Can a Klein Bottle be Constructed Using Rectangular Prisms and Circles?

[Anonymous217]

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.
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

 

[flatmaster]

That's certianly possible. You'd need to do some math to calculate the changing width of each mobius strip. Unfortunately, they won't all be the same. Also, a few strips will have a hole through them.

 

[Anonymous217]

Cool! That's what I was thinking. I'd have to ask my teacher if he accepts mobius strips instead of circles or rectangles as the base. Is it really possible to make an actual klein bottle though? I always thought that I might have to cheat somewhere and poke a hole where the handle part intersects with the cup so it wouldn't technically be continuous. It would certainly be a lot easier that way.

 

[Hurkyl]

I don't know if this is relevant, but the Klein bottle isn't a solid -- it's a surface.

(Like the sphere isn't a solid -- the sphere is a surface that forms the boundary of the ball)

 

[Anonymous217]

Haha, yes, that's what's cool about it. I asked my teacher about that technicality and he said to just pretend that there is an imaginary lid on top of the bottle. The volume inside will count as the volume I'll need to calculate. I have to find the volume theoretically using a model with equations and experimentally after adding up the volume of each slice (and then compare, hopefully with a small variation). This is why it's harder for mobius strips, since there's no volume for them and would therefore be no comparison possible. I might have to cheat like I said and just build it by making a bunch of concentric circles with a hole in them. These circles will count as my slices, but I won't be able to connect the handle to the inside.