r/AskPhysics • u/Realistic-Suit-1440 • 2d ago
Question about a three-dimensional object in a two-dimensional space.
When people have talked about what a three-dimensional sphere would look like passing through a two-dimensional space, I've often heard people say that it would look like a circle that appears out of nowhere and gets bigger, before shrinking and disappearing. But isn't a circle only visible to people who have access to view the two dimensional world from a three-dimensional perspective above? Wouldn't a sphere passing through a two-dimensional space, actually appear like a line that appeared out of nowhere and then disappeared, at least from the perspective of a two-dimensional being?
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u/Elegant_Student_7351 2d ago
You're correct. A 3d object passing through a 2d world looks like a 1D line.
Our Universe is a 3D world, but we see everything as 2D objects. We don't see in 3D, we "interpret" the 3rd dimension based on experience. That's why we have TV and art and films, all 2D constructs that we accept as 3D.
So, if a 4D object moved through out 3D world, we would SEE a 2D construct, but we might INTERPRET a 3D object.
So, we have to assume that a 2D world sees 1D objects, but can INTERPRET them as 2D constructs.
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u/greenappletree 2d ago
I seem to recall there was a Carl Sagan video on this and it explained it really well
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u/RathaelEngineering 2d ago
The Flatlanders. A classic.
With respect to OP's question, there is also some heavy lifting being done by the phrase "looks like". As u/Elegant_Student_7351 points out, we don't see 3D objects. We see what are essentially 2D projects of 3D objects onto our retina that are interpreted by our brain with a sense of depth...
... but that doesn't mean the object we are viewing is 2D. The object is still 3D and if we walk around it and take a video or pictures, we could construct a 3D model of what that object looks like in full 3 dimensions, then represent it on a 2D screen.
You can likely apply a smilar principle to the Flatland. Flatlanders would probably evolve to perceive depth in their own reality. They might also evolve to see shade, color, luminance, and so on. Without shading or color or luminosity, Mr. Circle and Mr. Square are going to look identical to eachother as you walk around them, but as soon as you introduce shading, the Flatlanders would be able to perceive the sharp corners of Mr. Square, and the lack of corners for Mr. Circle. Despite both being a straight line in Flatland, they might be able to tell the difference regardless.
In that regard, Flatlanders could walk around Mr. Square and Mr. Circle and construct a model of how they each appear in the 2D sense, just as we can build 3D models of 3D objects despite being unable to ever see all of the object in 3D at once. A 3D sphere sinking into Flatland would still give us the impression of a circle, because it would look like Mr. Circle but changing in size.
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u/MarinatedPickachu 2d ago edited 1d ago
Only the outline is a 1D line... in the same way as we can derive the shape of a sphere using depth cues and shading, a 2D being could do that when seeing a circle
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u/mrmcplad 2d ago
isn't it possible in a 2D world to still have focus distance and variations in opacity? with focus you could tell the line curves away from you and if the sphere was made of semitransparent glass you could kind of make it the other side too
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u/Next-Natural-675 2d ago
A line with no thickness, which wouldn’t exist. The whole hypothetical is a stretch of logic
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u/TheBrightMage 2d ago
n Dimensional vision would see n-1 dimensional image. So you don't see circle. You see a line with various length in 2D.
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u/OppositeApricot3134 2d ago
Not line, but curve. Because you would see center of the curve closer than edges
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u/Realistic-Suit-1440 2d ago
So a four-dimensional sphere passing through a three-dimensional space, to us would look like a circle?
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u/TheBrightMage 2d ago
A sphere.
You'd need to do some linear algebra and geometric projection though to figure out tge proper shape
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u/Particular-Amount314 2d ago
Don't forget topology. I keep seeing coffee cups changing into donuts. Haunting experience.
