Torii and not only

[moonlord]

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!

[Marek14]

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!

One of the possibilities simply is that torus-0 is ONE point, not two, then it works. Note that when you slice a circle, the first slice is also one point, and not two - just as first slice of doughnut is one circle, while all the others are two.

It seems that this construction makes a certain subset of toratopes, namely xn=([xn-1]1), torus-1 is (2), torus-2 (21), torus-3 ((21)1), torus-4 ((21)1)1). These are also what is called circle^n.

The way we originally looked at toratopes involves parametric equations, and can be found somewhere in the forums.

Your assessment of slicing is correct, but you have to take note that "two torii" is quite ambiguous. There are four different way a toratope can be sliced in two torii:

1. Two torii with different centres, separated in their symmetry plane.
2. Two torii with different centres, separated along their symmetry axis.
3. Two concentric torii with different outside radii.
4. Two concentric torii with diferent in-radii.

Each of these arises with one 4D toratope. Your torus-3 is the case 4, where one torus is completely inside the other.

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.

P.S. I assume that by "replacing each point with a circle in XW, YW, or ZW" you don't mean choosing one and replacing each point with it. You have to replace each point with circle in (f(x,y,z),W), where f(x,y,z) gives normal vector at that point.

[wendy]

A torus is typically rated by what is required to span the hole, literally, to prevent non-vanishing loops forming anywhere.

That is, a vanishing loop is one that can go to zero or go to infinity, without crossing a surface. A non-vanishing loop is somehow trapped by the surface. The nature of the loop is the solid sphere that contains the loop.

So, for a point pair, AA we have the loop aa, bb, cc as

c A a b b A a c

You can see that c can go to infinity, and b shrink to zero, without crossing A, but a--a can not vanish. Therefore aa determines a hole. The actual hole is AA, and spanning AA prevents aa and bb from forming.

The dimension of the surface AA is then 1d, and so is a 1d hole.

A given surface divides space into an interior and an exterior, and both regions must be tested. A typical torus admits loops inside and outside the surface, which do not vanish. You need two 2d disks to close off these possibilities, and so it has an interior hole of one hedron, and an exterior hole of 2d.

If you were to divide the interior of the torus up into eight cubes, by way of example, you see that the general equation becomes

n + 18v + 64e + 40h + 8c

Euler's defect is then -1+32-64+40-8 = -1. This means that it needs an extra 2d element to close the thing, ie a square spanning the hole.

Note that holes link, if the sum of dimensions add to N+1. eg in 4d, one can make a chain of hedrous and chorous links, since 2+3 = 5.

You can apply the torus product, to any set of polytopes, such that the sum of surface-dimensions gives the surface-dimension of the result. So there exists a pentagon-dodecahedral torus, because 2-1 + 3-1 = 4-1. You can also take a tripple torus-product of three polygons, since (2-1).3 = 4-1.

W

[moonlord]

:? Well, I could follow Marek, but, wendy, you totally lost me. What I know about topology is insignificant...

[bo198214]

but, wendy, you totally lost me.

I didnt understand her either, though it sounded really interesting.

[moonlord]

Now we're two, so it's a class...

[wendy]

It's pretty simple, really.

You can tie a string around a car-tyre, so that it is linked. You can't remove it by pulling it away.

If you put something to fill in the hole in the car-tyre, then it's pretty hard to put a string through the hole. To do this, you need to use a 2d closure. So that's a "hedron-like hole".

If you treat two objects as one, you can put a peice of string around them, but this can be easily removed. You have to put one of these in a box (ie a 3d solid with a cavity inside). So the fact that you can put one in a box, and not be able to remove that box, means that there is a hole there. The thing you need to prevent the use of a box is to connect the two object.

So these two objects behave as a one-dimensional hole, since it is missing the one-dimensional string to join the two bits.

On the other hand, the box is a kind of hole too, because you can put one end of a one-dimensional hole into it (one of the two objects). Accordingly, to prevent you using the box, you need to fill in the 3d cavity with something. So it's a 3d hole.

You can make a chain of 2d holes, this is the usual chain. You can also make a chain of alternately 1d and 3d holes. You normally see this as the sphere-and-link kind of chain, normally seen holding bathroom plugs and pens at banks.

W

[PWrong]

It's pretty simple, really.

