What is topology, and what's cool about it?

[billfish678]

@Thudlow_Boink

We do have a Doper named Topologist.

If only we had one named Mr.Right, these things would always resolve so much smoother and quicker

[bordelond]

@The_Seventh_Deadly_Finn

I know the joke about how a topologist is a guy who doesn’t know the difference between a cup of coffee and a donut, and I understand the concept (both are volume-occupying forms with a single hole through them).

What I don’t get is how you can turn that observation into math, and I especially don’t get how you can use it to solve real-world problems whose answer isn’t obvious by looking. Can anybody give a specific example of such a problem?

This is more applied topology than anything else, but consider the humble Möbius strip:

There have been several technical applications for the Möbius strip. Giant Möbius strips have been used as conveyor belts that last longer because the entire surface area of the belt gets the same amount of wear, and as continuous-loop recording tapes (to double the playing time). Möbius strips are common in the manufacture of fabric computer printer and typewriter ribbons, as they allow the ribbon to be twice as wide as the print head while using both half-edges evenly.

A device called a Möbius resistor is an electronic circuit element which has the property of canceling its own inductive reactance. Nikola Tesla patented similar technology in the early 1900s: “Coil for Electro Magnets” was intended for use with his system of global transmission of electricity without wires.

The Möbius strip is the configuration space of two unordered points on a circle. Consequently, in music theory, the space of all two note chords, known as dyads, takes the shape of a Möbius strip.

In physics/electro-technology:

[ul]
[li]as compact resonator with the resonance frequency which is half that of identically constructed linear coils[/li][li]as inductionless resistance[/li][li]as superconductors with high transition temperature [/li][/ul]

In chemistry/nano-technology:

[ul]
[li]as molecular knots with special characteristics (Knotane, Chirality) [/li][li]as molecular engines[/li][li]as graphene volume (nano-graphite) with new electronic characteristics, like helical magnetism[/li][li]In a special type of aromaticity: Möbius aromaticity [/li][li]Charged particles, which were caught in the magnetic field of the earth, can move on a Möbius band[/li][li]The cyclotide (cyclic protein) Kalata B1, active substance of the plant Oldenlandia affinis, contains Möbius topology for the peptide backbone.[/li][/ul]

[bordelond]

@suranyi

I don’t want to get into a lot of details, but it works like this:

At the bottom level, a map can be represented as a set of nodes, edges, and faces, each of which carries information on it. (For example, an edge can represent a road, but at the same time it can represent a county boundary, and the side of a park.)

…

This way, in our work, if we need to write a program that (let’s say) finds all the roads that join ramps that lead to Interstate highways, we don’t have to figure out what roads are connected to each other – the information is readily available from the topology of the network. Similarly, if we want to find all parks bordered by a road that extends past the park boundary. Or any of thousands of other questions.

Many of the fundamentals that **suranyi **is describing were worked out by Leonhard Euler in 1735 – the famed Seven Bridges of Königsberg. Look at the map at that link and read below (the entire article is worth the OP’s read):

The city of Königsberg in Prussia (now Kaliningrad, Russia) was set on both sides of the Pregel River, and included two large islands which were connected to each other and the mainland by seven bridges.

The problem was to find a walk through the city that would cross each bridge once and only once. The islands could not be reached by any route other than the bridges, and every bridge must have been crossed completely every time (one could not walk halfway onto the bridge and then turn around to come at it from another side).

… Euler proved that the problem has no solution.

Significance in the history of mathematics
In the history of mathematics, Euler’s solution of the Königsberg bridge problem is considered to be the first theorem of graph theory, a subject now generally regarded as a branch of combinatorics. Combinatorial problems of other types had been considered since antiquity.

In addition, Euler’s recognition that the key information was the number of bridges and the list of their endpoints (rather than their exact positions) presaged the development of topology. The difference between the actual layout and the graph schematic is a good example of the idea that topology is not concerned with the rigid shape of objects.

[Indistinguishable]

@suranyi

Am I the only one who has actually used topology in my (non-academic) work?

@Thudlow_Boink

We do have a Doper named Topologist.

Who has probably only used topology in academic work…

[Indistinguishable]

@billfish678

But wait you say, what if the temp distribution is not anything nice like you describe above. This is harder to grok, but if you start with the distribution as describe above and slowly warp it to what you want, you will NOT break that perfectly circular ring. You will warp that ring into some other shape, but it will still be a loop/ring of some sort. The same goes for pressure. And no matter what shape those two loops end up as, they still have to cross at at least two points if they encircle the earth.

I could still be missing something here though.