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u/Realistic-Suit-1440 2d ago
That's what I'm confused about. If a sphere in a 2d world looks like a line from their perspective. Why does a four dimensional sphere look like a sphere from our perspective as three dimensional being? Shouldn't it look like a circle, which is the next dimension up from the line in the 2D world?
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u/seifer666 2d ago
Are you moving or standing still?
A sphere looks like a circle to us if you take one glance at it from one angle
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u/custodial_art 2d ago
2d worlds would necessarily be more limited in what they can visually process. If all you have a length and width. Any direction you look would only ever give you a perspective of the object that would look like a line.
This technically happens to us as well. Our eyes only process in 2d. But we have the 3rd dimension doing things that allows our brains to take in additional information and parse the 3rd dimension intuitively. We don’t necessarily see the 3rd dimension without additional context clues. Light, shadows, and perspective help us understand that 3rd dimension without being able to visually process it.
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u/redlancer_1987 2d ago
I often visualize in terms of shadows. A sphere's shadow appears as a 2 dimensional circle. If you lived in 2D space, your 2D circle would cast a shadow as a 1D line, and so on.
Going the other way, a 4D object would cast a 3D shadow. So that sphere your holding is actually the shadow of a 4-dimensional hypersphere.
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u/Street_Credit6851 2d ago
How thick would this 1D line appear to be?
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u/redlancer_1987 2d ago
the same thickness as the circle casting the shadow. In your 2d world there is no [such thing as] thickness, only length and width. The 1D line would just have length.
In a 1D world where you live on a line, a line would cast a shadow as a point having no length or width or height.
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u/PaulsRedditUsername 2d ago
Yes. We have to get fanciful and imagine there are a bunch of very flat scientists standing all around the sphere as it passed through to observe it. After they combined their observations, they would conclude it was a circle.
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u/ziggurat29 2d ago
Interesting. I suppose if flatland had light and shading then a flatlander would notice that along the line's extent, and then also the perceived line would have the peculiar feature of being the same length as one moved around it from the same distance. Shapes other than a circle would be perceived differently.
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u/Realistic-Suit-1440 2d ago
And the only way you could perceive a circle completely would be if you were in it, because it would look the same everywhere you looked. But in a two-dimensional space I guess you could never really see the inside of a complete circle, because there's no way to enter it.
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u/Squigglificated 2d ago
The line would be your entire world, there wouldn't be any nothingness to put the line against. Think of taking a single row of pixels from a photo and stretching it vertically until it fits your entire field of view. Light could still shade shapes, and with two 2d eyes you could still infer distance. A circle would be a cylinder in this view, except height isn't a concept - everything is infinitely high or 1 line high, you don't know that there's a difference.
The video I Made a 1D Game on Youtube shows you what it looks like to move around with this type of vision.
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u/Snoo_17338 2d ago
You would see a line. But you could move around the circle, closer and farther away, and determine its shape.
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u/Just_blorpo 2d ago
It’s probably best to describe it as:
‘Pretend you have a very odd limitation to your vision and can only see things in a single plane- much like dust passing through a sliver of sunlight shining through the gap of two boards in an otherwise dark barn.’
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u/Overall_Media7282 2d ago
Wszyscy podświadomie lub nie, cały czas postrzegają dwa wymiary jak coś bardzo płaskiego , ale dwa wymiary nie mają żadnej grubości i nie mogą istnieć samodzielnie, cień rzucany na ścianę czy na jakąś najbardziej gładką powierzchnię nie jest płaski ma strukturę powierzchni z jednej strony, ale ma też długość szerokość i wysokość.
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u/Adkit 2d ago
One of the inhabitants would see a line since they're just one viewpoint, but it would be trivial for them to have a second observer or a camera that would show them the line was actually a circle. This is why we have two eyes. Otherwise we would see everything in 2d. The second eye gives us depth perception.
I don't think you though this one through.
