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Mathematicians just make stuff up. And as you can see in my question just above yours, they cannot really explain why things like Klein bottles are part of their field, while pink unicorns are (apparently) not. --Bumptump (talk) 08:47, 27 November 2021 (UTC)[reply]
The Klein bottle is the simplest example of a non-orientablemanifold, and a counterexample to the naive but understandable assumption that all manifolds are orientable – understandable because this is so for all manifolds that can be embedded in R3. That already makes it an interesting object. Can you name any mathematically interesting properties of pink unicorns? --Lambiam09:18, 27 November 2021 (UTC)[reply]
Mm, the Möbius strip is arguably simpler, is embeddable in R3 and is unequivocally a manifold, if you leave off the edges. (If you include the edges, it's still a manifold with boundary.) What you say would be true if compact were part of the definition of manifold, but it isn't. (It's true that "compact" is frequently understood.) --Trovatore (talk) 04:47, 30 November 2021 (UTC)[reply]
The idea of letting "answers I got to a mediocre question on the Wikipedia Reference Desk" stand in for "what mathematicians are capable of explaining" is horrifying. --JBL (talk) 23:46, 27 November 2021 (UTC)[reply]
I think what you're trying to say is that you can't embed a Klein bottle in 3 dimensional space. But a Klein bottle, 3 dimensional space, and the notion of an embedding are all mathematical abstractions. You might think of the surface of a marble as a sphere, but a real marble is just a collection of atoms, which are in turn collections of subatomic particles; as such a marble doesn't really have a surface. All mathematical entities are abstractions with no "real" existence; but sometime they serve as convenient models of what you might observe in the real world. Any construction is allowed mathematically as long as it doesn't create a logical contradiction; whether it's similar to anything that takes place in the actual universe is a matter for physicists to decide. --RDBury (talk) 23:26, 26 November 2021 (UTC)[reply]
In The Elements, Euclid defines the geometric concept of "line" by: "A line is length without breadth." In the real world, something with length without breadth is impossible. Imaginary numbers are not real – except 0, but does 0 exist in this world? If so, where can we find it? Mathematical entities may serve as (imperfect) models of things in the real world, and, conversely, things in the real world can be used as (imperfect) models of mathematical entities. We can use our imagination to abstract the imperfection away. In M-theory, the universe is modelled as having ten spatial dimensions. For all we know, one day a physicist may discover that the elementary particles can be described as a set of closed branes, one of which has the topology of a Klein bottle. --Lambiam00:33, 27 November 2021 (UTC)[reply]
First of all, I must apologize to the OP, @Georgia guy:, for the simply dreadful and frankly wrong and misleading initial answers they were given. It was insulting for you to be subjected to them, and you did not deserve that. Secondly, it is important to understand that mathematical objects exist under a number of different formulations, and a formulation does not have to be visualizable in order to be useful. For example, you can't actually count to 1 trillion in your lifetime, and yet the number 1 trillion is a useful mathematical object. Just because you can't experience it directly with your senses does not make it "not useful" or "made up", at least any more "made up" than all of mathematics is. The question is not whether any bit of mathematics is real, but rather whether it is consistent, logical, provable, and useful, and klein bottles are all of these.
To do a little more details on the klein bottle, we need to start with a simpler object, a circle. And we'll start with the simplest circle of all, the "unit circle", which is defined as a circle with a radius of 1 unit. We can formally define such a circle in multiple ways. We can draw it on a paper, but we can also just describe it in words, or we can define it algebraically; in this case the unit circle is defined as the set of all points x and y that satisfy the equation x2 + y2 = 1. We can also define a shape in three dimensions which is analogous to a unit circle, let's call it a "unit sphere". We can't actually draw that, but we could construct it and hold it in our hands. This is a sphere of radius 1 unit, or defined algebraically as x2 + y2 + z2 = 1. Notice that we've established a pattern; we've defined a family of objects which are all defined as having a radius of 1 unit, and which also have the general formula of the sum series of variables squared all equalling 1. We don't need to actually draw a picture or construct an object to do math with these equations, we can still use the equations without drawing anything; things like algebra and calculus are certainly made easier when you can draw pictures, but you don't have to. The math itself doesn't require pictures to work. So, there's nothing to stop us from defining circle-or-sphere-like objects in higher dimensions, say w2 + x2 + y2 + z2 = 1 or v2 + w2 + x2 + y2 + z2 = 1. These "hyperspheres" have many of the same properties as circles and spheres, and we can still do all sorts of math with them by manipulating the equations using the rules of algebra and calculus. That's why math is useful in the first place: it allows us to understand things we can't actually see anyways.
