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## Main Question or Discussion Point

Hi guys. I'm doing a high school project for AP Calc BC. We're supposed to build a 3-D solid using Riemann approximations. Basically, we have to build an object using solely rectangular prisms, or circles.

For example, let's say I want to build a cone. On a coordinate system, that would be a right triangle revolved around one if its sides. Therefore to build it, it would be pasting together tiny circles that increases in radius and become bigger.. The width of each circle is the constant thickness of the material used (this is dx). The smaller the width, the more like a cone it is. Starting from the top to the bottom, this will create a cone.

I already self-studied all the AP Calc BC information during the summer, some multi-variable from MIT OCW, a differential equations course, and a few lectures on topology. I want this project to be something that I can be proud of when looking back into my high school senior year. And when I was thinking of a solid to do it on, I went, "Well, why not some shape that uses topology?". I then thought about building a Klein Bottle. Do you guys think it's feasible? Too hard?

I was thinking of just creating a bunch of varying mobius strips that when pasted together, will create the klein bottle. However, that seems a bit too hard so I'm also been thinking about using a bunch of concentric circles with a hollowed out circle inside of it to create the tubing for the handle for example. Then I can expand the circle and create the top. The geometry inside might be hard to make with simple circles though. Any ideas? Thanks for reading by the way. I might have confused you with my explanation though.

Here's an image of what I'm talking about:

http://www.ima.umn.edu/2005-2006/gallery/polthier/kleinBottleNormalShowStill_med.jpg [Broken]

**Prompt: Make a physical model of a solid with a known cross-section.**For example, let's say I want to build a cone. On a coordinate system, that would be a right triangle revolved around one if its sides. Therefore to build it, it would be pasting together tiny circles that increases in radius and become bigger.. The width of each circle is the constant thickness of the material used (this is dx). The smaller the width, the more like a cone it is. Starting from the top to the bottom, this will create a cone.

I already self-studied all the AP Calc BC information during the summer, some multi-variable from MIT OCW, a differential equations course, and a few lectures on topology. I want this project to be something that I can be proud of when looking back into my high school senior year. And when I was thinking of a solid to do it on, I went, "Well, why not some shape that uses topology?". I then thought about building a Klein Bottle. Do you guys think it's feasible? Too hard?

I was thinking of just creating a bunch of varying mobius strips that when pasted together, will create the klein bottle. However, that seems a bit too hard so I'm also been thinking about using a bunch of concentric circles with a hollowed out circle inside of it to create the tubing for the handle for example. Then I can expand the circle and create the top. The geometry inside might be hard to make with simple circles though. Any ideas? Thanks for reading by the way. I might have confused you with my explanation though.

Here's an image of what I'm talking about:

http://www.ima.umn.edu/2005-2006/gallery/polthier/kleinBottleNormalShowStill_med.jpg [Broken]

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