Edit: Sorry for “spacial” instead of “spatial”. I always get that spelling wrong.
Let’s call the four spatial dimensions W,X,Y, and Z, where X,Y, and Z are the 3 familiar directions, and W is our fourth orthogonal direction.
Suppose a simple 3 dimensional sphere of radius 1 (size 0 in W) has the positional coordinates W0, X0, Y0, Z0.
If the sphere is moved to any non-zero coordinate along W, it disappears from 3-space instantly, as it has no size in W. By analogy, if we picked up a 2D disk into Z, it would disappear from the plane of 2-space.
Now nudge the sphere over to W1. The sphere no longer intersects 3-space, but retains the coordinates X0, Y0, Z0. Right?
So, while the sphere is still “outside 3-space” at W1, it can be moved to a new location in 3-space, say X5 Y5, or whatever, and then moved back to W0 and “reappeared” at the new location.
Am I thinking about that correctly?
A 3-space object can be moved “away” in the 4th, moved to a new location in 3-space without collisions, and then moved back to zero in the 4th at the new 3-space location?
What does it even mean to move an object in 3-space while it has no intersection or presence with said 3-space?
What would this action “look like” from the perspective of the 3-space object? I can’t form a reasonable mental image from the perspective of a 2-space object being lifted off the plane either, other than there suddenly being “nothing” to see edge-on, a feeling of acceleration, then deceleration, and then everything goes back to normal but at a new location. Maybe there would be a perception of other same-dimensional objects at the new extra-dimensional offset, if any were present, but otherwise, I can’t “see” it.
Edit: I guess the flatlander would see an edge of any 3-space objects around it while it was lifted, if any were present. It wouldn’t necessarily be “nothing”. Still thinking what a 3D object would be able to perceive while displaced into 4-space.
Bonus question: If mass distorts space into the 4th spatial dimension… I have no intuition for that, other than that C is constant and “time dilation” is just a longer or shorter path through 4-space…. eli5
In: 296
Consider the W dimension as time, if you’d prefer a proxy that isn’t the 2d -> 3d mapping. If an object was temporally displaced such that it always existed forward or backwards in time from when you are, it would instantly disappear from the 3d space you currently inhabit, but it’s position could still be the same position in X, Y, Z. You could overlap with it in those dimensions, and as long as you do not meet it in time, it will never affect you. That distance could be the smallest instance of time we could possibly measure, but you would never interact with the object, as it exists in a new 3d-space. You likely wouldn’t even feel the “acceleration” if it was you that moved in this way, as the sensation of motion we experience is limited to the 3 dimensions which we perceive.
Where it gets a bit tricky is the part about seeing the edge of 3d objects in a 2d analog. If the objects around you existed in the 4th dimension through which you were being moved, you would obviously still see them. Objects which may not normally intersect our 3d space may briefly intersect with yours as you move through W. Hopefully our environment would exist in W in the direction in which you get pulled.
> A 3-space object can be moved “away” in the 4th, moved to a new location in 3-space without collisions, and then moved back to zero in the 4th at the new 3-space location?
> Am I thinking about that correctly?
Yes, you do.
> What does it even mean to move an object in 3-space while it has no intersection or presence with said 3-space?
Technically it isn’t in the same 3-space during that move, but a parallel copy. You shift the sphere into a parallel 3-space, move it there as usual, then shift it back to its original (hyper)plane of existence.
> What would this action “look like” from the perspective of the 3-space object?
The rest of 3-space sees it popping out of existence and then back into it at another location. The sphere itself sees kind of the reverse: everything else pops out of existence, and if the parallel 3-spaces it moves through and to are not empty, it sees their content pop into and out of existence. The middle “lateral” movement will look just as usual.
You mention feelings of acceleration, but that is unlikely: we and no other purely 3D things do not have any receptors to sense motion in the 4th direction of space. Our ears for example feel motion by inertia, things lacking behind and needing a little bit of time to reach speed and keep up; but there is zero width in the new direction, whatever small distance anything would lack behind means it just vanishes for all that matters, like part of your ear being replaced by vacuum (sounds painful).
> Bonus question: If mass distorts space into the 4th spatial dimension… I have no intuition for that, other than that C is constant and “time dilation” is just a longer or shorter path through 4-space…. eli5
I am not entirely sure if this speaks about the typical image of masses bending spacetime “into” another dimension, as if lying on a sheet. If so, that’s a simplification only. The actual bending happens within spacetime itself; it takes no extra dimensions, instead it rather figuratively stretches and thins the fabric without any “bumps” into a new direction. That however doesn’t make a good analogy for why it would cause what we call _gravity_.
If anything, it might need more than one extra dimension to even properly draw the bent shape of space(time) into it. Mathematics of what we call _embeddings_ says we need about as many extra dimensions as we already have normal ones, so 3 or 4 more, not just one. You can try to imagine it as one dimension not being enough to accommodate all the potential ways things can be bent: think of a knot, where a line (1D) uses all of 3 dimensions, while it is completely impossible to make any knots in 2D alone.