A) What exactly do you mean by warping one distribution into another, and how do you know you can warp this starting distribution into every other continuous distribution?

B) What exactly do you mean by a loop/ring “encircling the Earth”?

Incidentally, even though I was the first to bring it up, we don’t really need to show the existence of a loop of points whose temperature is equal to their antipode; it suffices to just have a path of such points from one point to its antipode (though this, of course, suffices to construct a full loop anyway, either by coming back the same way one went, or by coming back via the antipodes of the way one went). Once we have such a path, we can do what Chronos said and find a point on it whose pressure is equal to its antipode by the IVT. But I still don’t think anyone here has given a proper proof that such a path exists.

[billfish678]

@Indistinguishable

A) What exactly do you mean by warping one distribution into another, and how do you know you can warp this starting distribution into every other continuous distribution?

B) What exactly do you mean by a loop/ring “encircling the Earth”?

.

As for part B, isnt that the more obvious part ? As described, all points on the equator meet both the requirement of equal values and anti-podal. That “ring” encircles the earth does it not ?

The real crux of the matter is part A as you have said.

Imagine I decrease (compared to the original distribution) the temp of a region that spans north south across the equator as well as some distance east west. A smooth, depressed (compared to the surrounding areas) temperature dimple if you will.

Our ring/line obviously CANNOT still “run” straight across that dimple, as the temps antipodal to that dimple along the equator no longer are equal. It will have to divert either to the north or the south. If we pick a path that runs north of the equator (but still in the dimple and at a “lower” temp), can we not find a lower temp on the opposite side south of the equator that can match that and IS STILL antipodal ?

Without being rigorous, its appears to me you can.
Moving that line north/south only one point at time where it crosses the dimple while finding the corresponding change to the point/ line on the antipodal side “seems” to me be an application of the 1 D IVT.

Or in other words, we have taken what seems like a clear answer (equator ring) and used a 1 D IVT to modify it.

Best I can do at the moment.

Note, at this point, we are ONLY talking about the temp distribution.

[suranyi]

@bordelond

Many of the fundamentals that **suranyi **is describing were worked out by Leonhard Euler in 1735 – the famed Seven Bridges of Königsberg. Look at the map at that link and read below (the entire article is worth the OP’s read):

Indeed; what I described can’t be considered mind-blowing to anyone knowledgable about this area of math. Though in our internal docs we trace it more specifically to the work of Poincare.

[Indistinguishable]

@billfish678

As for part B, isnt that the more obvious part ? As described, all points on the equator meet both the requirement of equal values and anti-podal. That “ring” encircles the earth does it not ?

It’s clear that when you say “encircles the earth”, you mean some condition which holds of the equator. It’s not clear what other loops you intend this condition to hold of; in particular, in your proof, you assert that, even post-warping, the equator turns into some loop which still “encircles the earth”, and that any two loops which “encircle the earth” meet somewhere. But we cannot consider either of these assertions a given, or even begin to provide a proof for them, until we are more clear about what you mean by “encircles the earth”. (E.g., how about non-Equator lattitude lines? Do they encircle the earth? I can certainly get two of those which don’t intersect anywhere)

Without being rigorous, its appears to me you can.

I would not be inclined to call this sort of thing a proof.

[billfish678]

[

Opps hit submit too early.

Lets try this again.

[billfish678]

@Indistinguishable

I would not be inclined to call this sort of thing a proof.

If you can’t tell me how I am wrong, you aint helping things here any either

Lets step back a bit to make sure we are on the exact same page first.

A First, as I have originally described the temp distribution, with the south pole coldest, north pole warmest…blah blah blah, is it NOT a ring that happens to coincide (because of the way I set up the initial conditions) with the equator?

B And does it not “encircle the earth” ?

C And is the ONLY one ring that does so.

DAnd this ring outlines a circular plane that has the center of the earth at its center?

Note, rather than a long assed posts back and forth by both you and me, lets run through this detail by detail okay? Starting with the ones just mentioned. Might take longer, but will probably be less confusing I think.

Not to say as some step I myself am not going to have a :smack: moment about this

[Chronos]

How did you get a loop of antipodal points for temperature?

Ah, OK, I thought from your post that you were accepting that from the IVT. I probably could come up with a proof of this, but it’ll have to wait until I’ve had more than six hours of sleep. For now, me being a physicist rather than a mathematician, I’m willing to let it go as “obvious”.
billfish, even if you start with the poles being extremes of temperature, that doesn’t necessarily mean that the Equator is going to be constant temperature. The constant-temperature line might well be wiggly.

