moonlord wrote:The torus-0 is, I believe, two points at radius R from origin: x^2=R^2. The torus-1 is a circle (x^2+y^2=R^2), the torus-2 is, well, a daughnut.
What I don't understand is the following method for building a torus: Given a torus-n you can construct a torus-(n+1) by replacing any point in the torus-n with a circle lying in a plane (say, X_n, X_(n+1)), so that only one of the directions is new. Otherwise said, X_n already exists. This works for the first (1,2) torii, but what afterwards? Using it on a torus-0 gives me two circles, and not one. Why so? Is the method correct for higher torii?
Let's look at the torus-1 in XY. Replacing every point with a circle in XY is not possible, as they overlap. Replacing every point with a circle in XZ or YZ gives a torus-2. Replacing every point with a circle in ZW or any other dimensions not already used gives a duocylinder (not sure about this).
What about a torus-2? Replacing every point with a circle in XY, YZ or XZ is not possible because they overlap. Replacing every point with a circle in XW, YW or ZW should give a torus-3. But does it? I will try to do a cartesian verification later this evening. Replacing every point with a circle in, say, WT should give a 23, a 32, a (23) or what? I can't follow this...
Why I'm bothering with this is the fact I wish to have an idea of how a torus-3 would look like. By dimensional analogy, it should be something that, when passing through a hyperplane (3D) with the symetry plane first, is a torus, then splits into two torii, and they merge afterwards back into one torus. Is it correct? Does this even make sense?
Thanks in advance for your help!
moonlord wrote:but, wendy, you totally lost me.
It's pretty simple, really.
Using it on a torus-0 gives me two circles, and not one. Why so? Is the method correct for higher torii?
As an aside, so far I can only find the slices through coordinate hyperplanes. It would be interesting to find general slice with hyperplanes parallel to them.
:?That's why proving theorems is so much harder than solving problems.
For instance, today I learnt the definition of a limit in n dimensions. To define it, you need the definitions for a set, an "open ball", and the boundary of a set.
Furthermore, the only reason we need limits is to define partial
derivatives, which are relatively easy to understand.
Its not necessary. You simply can say that a a sequence of tupels converges if each component sequence converges.
The other area is algebraic topology what is usually referred to topology, i.e. the investigation of deformation. For example there is an interesting result that any closed 2-manifold is (deformable to) either the sphere with attached loops, or the crosscap with attached loops. On the other hand the 3-manifolds are not completely categorized yet.
The only reason for YOU. I for example need limits to define sqrt(2).
PWrong wrote: Anyway, if I understand you correctly, that's the definition for the limit of a vector function r(t), not a function of several variables f(x,y,...).
All the words you used here are defined in terms of general topology, which involves calculus. The words themselves aren't satisfying. You need definitions and mathematical notation to actually do anything with topology.
`No, number theory is more useless than topology. And it is called the queen of mathematics. (Saying this knowing the danger of completely taken wrong.)
wendy wrote:bo198214 schreibt
On the other hand, keeping number theory in mind has helped me thin through much of the elsewhile infinite possibilities of hyperbolic geometry in 3 and higher dimensions.
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