Whitney! is going to love this

two words: Klein's bottle.

Not enough?

another two words: Alice Universe

THIS STUFF IS RIDICULOUS!!!!!

here's the breakdown. I wander into Calc 3 with Dr. Kiguradze ready to learn about Surface Integrals which are certainly interesting in and of themselves, but then the real magic happens. When integrating a Surface it's important to know its orientation, therefore a teacher must explain oreientation to the students, and whats the best way to describe something? with a counterexample. in this case the question was posed, does every surface have an orientation? answer from person, who probably sits in the back of the class so when he answers these kinds of questions everyone has to turn and look at him or if they're too lazy to turn can for a moment wonder if it's the voice of God answering questions so Kiguradze doesn't just stare at us for ten minutes, (and i quote) , "no because of this little thing called a Mobius Strip."
first time Kiguradze's actually looked surprised by an answer. but he agrees. A Mobius Strip (which i spelled as Noobius when i first heard it prounounced,i love terrific accidental puns!) is a 2D object that requires a 3D space to be modeled. It's basically a strip of paper with a twist in it, here's a picture...

Now some of you may say, "Wait a minute, that looks like a 3D object! " (If you've ever taken or had the pleasure of sitting in on Dr. Sawyer's Lectures, imagine this in his imitation of student's angry objections) but you'd be wrong because it only has one surface. Prove this by starting anywhere along the strip and tracing your finger along it, eventually you'll end up on what APPEARS to be the other side of the strip at the exact same location that you started. Pretty Crazy. but then it got BETTER...
Anyone familiar with Calculus knows, that once people discover things like this the only reasonable thing to do is add a dimension and see what happens...
You get Klein's Bottle! This is a two dimensional object just like the Mobius Strip except it needs 4 dimensions to be modeled. those of you now angrily pointig at the picture that i've so kindly posted for you will allow me to explain that this is only a conceptual drawing. Its not accurate because Klein's Bottle by Definition (something Dr. Martin knows nothing about, *cough Magnitude cough*) requires 4 dimensions to be portrayed. (also keep in mind, that although time is often considered the 4th dimension, that is not what I'm referring to in this instance, but rather an as of yet unimagined, "spatial" fourth dimension) the idea of this bottle is that if you were walking somewhere along the outside of the surface (similar to how we were earlier tracing our finger along the Mobius Strip) you could end up on the inside . Again keep in mind that inside and outside arent exactly correct terminology because there is only one side hence the lack of orientation!

I love that we can hypothesize and create things in math that are literally unimaginable, and yet we know them to be accurate. CRAZY.

I'd like to give mad props to Wikipedia for the pictures, and because while on their page for the Klein Bottle i accidentally stumbled across the Alice Universe just as Alice fell into the rabbit hole (or did she follow the rabbit? I may have been following an instinctual rabbit, so the metaphor still fits )

"A universe that includes one of these "non-orientable" connections does not allow a global definition of whether a particle is "really" matter or antimatter, and this sort of universe, with no global definition of charge is referred to in research papers as an "Alice universe." "
-wikipedia


this probably makes no sense but basically what happens is : you have whats considered an inverted wormhole(!!!!!!!!), so when anything passes through it everything in it flips. (i.e. Right handed becomes Left handed,) Most importantly Positive charges become Negative charges, matter becomes anti-matter, etc. As a result, nothing can be defined to be Positive or matter because in fact its only that way for the moment and it need only pass through an inverted wormhold to become the opposite. Now imagine an alternate universe in which this wormhole exists and TaDa you have an Alice Universe, (named for Lewis Carrol's character).

Now the connection:
Because of this changing property of everything within it, its impossible to determine which is the normal and which is the reversed except locally, just as in a Mobius strip its impossible to determine which side is which except when you're at a point and are convinced that there is another side to the strip. On a Mobius strip you can end up on that other "side" and in an Alice universe you can end up with that charge just by following the path!!!!!
Furthermore, The Alice Universe is kinda the 3D version closed volume version of a Klein bottle...in wiki's words "What a Klein Bottle is to a closed two-dimensional surface, an Alice universe is to a closed three-dimensional volume[!!!!!!!!!!!!!!] "

to better get a grasp of all this check out
http://en.wikipedia.org/wiki/Alice_universe#Alice_universe
http://en.wikipedia.org/wiki/Klein_bottle
http://en.wikipedia.org/wiki/M%C3%B6bius_strip

to clear your mind if all this confused you, check out
http://www.youtube.com/watch?v=eyqUj3PGHv4

If I explained anything incorrectly, I apologize, please correct me and I'll edit this post.
Feel free to comment with your own theories, discoveries, or weblinks that relate either directly to these three topics or anything else that just blows your mind and relates to science and or math.

p.s. INVERTED WORMHOLE!!!! my life will never be the same