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I'm reviewing and expanding my knowledge of dimensions.

We live in three spatial dimensions but, apart from volume, we also have the concept of surface and curve. However, if you write a line on paper, it won't be a true two-dimensional object since you can assign a volume to the ink. Similarly, the paper itself has some non-zero width.

Therefore, it seems to me that we can't have a two-dimensional object in a three-dimensional environment. Maybe the best realization of this idea is to consider the image from a screen: we can see two-dimensional output on computer monitors; each pixel has length and width. However, is there a true one-dimensional object?

What I mean by "true" is something that can be observed in our 3D world, as opposed to a false example, such as a line on a piece of paper that becomes a 3D object when zoomed in.

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    $\begingroup$ There isn't really a need to include anything from ChatGPT here, it is a Language Learning Model. The best it can do is fit words together to sound correct while not actually teaching any real physics. $\endgroup$
    – Triatticus
    Commented Apr 29 at 17:40
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    $\begingroup$ Are you asking if there are 1D objects in our 3D world? $\endgroup$
    – Ghoster
    Commented Apr 29 at 17:46
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    $\begingroup$ Please clarify what your definition of "true" is. And also clarify what scope of things you are considering. Also note that the concept of dimension is not as straightforward as it seems. Dimension need not even be an integer. $\endgroup$
    – Vincent Thacker
    Commented Apr 29 at 21:31
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    $\begingroup$ Not only are there no one-dimensional objects; if you look close and squint a bit, there are no objects. $\endgroup$
    – Peter - Reinstate Monica
    Commented Apr 30 at 15:31
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    $\begingroup$ But are there 3 dimensional objects in the world then? How can you tell if the object is in three dimensions or if your spatial understanding is? Is there a difference? Can you verify one or the other? $\endgroup$
    – Stian
    Commented Apr 30 at 19:47

8 Answers 8

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As far as we know, there are no one-dimensional objects in the real world. A one dimensional object (an object that has length but no width or height) is a mathematical abstraction.

Having said that, there are objects in the real world (like long strings or wires) that are approximately one dimensional, because their length is very much greater than their other two dimensions. And it can be useful to use a one-dimensional model to describe and think about such objects. But do not confuse the model with reality.

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    $\begingroup$ A Kerr black hole has a 1-dimensional singularity. $\endgroup$
    – Vincent Thacker
    Commented Apr 29 at 21:38
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    $\begingroup$ @VincentThacker Our mathematical model of a Kerr black hole may have a 1-dimensional singularity. But we don't actually know what lies behind the event horizon of any black hole. You are confusing the model with reality. $\endgroup$
    – gandalf61
    Commented Apr 30 at 5:50
  • $\begingroup$ Comments have been moved to chat; please do not continue the discussion here. Before posting a comment below this one, please review the purposes of comments. Comments that do not request clarification or suggest improvements usually belong as an answer, on Physics Meta, or in Physics Chat. Comments continuing discussion may be removed. $\endgroup$
    – ACuriousMind ♦
    Commented May 1 at 8:30
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    $\begingroup$ @gandalf61 in your answer you say "there are no one-dimensional objects in the real world," but in your comment you say "we don't actually know [whether a specific hypothesised one-dimensional object exists in the real world]". These two statements contradict each other, and it's the one in the comment that is correct, not the one in the answer. I would note that the strings in string theory are a similar case. $\endgroup$
    – N. Virgo
    Commented May 3 at 2:43
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    $\begingroup$ @gandalf61 I don't think strings or singularities have the same status as pink unicorns. You might feel there are good reasons to think they don't exist, but other qualified people see good reasons to think they do exist - I don't think "we are fairly certain that there are not [one dimensional objects]" is a fair characterisation of our current state of knowledge. I think it's more like we just literally don't know one way or the other. $\endgroup$
    – N. Virgo
    Commented May 3 at 6:51
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While other answers have pointed out that there is no 1D or 2D “physical object” in the real world, you could make the argument that they do exist depending on what you mean by "object".

If you accept that points in space exist then there is no reason you couldn’t specify some set of points which define a plane or line segment. Sets of points like this can correspond to meaningful physical boundaries too. @Vincent Thacker has pointed out that a Kerr black hole has a 1-dimensional singularity; additionally, the black hole event horizon is arguably a 2d surface. The edge of a wavefront traveling through space is a 2d surface, and relatedly an observer's "light cone" is a 2d sphere expanding at the speed of light.