Back to the klein bottle. In topology, there's an object called a moebius strip. It's an twisted two-dimensional object that has 1 side and 1 edge. It has all sorts of interesting topological properties. You can actually make a moebius strip by taking a piece of paper, giving it a half-twist along its long axis, and taping the edges together. That's fine and all, but for mathematicians, the physical object is not the important part. It's the mathematical abstraction, the equations and formulations that you can manipulate using the rules of mathematics, that they work with. For them a moebius strip is no different than a circle or a sphere in that regard, the mathematical part of it is not the twisted paper, it's the "mathematical objects" that represent the general shape of the twisted paper, and what you can do with it. A klein bottle is to a moebius strip as a 4-dimensional hypersphere is to a 3-dimensional sphere. It's the next step in an infinite class of objects based around the same definition; in the case of a klein bottle, just as a moebius strip was an twisted two-dimensional object with one edge and one face, a klein bottle is a twisted three dimensional object with one surface. It has no inside or outside, it just has one surface. We can't picture that (because you need a fourth dimension to "twist" a three dimensional object into), but we can work with it, in the sense that we can take the same mathematical formulations that allow us to define a moebius strip without building it, and do the same thing with a klein bottle.
Jayron's answer is pretty good, though with what I see as somewhat unjustified criticism of other posters, with a single hopefully obvious exception. That said, you actually can visualize the Klein bottle with just a little effort. Everything is unproblematic except for the self-intersection, where the narrow "handle" meets the base of the "flask", right?. Well, there, you just have to use a little imagination. Think of yourself as an ant crawling along the surface. If you're on the "handle", you can keep crawling along the "handle", even when it goes "inside the flask" — the walls of the "flask" are insubstantial; you can pass through them without resistance. Conversely, if you're on the surface of the "flask", you can pass through the "handle" without resistance. However, you can't jump straight from "handle" to "flask", or vice versa, at the point they pass through one another. Does that help? --Trovatore (talk) 06:33, 2 December 2021 (UTC)[reply]
Also, I've never really been satisfied with the explanation that essentially says "This place where the klein bottle intersects with itself? It doesn't actually do that". Because a klein bottle does not pass through itself like you see in the glass bottle in the picture you showed. That's the bit which is impossible to work out in your head, visually. It's also, as I note, not a problem for mathematicians. These problems are fundamentally "engineering" problems, not "mathematical" ones. Mathematicians deal primarily with symbolic abstractions and generalizations. That's what mathematics is. Even basic arithmetic like 2+3 = 5 is a symbolic abstraction. --Jayron3212:20, 2 December 2021 (UTC)[reply]
Ah, but that's the thing — it's not actually impossible to work it out in your head visually. It just takes a bit of effort and practice. You just have to remember that handle points are not close to side-of-flask points on the surface, even if they happen to share an exact location in 3-space.
An aid that works for some people is to use color. Let the handle be red. As you move up or down, the red fades, shading into orange and yellow as you approach the flare-out from the inside at the very bottom of the flask or at the very top, then turning green as you pass towards the side of the flask.
