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When discussing a 2D universe, many assume that an object would only be seen as "a line". This would imply that you are seeing the "edge" of the object. But, if there are only two dimensions, height and width, then there would be no edge, since there is no depth. So there would be nothing to be seen.

In a similar manner, it would seem that objects could not "bump" into each other, since they have no edge.

Could objects be "seen" in a 2D world?

EDIT: By "edge" I mean in the colloquial sense, such as the "edge on" view or flat side of a piece of paper.

Using the conventional idea of a drawing of a piece of paper being representative of 2D, we have to add the caveat that both the paper and the medium in which the drawing is done will have depth. But in a 2D world an object would, of course, have only two dimensions. Assuming we use the terms height and width for these two dimensions, as we use when describing a drawing, that would mean there is no depth. In which case an object could not be seen as a line or line segment. Perhaps it could be sensed in another way, such as magnetically, but this would seem to bring up other questions about laws of physics in a 2D universe.

So an 8x11 piece of paper would be 8" wide (width) by 11" high (height) and .0004 inches thick (depth). So the "edge/side" of the paper is .0004 inches thick in our 3D world but would be .0000 inches thick in the 2D world.

My question is whether we would actually be able to see a 2D object as a line or line segment, which is the usual interpretation, when it has no depth.

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    $\begingroup$ Why do you think that something like a 2D disk has no edge? $\endgroup$
    – G. Smith
    Feb 6 at 17:36
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    $\begingroup$ Perhaps I am using the word "edge" in a different manner than the mathematical concept. A drawing on a piece of paper is often used to represent a 2D concept, with the caveat that the piece of paper is actually 3D, since the paper has a thickness. But in a 2D world there would be no thickness to an object, so there would be nothing for light, sound or any type of radiation to bounce off of, so the object would not be seen or sensed in the way that we typically do so in our 3D world. A 2D object would not be seen as a line because you would not see the "side" of an object with no depth. $\endgroup$
    – Rob
    Feb 6 at 20:08
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    $\begingroup$ You might really enjoy reading Flatland. The gender roles are a bit dated, but the geometry and concepts like "edges" are spot on. $\endgroup$
    – Cort Ammon
    Feb 6 at 22:17
  • $\begingroup$ @Rob: actually, there is thickness in 2D. Your paper becomes just a very narrow rectangle (almost a line), while it was a very thin cuboid (almost a rectangle) in 3D. $\endgroup$
    – oliver
    Feb 6 at 22:50
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Yes a 2d object has an edge. You can draw a square in 2d and it will have 4 edges. A circle in 2d has one edge.

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  • $\begingroup$ In a 2D Universe all shapes would be line segments. $\endgroup$ Feb 6 at 18:10
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    $\begingroup$ @AdrianHoward Filled shapes are not line segments. $\endgroup$
    – G. Smith
    Feb 6 at 18:44
  • $\begingroup$ Yes, I used the term "edge" in a more colloquial sense, as in "edge on" view of a piece of paper. I am actually thinking of "depth". As in a 8x11 piece of paper is 8" high (height) x 11" wide (width) with a thickness (depth) of .004 inches. Without depth, how could you see an object as a line segment? $\endgroup$
    – Rob
    Feb 6 at 20:49
  • $\begingroup$ @G.Smith It would require the third dimension to look upon it to see that it is filled. A 2D being with no access to the 3rd D could only observe line segments. I respect your knowledge please let me know if this is incorrect or miscommunicated. $\endgroup$ Feb 6 at 21:29
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    $\begingroup$ @AdrianHoward If you meant that they appear to be line segments, then I agree. $\endgroup$
    – G. Smith
    Feb 6 at 21:49
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To imagine a 2D world as a 3D person, you can imagine you are floating in a maze with walls that infinitely go up and infinitely go down.

You are also an extended being that goes infinitely up and down, but can move around in the other two dimensions. This situation of symmetry along the $z$ axis in the 3D world is equivalent to the 2D world.

For a physics example, the electric field derived from Gauss's Law in 2D around charged point matches the electric field derived from Gauss's Law in 3D around a charged line, which is a point extended infinitely up and down along an axis.

Can you see things in this world? It's a toy model of a world, but clearly at least in theory you can, just like in a maze you can see the walls.

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You are using anthropomorphisms ("seeing") to describe what answers you expect from a 2D world. However this assumes that it is possible that something like an eye (or a human being, or a brain) can even exist in a 2D world. This is highly questionable, because for example in 3D electric charges have an electric potential that is inversely proportional to the radial distance to the charge. In 2D Gauss' law dictates that the electric potential is basically proportional to the logarithm of the radial distance (see e.g. Electric field and electric potential of a point charge in 2D and 1D), which is a completely different thing. Given that the possibility of life is depending on atoms and molecules, or more generally very complex and delicate balances in various nonlinear interactions, I doubt that you can even begin to prove that life is possible in 2D and hence, that the term "seeing" can be given any meaning in 2D.

But, if you just mean egde in the sense of the mathematical boundary of a 2D solid volume when projected to a 1D plane/line (which would be the mathematical correlate of a photographic plate in 2D, for example), then of course such a boundary exists (see the drawing below). A compact solid would be projected to a compact (finite) line, and the (disconnected) boundary of a line consists of two points.

Actually, in geometrical optics, people often use ray tracing diagrams (like this) in order to visualize the lens images of objects in 2D (because it is easier to draw in 2D and lens systems are often axisymmetric). In these diagrams, objects are drawn as arrows which represent one of the two points the boundary of a line consists of.

Projection ("Seeing") in 2D

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  • $\begingroup$ I wanted to limit the scope of the question so as to not make it too long, but in my mind the second part of this question was whether the physics of a 2D universe would allow"seeing" in the way that we do in 3D, as in physically processing reflected light. I decided to see if anyone would take the next step and discuss the possibility of physics as we know it not being supported in a 2D universe. Your response mirrored some of the thinking I have done. Thank you for your response. $\endgroup$
    – Rob
    Feb 8 at 3:45
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In our 3D plus time Universe we cannot actually see a dimensionless point, a 1D line, or a 2D plane in a 2D Universe. To us these are geometric concepts. We can draw representations of them on a sheet of paper, for instance, but these are only representations to help our conceptualization of them. In a true 2D universe a circle would appear as a line segment of the same length from any angle, and a square would be a line segment with a varying length from different angles because only the edges could be perceived as there would be no third dimension from which to look up or down at them.

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  • $\begingroup$ Yes, but how would you see a line segment that has no depth? The 2D circle or square would have height and width, but no depth. As such, it would seem to have no "side" that could be seen. In this case I am actually referring to "seeing", which is how it is often conceptualized, as opposed to "sensing" in some manner that might be possible in a 2D universe but unfamiliar or unknown to us in our 3D (4D spacetime) universe. $\endgroup$
    – Rob
    Feb 6 at 20:53
  • $\begingroup$ We would not see it at all. That would require the third dimension. $\endgroup$ Feb 6 at 21:21

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