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    $\begingroup$ Yes, which is why the OP's question is a little too vague. $\endgroup$
    – Vincent Thacker
    Commented Apr 29 at 21:41
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    $\begingroup$ Points do not exist in the physical universe: When you get close, it's all fuzzy. $\endgroup$
    – Peter - Reinstate Monica
    Commented Apr 30 at 15:33
  • $\begingroup$ @Peter-ReinstateMonica I do not think we really know enough to say whether or not that is the case. QFT models still have fields distributed over manifolds, and those manifolds have points, but I do not know the limits of the theory here $\endgroup$
    – Jbag1212
    Commented Apr 30 at 15:53
  • $\begingroup$ @Peter-ReinstateMonica that relies on a strong assumption that our fuzzy description (which is incomplete, problematic and can contain inconsistencies) of the universes is correct in the first place $\endgroup$
    – ioveri
    Commented May 1 at 2:13
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Ideal mathematical objects only exist in human imagination, not in the real world. It sometimes happens that a particular mathematical object is a useful (if imperfect) model for something in reality.

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  • $\begingroup$ I'm pretty sure mathematical objects exist in much more place than human imagination. Animals perform abstraction just like we do. They use tools like we do, they count like we do. I think you mean abstraction isn't the reality but even the reality is an abstraction created by your mind because you can't sense it completely, you have to fill the gaps with a model. $\endgroup$
    – xryl669
    Commented Apr 30 at 17:20
  • $\begingroup$ @xryl669 Can animals really count? Can they abstract the concept of 5 from the concept of 5 specific things? In any case their abilities are extremely limited here. $\endgroup$
    – John Doty
    Commented Apr 30 at 18:30
  • $\begingroup$ mathematical objects are a-priori abstractions. Abstractions in the sense that they are a representation, not a thing in itself, and a-priori in the sense that they are invokable independently of a thing and dont even need a thing to be invoked. Following from that because of that independence, the brain invoking it is irrelevant. We dont know if an animal actually can count, but if they can that doesnt reify the number into a reality, platonic or otherwise. Beyond that, well philosophers have been tearing each others hair out over it for many many centuries. $\endgroup$
    – Shayne
    Commented May 8 at 23:45
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    $\begingroup$ @Shayne How can a story exist without a storyteller? $\endgroup$
    – John Doty
    Commented May 9 at 0:05
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    $\begingroup$ @Shane You're claiming no mathematician is necessary to create mathematics. This is nonsense. The story needs a storyteller. $\endgroup$
    – John Doty
    Commented May 10 at 11:31
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In reality there's no 1D objects, nor 2D objects, everything has width, height and depth. Even molecules, atoms and electrons (although many particles in some branches of physics are modeled as zero dimensional objects (dots), where their shapes can be considered irrelevant).

Pixel is like rectangle

No. Here's your zoomed-in pixel of LCD display:

enter image description here

Blue spheres are LCD crystal molecules which when controlled by electrodes current changes configuration and alignment and by that polarizes light in one or the other way.

Green-transparent blocks are electrodes which manages electric current.

Red-transparent blocks are orthogonaly polarizing filters.

So LCD display smallest controllable element "pixel" is like everybody else in this world,- a 3D patch.

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    $\begingroup$ A pixel is not a little square (or, for that matter, a 3-D patch): alvyray.com/Memos/CG/Microsoft/6_pixel.pdf $\endgroup$
    – WillO
    Commented Apr 30 at 4:15
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    $\begingroup$ This memo has little physics, but more about image processing/geometry. I find only interesting part where it speaks about electron gun in CRT monitors (where it is a bit outdated, because CRT is past). Note that electron gun has gaussian profile and so the region of phosphor layer molecules where it hits the screen will have a radius. Also phosphor coating is probably composed from multiple layers of molecules which can be activated in E-gun shot. So activated batch of molecules in a photoluminescence of "1-pixel shot" WILL have some 3D pattern (like cylinder or something related). $\endgroup$
    – Agnius Vasiliauskas
    Commented Apr 30 at 8:13
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It really depends on what you mean by "object". There are mathematical objects of 1 (geometrical) dimension, like curves. Also, there are mathematical objects of 0D, like points, of 2D, like surfaces and also of higher-than-3D, which we cannot visualise, like hypercubes (an example is the 4D tesseract). And all these can realate to reality, e.g. a line representing the distance over which to measure a height, or a surface representing the surface area of a roof, and so on.

But if you by "object" mean a physical, real-life, tangible object or maybe particle, then there really is nothing that is not 3D.

Sometimes, certain categories of objects can be considered as lower-dimensional to all practical purposes in particular contexts. E.g. when working with human-scale sizes, an atom can effectively be considered as point-like, so 0D, when working with astronomical distance scales, a star or planet can effectively be treated as point-like, when rolling out a long wire, it can maybe effectively be considered line-like, so 1D, or when considering the interactions between charges in a capacitor, the capacitor plates themselves might be effectively considered a surface-like to an infinite extend, so 2D, and so on.