Now you just remember that red points and green points are always distinct and not close to one another, even though some of them may share an exact location in space. In fact, you can deform the thing, jiggle the handle around, and its red points move without disturbing the green points they pass over and through. --Trovatore (talk) 16:58, 2 December 2021 (UTC)[reply]
By the way, this use of color is a more general trick for visualizing four-dimensional objects, with color being one of the axes, and is essentially a proof that the Klein bottle can be embedded in R4. --Trovatore (talk) 17:04, 2 December 2021 (UTC)[reply]
There are ways to represent the concept. But you can't actually not see where the handle intersects the body, that's an artifact of the 4-dimensionalness of the object being represented in three dimensions. All of those visualizations are still happening in 3-space. Just like we can use things like Perspective (graphical) to represent three dimensions on a flat surface, a flat surface is still not three dimensional. The sort of tricks you are describing are like perspective drawing; it's just psychology tricks you play to get some handle on it. Ceci n'est pas un pipe and all that. The thing about mathematical abstraction is that you don't need to do anything like that. Want to understand how the 17-dimensional analogue of a klein bottle works? The mathematical formalisms make it no trickier than a moebius strip is. Math is all about generalizing the patterns, so we have algorithms for determining, for example, how a 1-surface object in n-dimensional space would behave if represented in n-1 dimensional space. I have no idea how to even begin to approach visualizing a 17-dimensional klein bottle analogue represented in 16-dimensional space. But mathematics can tell us. The usefulness of mathematics comes from expanding human understanding beyond human senses. --Jayron3217:10, 2 December 2021 (UTC)[reply]
Intuition is fundamental to progress in mathematics. It's rarely productive to generate formal arguments at random and see if something interesting happens to pop out; the search space is too large and too branching, and it's not clear that you'd even recognize an interesting result if you did happen to prove it. The "color" picture I described above is, as I said, "essentially" a proof that there's an embedding of the Klein bottle in R4. Turning it into an actual formal proof would be extraordinarily tedious, and honestly not very useful. I think if you talk to low-dimensional topologists they'll tell you that they do this sort of thing on a daily basis, that it's like breathing to them. --Trovatore (talk) 17:37, 2 December 2021 (UTC)[reply]
The representation of the Klein bottle as a three-dimensional glass object is no more problematic than the representation of a knot as a two-dimensional image - you just have to understand the conventional tricks by which the artist indicates that certain parts of the object are not in fact in contact, as the physical representation naively shows, but are to be understood as passing over/under each other without touching. Anyone for a tesseract?--Verbarson (talk) 13:30, 3 December 2021 (UTC)[reply]
The question was never whether or not such representations were either useful or reasonable. The glass klein bottle, and your knot image are clearly both. The question is whether such visualizations are necessary; the OP posited a question about whether or not mathematics is "believable" if an object cannot be visualized in the full dimensions in which it exists; i.e. the OP expressed incredulity over the believability of klein bottles as objects in 4D space, since humans can't actually operate themselves in 4D space. Lower-dimensional representations of such objects are possible, useful, and reasonable, but mathematicians also don't need such lower-dimensional representations for to do their work. Mathematics expands the ability to know things beyond what can be reasonably visualized, and the ability to visualize in any form is not a prerequisite for mathematics to be believable. --Jayron3214:41, 3 December 2021 (UTC)[reply]
Jayron certainly makes some good points. He's quite right that the meaningfulness of a mathematical concept does not depend on the ability to visualize it.
That said, I don't really agree that "mathematicians also don't need such lower-dimensional representations for to do their work". Or at least it depends a lot on what you mean by "need". As a practical matter, I doubt you could get hired as a low-dimensional topologist (actually I doubt you could even finish your degree) if you didn't use mental aids of some such kind.
Even in my own (former) field, set theory, one of the most abstract branches of mathematics, we relied on mental pictures. They aren't very faithful pictures, of course — the sets may well have more elements than there are actually points. But you have to develop an intuition somehow, and the visual faculty is one of the most powerful aspects of cognition. Purely symbolic reasoning is simply not efficient enough to get you where you want to go. --Trovatore (talk) 08:10, 4 December 2021 (UTC)[reply]
If you want to describe efficiently the set of the states of a complex mechanical object like e.g. a cartesian coordinate robot, you will model it by a manifold. In particular, a surface, if it has only 2 degrees of freedom. Imagine e.g. a symmetric object B——(A)——O—— (A’)—— B’, that rotates on a plane around it medium point , which is fixed, always keeping equal the angles and At the joints and . Any position of it could be fully described by the two angles at and ; thus, the set of its states make topologically a torus. But if we don’t want to distinguish between and , we may identify symmetric pairs of physical positions, as after all they gives the same picture. If you think a little, the resulting model is a Klein bottle. We use mathematical objects to model many concrete situations of the real world pma08:00, 4 December 2021 (UTC)[reply]
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