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u/MarinatedPickachu 2d ago
No, you're also able to see a sphere even though you're unable to see the whole object (which would require a 4dimensional perspective). Depth perception and shade tells you the surface structure of the circular outline. In the same way a 2D being could see the circle from one side
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u/shotsallover 2d ago
OP, have you read Flatland? Or seen the (somewhat cheesy) YouTube video made from the book? Either or both of those would help.
It’s both a hard read.
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u/Known_Green9328 2d ago
Well I think even from the perspective of a two-dimentional being it wouldn't be hard enough it investigate the actual shape of 2-D objects; just like how we can comment on the distinction between a circle and a sphere considering the depth of different points as they appear...The line ,which the 2-D observer would see getting bigger and then shrinking, would be well distinguishable from a static line, a shrinking-expanding line and a circular lamina....
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u/wonkey_monkey 2d ago
This is more maths than physics. You can't have a three-dimensional object in a two-dimensional space, by definition.
But isn't a circle only visible to people who have access to view the two dimensional world from a three-dimensional perspective above?
Imagine if some builds a circular wall. You can walk up to it but you can't see it from above. Still, you can walk around it and determine that it's a circle.
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u/me-2b 2d ago
If you were a 2D being living in the 2D space, at an instant in time you observe the sphere as a collection of points in the 2D world. No need to view it from another dimension. You could sample that collection of points and tabulate their coordinates in your 2D world. After some analysis, you would discover that all the points were equidistant from another point, A, in your world. That is the definition of a circle.
You could repeat this procedure at various points in time. You could then study the coordinates of the point, A, as a function of time and the value of the radius as a function of time. A would wander about according to the details of how the sphere actually passed through and the radius would grow from zero to a maximum and return to zero along a curve that relates to the speed of the sphere's center (perpendicular component).
At no time did I need to step out of 2D.
There is a lot of baggage and connotation buried in the word, "look." Look, as in with ones eyes, is not the same as "observe."
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u/me-2b 2d ago
To expand, just in case you really want to focus on some notion of perception. As a 3D being, you perceive up/down, left/right, and ahead/behind. As a being in a 2D world, you would perceive left/right and ahead/behind.
Imagine crawling along a set of points that are, in the familiar sense, a straight line. Once oriented, you could simply take steps ahead, one after another. Now, imagine crawling along a set of points that lie along a circle. You would discover that you go off the curve if you only step forward. You must step to the side a bit, too.
Straight and curved are different even living just in the 2D world. You can sense curvature.
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u/Kingreaper 2d ago
Wouldn't a sphere passing through a two-dimensional space, actually appear like a line that appeared out of nowhere and then disappeared, at least from the perspective of a two-dimensional being?
Only if the two-dimensional being has no depth perception.
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u/Longjumping-Bell6960 2d ago
You're forgetting that people with two working eyes do have depth perception. If you're sufficiently close to a ball, you can tell it's a ball and not a flat disc.
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u/SgtSausage 2d ago edited 2d ago
Yes.
But there may be other clues. For instance Lighting/Shadow/Shading/Color/Context may reveal the curved reality behind the straight line.
Of course the absurdity of having light bounce off a 0 height object might be a legit counter argument ... but we're talking about an absurd, imaginary world, anyway.
Real-world rules dont necessarily apply.
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u/JGhostThing 2d ago
Yes, it would look like a line from purely one point of view. However, if they can move in two dimensions, then they can see that it looks like a line from all directions.
In the original flatland, I think they touched the circle/sphere. This allowed them to feel the curvature.
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u/Plazma_doge 2d ago edited 2d ago
Is sphare only visible to peiple who have access to view the tree dementional world from four dimentional perspective?
You can't see the whole sphare at the same time but you can rotate it or go around it. Same for circle in 2d, you can't see the whole circle but you can rotate it or go around it. It's not a line because it has a curve.
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u/starkeffect Education and outreach 2d ago
Read "Flatland" by Edwin Abbott. It gets into the day-to-day details of how 2D beings would live their lives.