But there will of course always be an extend in all three dimension of any real-life object if your zoom-level is chosen fittingly.

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Actually, if you think about it, there isn't anything that's 3 dimensional either.

Any physical object is made of atoms. The frontier of such object is undefined (since atoms & electrons do move and interact with other non object atoms). Atoms are made of quarks and bosons and leptons and none of them have an actual position in space (you can estimate the most probable position by using a quantum wave function)

So you have to define a limit to your mental model.

If your limit is: the set of atoms for this object at a given time, then a 1D object does exist. You can make a single layer of atoms (think graphene) or a single chain of atoms.

In reality even the 3 dimensions are just a convenience representation of an overall infinite distance in 2 dimensions or less.

Take this thought experiment:

  1. I'm using a spiral. It's a 1 dimensional object that's defined by its step p and the angle of rotation w. Every (2 pi + w) rotation, the point on the spiral is one step away from the point that was at w rotation.
  2. Yet you can locate any point on the spiral by using a single dimension: the distance from the origin l. That's the distance an ant would have traveled from the origin to your point.
  3. A spiral lives in a 2 dimensional plane where you normally need 2 coordinates to locate any point.
  4. Now, choose any point A in this 2 dimensional plane and find the closest point on the spiral. Let's call the distance between this point and the spiral d
  5. It's very easy to demonstrate that this distance d will be zero when the step p of the spiral decreases to zero.
  6. There you have a mind blowing issue: the point A that's requiring 2 coordinates in 2 dimensional space can be found with a single coordinate l: the distance traveled on the spiral (that will reach an infinite distance when the step decreases to zero)

Since there is a minimal distance (the Planck's limit), where it's impossible to measure a difference in spatial position, it means that where d is zero, the step p isn't zero so l isn't infinite.

Thus the 2 dimensional coordinates can be expressed in a single dimensional coordinate that's not infinite but very very large.

Similarly, you can express any 3 dimensional position with a single coordinate: the distance traveled on a space filling curve with infinitesimal step => Everything is just 1 dimension anyway.

So the answer is yes, every object is a 1 dimensional object.

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  • $\begingroup$ An interesting approach. It is a pity that the natural metric on your space does not correspond to our intuitive notion of distance. $\endgroup$
    – Peter
    Commented May 1 at 3:43
  • $\begingroup$ @Peter: Yes, it's an awful metric to use for anything else than a demonstration since it isn't a Lie group. But it's convenient to prove that any dimension is just a commodity to avoid infinite single coordinates. Maybe someone can develop a space filling curve that's convenient to use? $\endgroup$
    – xryl669
    Commented May 3 at 13:12
  • $\begingroup$ This is by far the best answer. $\endgroup$
    – Peter Moore
    Commented May 5 at 4:34
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Well, a simple example of a one-dimensional object is a string (consider it infinitely thin).

This string might be deformed and curved as much as you want. Interestingly enough, you can join together the two ends of the string, to create a (deformed) circle. In these cases the string would still be one dimensional, but it would be "embedded" in more dimensions. An example of a two-dimensional object embedded in three-dimensional would be a crumbled piece of paper.

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  • $\begingroup$ but those examples are residing in 3 dimension they will have characteristic of 3d objects? $\endgroup$
    – jmazaredo
    Commented Apr 29 at 16:40
  • $\begingroup$ No. Wouldn't your 2D computer screen also "have characteristics of 3D objects"? Also, what do you mean by "have characteristics of 3D objects"? Also, if this helps, you could imagine a world with a straight string form. This would be one dimensional. $\endgroup$
    – Gabriel Ybarra Marcaida
    Commented Apr 29 at 16:43
  • $\begingroup$ so in a computer the screen is 3d, but inside where the display is shown the pixel is flat no depth length and width only so its 2D . What i mean by 3D is even if you write a line on a piece of paper if you check on microscope there will be like still height. $\endgroup$
    – jmazaredo
    Commented Apr 29 at 16:50
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    $\begingroup$ Yes, precisely, there are no 1D things in our world (nor 2D). You can only have approximate 1D things. Nevertheless, there are many "abstract objects" that are 1D or 2D in our world. For instance, in your example the pen follows a 1D path along space, even if the pen mark is 3-dimensional (as it is not infinitely thin). $\endgroup$
    – Gabriel Ybarra Marcaida
    Commented Apr 29 at 18:06
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‘1D’ is a mathematical abstraction. There are no ‘true’ physical embodiments of mathematical abstractions.

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