lot of discussion about how we would do this, what parts of the cube we should cut off, and how much each method would result in. From this, I feel like I learned a lot about the differences between volume and area that I never really gave much thought into before. I remember debating with Duncan about why cutting off the 8 corners of the cube like we did to the square would result in much more error. We talked about taking infinite 2D slices of a 3D shape and how if we only cut corners, the cross section of each slice wasn't guaranteed to be an octagon. Especially where the sphere touches the face of the cube, a cross section of that part would result in only a point in the middle of the square being part of the sphere resulting in a lot of extra area.

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