When you think about these things you have to think about the 3 dimensional space you exist in as a hyper surface, the 3 dimensional sphere is also apart of this hyper surface.
If some being in a 4th spatial dimension came and plucked the sphere from the hyperplane it would seem to just “disappear” from your perspective. From the perspective of the 4 dimensional being, it just moved it to a 3 dimensional hyper-surface adjacent to it along the w axis.
I’m not sure if this answers your question precisely.
If we dial down to 3D objects, for an observer in 2D space (which is one dimension less than our 3D world) we must first select the point of observation. Let’s say (0,0,0). Since the observer is in 2D, they can only “view” slices of the 3D object. Let’s say a sphere exists at (0,0,0) at time T0. This sphere now moves to (1,0,0) at time T2. For the observer who only sees 2D, he observes that there was a circle at time T0 and this circle becomes smaller. This is only when the observer is on a plane without the x coordinate, so necessarily the y-z plane. If the observer was on the x-y or x-z plane, they would observe a circle moving in the positive x direction.
Similarly, when this 4D object is at (w,x,y,z) of (0,0,0,0) and the observer is at (0,0,0) in the x,y,z dimension, if the object moves to (1,0,0,0), the observer will see a shrinking sphere. The velocity of the shrinkage depends on the velocity of the movement. If this observer was in any 3D space with “w” coordinates, they will observe a non-shrinking sphere move one unit in the positive “w” direction.
Now if we transport 3D objects in 4D space, with the same analogy, let’s move 2D objects in 3D space. A disc (in y-z plane) at (0,0,0) moving to (1,0,0). If the disc has NO thickness, it would seem to disappear at 0,0,0 for observer at 0,0,0 and appear to the observer instantly at (0.1,0,0) and then disappear again and so on until it reaches (1,0,0). But to the observers in x-y or x-z, they will see a straight line moving through their space. Depending on their position with respect to the disc they might observe different sizes this projected line.
Similarly, if we transported 3D objects in 4D space, a similar event takes place depending on which observing coordinates you pick.
I hope this helps.
Carl Sagan did a video that was really neat.
The great thing is we actually live technically in a 5D world, and the lowest theoretical dimensions we can get is 3.
This is because of time and gravity.
Those are largely considered dimensions, but temporal ones.
we live in a 5D world with up down, left right, forward backward, and time that is relative to the observer, as well as gravity containing us all together.
It’s weird to think about but basically the spatial dimensions are what we do, the temporal are how we interact
Yes that is correct. While we’re thinking in the fourth dimension another fun fact is that if the 4d being decided to rotate you into the fourth dimension (instead of trying to translate you as with your example), only a 2d slice of you would remain in normal 3-space. Anyone around could see into your stomach or poke you in the brain.
[4D Toys: a box of four-dimensional toys, and how objects bounce and roll in 4D](https://www.youtube.com/watch?v=0t4aKJuKP0Q) this video have all answers to your questions.
It was modeled using math and then computers drawing only 3D parts, so it exactly how 4D object would look and behave in our 3D world.
> So, while the sphere is still “outside 3-space” at W1
This is not true. If it has an X, Y and Z coordinate, it’s not outside of 3d space.
There is no “**the** 3d space” as your phrasing seems to imply. In the 4d space you describe, there is an entire continuum (collection) of 3d spaces. It’d disappear from one of those and appear in another. Think of moving across W as going to parallel dimensions/universes as depicted in popular media.
The best way I have found to think of this as slices. The same way you can make a 2d slice of, say, a 3d apple, you can make a 3d slice of a 4d object.
Similar, if any 2d object with no depth were moved even slightly out of the 2d space it exists it, it’d vanish entirely. If it had depth, an inhabitant of that 2d space would see it cycle through the various cross-sections until it vanished entirely.
A 4-sphere is a collection of many 3-spheres. So if you were to push a 4-sphere across the W direction and observe it from our point of view you would see a small sphere appear out of nowhere, grow in size, then shrink back down and vanish. These are all the 3d slices with increasing, then decreasing radii you see as it moves along the W direction.
As to your bonus question: I’m not sure you need a special intuition. You may conceptualize it as a distortion of space, but you may also think of it as a simple number attached to every point in space. Similar to the strength of the electric field.
Imagine a pond with no wind and no waves. The surface is a 2D surface in a 3D world. Imagine something like paper floating on the surface. That piece of paper would exist in the 2D world of the surface. 2D creatures that can only see the surface would be able to see that paper. If you lifted or submerged the paper, it would disappear from that 2D world and those 2D creatures would not see it. After the paper is lifted it, it could be parallel or it could be tilted. I use the pond analogy, because once it’s lifted it, it is no longer in the same type of universe it was in.
Paper is flexible, so once it’s out of the 2D world, it could bend. Half of it could be lifted out of the 2D world and 2D creatures would only see half of it. The paper could intersect, so that 2D creatures would see a line instead of a circle. I don’t know if a 3D sphere in 4D space would be flexible, but it might be.
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