No, the analogy is simple. The maths of topology is much more difficult.

Using it on a torus-0 gives me two circles, and not one. Why so? Is the method correct for higher torii?

That's the torus product of a 0-sphere with a 1-sphere (circle). The standard torus-n is always a product of several circles. The more general torus is a product of spheres of various dimensions (although we usually don't use 0-spheres).

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.

I'm not sure how to do this with a surface in 3D. I suppose you would find the intersection of the surface with the plane. But then you'd need a new coordinate system for the plane. Maybe we need tensors to do this.

[moonlord]

Do you happen to know a good intro to topology? Online, preferably.

EDIT: Oh yeah, I just realised I was saying a stupid thing, as Marek pointed out. He made a good assumption, though... I believe that's what I was meaning...

[PWrong]

All the sites I've seen on topology are either too simple, like this one or too complicated, like this.

I think topology is something you learn very gradually as part of calculus. It's all the abstract stuff from calculus that you don't need in the exam. To prove any theorem in calculus, you need to understand topology. That's why proving theorems is so much harder than solving problems. Topology is what makes us mathematicians smarter than engineers .

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. These are all topogical concepts (I think). Furthermore, the only reason we need limits is to define partial derivatives, which are relatively easy to understand.

[bo198214]

Hoho, much provoking statements but not so much sense

First there are two different areas called topology. What PWrong was mainly talking about is set theoretic/general topology, i.e. what are the least concepts that one can speak of limits. Main terms are: open/closed sets, separable, environment, compact, filter, and based on that, more abstract definitions of (uniform) convergence, limit and continouity.

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.

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.

Its not necessary. You simply can say that a a sequence of tupels converges if each component sequence converges.

Furthermore, the only reason we need limits is to define partial
derivatives, which are relatively easy to understand.

The only reason for YOU. I for example need limits to define sqrt(2).

[PWrong]

Its not necessary. You simply can say that a a sequence of tupels converges if each component sequence converges.

That statement requires a definition for sequences, component sequences and convergence. 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,...). The definition I learned today was the epsilon-delta kind of definition.

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.

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.

The only reason for YOU. I for example need limits to define sqrt(2).

If you're working in the rational numbers, then you need limits to define irrational numbers. I was only talking about limits of functions of several (real) variables. There are certainly other uses for limits, but their purpose in my calculus unit is to help define derivatives. I'd prefer to learn some more topology, but I guess they want to teach us stuff we can use.

[bo198214]

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,...).

"In n dimensions" I understood as in Rⁿ. But you mean in n variables (i.e. R^m-> Rⁿ). Then you need at least a norm of course.

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.

You can also say, to explain anything in math you need set theory, so everything is set theory? To understand algebraic topology you dont need general topology, not even differential geometry. The base operations are glue and cut. Its more about showing whether two algebraic structures are equal.

[moonlord]

So, I should gather that, in very brief, topology is one of the most useless, but most fascinating aspect of maths... Right?

[bo198214]

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]

bo198214 schreibt

`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.)

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.

I should hardly call it useless.

W

[bo198214]

bo198214 schreibt

Are you using some e-mail program that knows that I am german or do you speak yourself?

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.

Of course mathematics is useful for mathematicians to do mathematics.
And in some rare cases also for other people (1st to call the physicists and computer scientists).
But try to explain to normal people the usefullness of discussing 4 spatial dimensions.
And I like it!

[wendy]

i speak a little german and french. enough to glark through german, french, danish and a few other languages. i know enough to listen to the news auf deutsch, ohne subtitles. I use a number of german abbreviations, aslo, eg usw (und so weiter), zB (zum Beispeil), früh (earlier), &c.

Actually, i do lots of experiments in number theory, and polytopes and higher dimensions get used to provide some focus for this, so it kind of echos one way to another.

using number theory, for example, you can easily prove that the cube is the only regular polyhedron with a rational dihedral angle, and therefore the only regular figure with faces with even number of edges.

W

[bo198214]

But what I dont understand, Wendy, when you are so engaged in doing mathematics, why cant I read your results in some journals? :?

[wendy]

Much of what i did is largely a hobby, rather like a spot of bush-bashing or something. It is more that i largely have been to places where people have not been yet.

Also, a lot of it was not ready for prime time until i started to mingle on the net. Still.