mobius strip?

Higher-dimensional geometry (previously "Polyshapes").

mobius strip?

Postby jmichae3 » Sat Jul 14, 2007 5:25 am

how would one envision a mobius strip extended into R4?
public class z{public z(){System.out.println("Hi");} public static void main(String[] args){z p=new z();}}
Jim Michaels
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wikipedia Eulidian space plot

Postby jmichae3 » Sat Jul 14, 2007 5:29 am

http://en.wikipedia.org/wiki/M%C3%B6bius_strip

It is called the Sudanese Mobius Band.
alas, it requires complex numbers.
"To see this, first consider such an embedding into the 3-sphere S3 regarded as a subset of R4. A parametrization for this embedding is given by"

z1 = sin(eta*e^(i*phi))
z2 = cos(eta*e^(i*phi/2)).

and |z1|^2+|z2|^2=1 and "therefore the entire surface lies on S^3"
what does that phrase mean? does this have something do do with the fact that this lies on a circle instead of a flat line strip? it's beginning to look like it. x^2+y^2=r^2 is the equation for a circle, so if r=1...

eta runs from 0 to pi
phi runs 0 to 2pi
They said this is in the complex space C2 instead of R4.
I don't have my math books handy anymore and I'm a little rusty lately, so does anyone know how to put this into Euclidian space (R4)?

It will be interesting to try to decide where to draw lines to and from...
Last edited by jmichae3 on Sat Jul 14, 2007 9:00 am, edited 2 times in total.
public class z{public z(){System.out.println("Hi");} public static void main(String[] args){z p=new z();}}
Jim Michaels
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Postby bo198214 » Sat Jul 14, 2007 8:06 am

eta and phi are parameters (the mobius strip is 2 dimensional so we need two parameters).
On the other hand I dont understand the question. Why would I try to envision something in 4d if I can do it without hassles in 3d?
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Postby jmichae3 » Sat Jul 14, 2007 8:24 am

I wrote a viewer/rotator program. I just discovered those 2 parameters are angles myself at answers.com. they give a little more info than wikipedia on this object.
4D objects do strange things in 3D when you rotate them in 4D. I just wanted to get as true a representation as I could get (wireframe), and then once I've modeled it, I can see what it does and add it to my list of other objects and make it available to other people.


I'm thinking that to plot this in R4, I could treat i and j as x and y, but what do you do in c2?
public class z{public z(){System.out.println("Hi");} public static void main(String[] args){z p=new z();}}
Jim Michaels
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