[Indistinguishable]

Re: billfish, all those things you say are true, for the particular temperature distribution you assert them of (that where temperature strictly tracks latitude). Your proof has to work to establish its conclusion for any temperature distribution. And the burden of proof is on, well, the one claiming a proof. To the extent that I am pointing out where your proposed proof is unacceptably vague or making unsupported leaps in getting to the final, universal conclusion, I am pointing out where you are “wrong” (i.e., where your proposed proof fails to meet the standards of an actual mathematical proof). I can do no more than that.

To Chronos, yeah, it seems like it follows obviously from the IVT, but it’s the same dodgy logical leap I made in my nostalgic anecdote (and then had to make even dodgier logical leaps to try and prove when I realized it wasn’t automatic). All the IVT tells us (at least, so far as I’ve seen an argument for) is that on any path from a point to its antipode, we’ll find some equi-antipodal point; this gives us a whole bunch of equi-antipodal points, but we don’t automatically know anything about whether they form or contain a path of any particular sort, or whether they are rather disparately situated, or whether they do form local paths but these fail to cross hemispheres, or various things.

[billfish678]

@Chronos

billfish, even if you start with the poles being extremes of temperature, that doesn’t necessarily mean that the Equator is going to be constant temperature. The constant-temperature line might well be wiggly.

Read my description more carefully. I intentionally start off so that it IS NOT wiggly.

[billfish678]

@Indistinguishable

Re: billfish, all those things you say are true, for the particular temperature distribution you assert them of (that where temperature strictly tracks latitude). Your proof has to work to establish its conclusion for any temperature distribution. And the burden of proof is on, well, the one claiming a proof. To the extent that I am pointing out where your proposed proof is unacceptably vague or making unsupported leaps in getting to the final, universal conclusion, I am pointing out where you are “wrong” (i.e., where your proposed proof fails to meet the standards of an actual mathematical proof). I can do no more than that.

.

Do you wanna work with me to understand my thought process or do you wanna be a jerk about it ?

Chronos claimed the proof. I am trying to explain it. This aint a class on topology I am getting paid for nor a peer reviewed journal I am submiting to.

I am starting with a simple case anyone (or at least I thought anyone) could see was obviously true, and trying to legitimately modify it from there to any possible case.

Give me a fricking break here. This may well be my first topology proof/problem **EVER. **Besides whatever passes for topology in basic Calculus classes. I’ve never had any classes at any level on topology nor read anything on it more fancy/sophisticated than the “doughnut equals coffee cup” pop science article.

I could go search the web and probably cut and past some fancy sounding crap that makes me look good and shows that Chronos is right. But I aint interested in do that.

Again, do you want me to work through this step by step or not ?

Geez Louise.

[Haunted_Pasta]

@The_Seventh_Deadly_Finn

I especially don’t get how you can use it to solve real-world problems whose answer isn’t obvious by looking. Can anybody give a specific example of such a problem?

Knot theory is a sub-branch of topology that has found use in studying some aspects of molecular biology, as was noted in the post about Mobius strips. Google “knot theory” + “DNA” for a lot of information about… well, knot theory and DNA.

[Chronos]

Actually, I don’t think anyone here is disputing the result itself (either the final result, or the sub-result that there exist loops for any single quantity). The question is just whether those results follow easily from the Intermediate Value Theorem.

[billfish678]

@Chronos

Actually, I don’t think anyone here is disputing the result itself (either the final result, or the sub-result that there exist loops for any single quantity). The question is just whether those results follow easily from the Intermediate Value Theorem.

Hey there.

Not trying to pass the blame off onto you there Chronos Of course you did bring it up, so my sympathy is limited if you cant back it up yourself

I CAN see how this is at least possible.

If anyone is really interested, I’ll explain it as I see it. If somebody wants to turn this into an internet pissing contest, I am outa here.

My neighbor just died and my BS tolerance level is about at zero right now.

take care

[billfish678]

@Indistinguishable

. (E.g., how about non-Equator lattitude lines? Do they encircle the earth? I can certainly get two of those which don’t intersect anywhere)
.

Those arent antipodal now are they ?

[billfish678]

Heh…

You guys are busy surfing the web or cracking open textbooks or talking to folks arent you ?

The only input I have is one cat, two dogs, and a SO with a degree in the social sciences and I am pretty sure I have this figured out.

[Indistinguishable]

billfish678, I think you are taking my responses as much more personal than they are intended to be. My only purpose has been to point out how proving this is a more difficult problem than it may at first seem (the difference between “It seems rightish; this argument sounds plausible” and the standards of mathematical proof). It’s not my intent to disparage you or even make any comments about you rather than your proposed proof, but since it appears that I am coming off that way to you, let us consider the matter dropped, if you like.