# Mirror image inversion in the 4th dimension.

I don't really know where this topic belongs.

Let's say you're an ugly asymmetrical person, with your right hand much larger than the left.

A 4th dimensional being removed you and "flipped" you in the 4th dimesion, then put you back.

Would you come back with a large right hand but find your room inverted? Or would your room still look the same but you have a large left hand instead?

I recall something similar in Dionys Burger's novel Sphereland.

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Burger used a dimensional analogy of a 2-D being flipped by a 3-D being.

In 3D space objects are rotated about axes, in 4D presumably they are rotated about planes.

I hope its not too hard to grasp.

Are you talking about rotation, or reflection?

Are you talking about rotation, or reflection?
A reflection I guess.

BruceW
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err. maybe think about what happens to a 2d person, when we reflect him through our 3rd dimension. his right hand becomes his left, and his right eye becomes his left, so from his perspective, he looks the same. Also, his (2d) room flips in a similar way, so from his perspective, the room looks the same as well.

So in conclusion, everything should appear to stay the same?

err. maybe think about what happens to a 2d person, when we reflect him through our 3rd dimension. his right hand becomes his left, and his right eye becomes his left, so from his perspective, he looks the same. Also, his (2d) room flips in a similar way, so from his perspective, the room looks the same as well.

So in conclusion, everything should appear to stay the same?
But I mentioned in the OP that he's asymmetrical.

jbriggs444
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The physical layout of his brain, his eyes, all their interconnections and all of the nerves leading down to the fingers in his hands would have been reflected along with the rest of this ugly fellow.

We would expect that his constituent molecules have all been reflected into their corresponding stereoisomers. All of his dextrose has become levulose and he may be doomed to die of malnutrition.

It seems obvious that he would not notice that he himself had changed. Everything that he does and every observation that he can make about himself will match his [reversed] expectations. There is a right-hand/left-hand asymmetry in the weak interaction, but it takes careful experimentation to detect.

http://en.wikipedia.org/wiki/Parity_(physics [Broken])

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BruceW
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@tade: doesn't matter. His right eye 'flips' and his right hand 'flips', so from his perspective, his big right hand is still on the same side as his right eye, and the right side of his brain and so on.

If you think more about it, reflecting the 2d person through the 3rd dimension doesn't do anything. Imagine you drew a person on a piece of paper. Now you want to reflect that picture through the 3rd dimension. The picture lies all on one plane which is perpendicular to the 3rd dimension, so in other words, his '3rd dimension coordinates' are all exactly the same. Therefore, we have nothing that we can 'flip'. When we do a reflection through the 3rd dimension, it means we replace each 3rd dimension coordinate with its negative. But in the case of the 2d person, his 3rd dimension coordinate is the same at all points on his body. (call it 'a'), so when we do a reflection through the 3rd dimension, it becomes -a for every point. So this is equivalent to just translating the whole piece of paper through the 3rd dimension. i.e. in his world, nothing changes. And in our world, we are only changing where the piece of paper is, but no parts of his body are going to change with respect to each other.

@tade: doesn't matter. His right eye 'flips' and his right hand 'flips', so from his perspective, his big right hand is still on the same side as his right eye, and the right side of his brain and so on.
But would he come back to find his bedroom inverted?

If you think more about it, reflecting the 2d person through the 3rd dimension doesn't do anything. Imagine you drew a person on a piece of paper. Now you want to reflect that picture through the 3rd dimension. The picture lies all on one plane which is perpendicular to the 3rd dimension, so in other words, his '3rd dimension coordinates' are all exactly the same. Therefore, we have nothing that we can 'flip'. When we do a reflection through the 3rd dimension, it means we replace each 3rd dimension coordinate with its negative. But in the case of the 2d person, his 3rd dimension coordinate is the same at all points on his body. (call it 'a'), so when we do a reflection through the 3rd dimension, it becomes -a for every point. So this is equivalent to just translating the whole piece of paper through the 3rd dimension. i.e. in his world, nothing changes. And in our world, we are only changing where the piece of paper is, but no parts of his body are going to change with respect to each other.
I didn't understand this paragraph except the last line, which is the same as your first statement.

BruceW
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well, in our 3d world, a general point is given by 3 coordinates (x,y,z) So if we say we have a 2d piece of paper, which is perpendicular to the 3rd coordinate, then all points on that piece of paper must have the same value for 'z', so let's call this same value 'a'. So then, a general point on the piece of paper is (x,y,a) Which makes sense, because in the 2d 'world', you can only move in 2 directions, since 'a' must remain the same.

Now, reflection in the z direction is given by the operation: (x,y,z) becomes (x,y,-z) And this happens for every point in 3d space. So for our 2d 'world', all the points have the same z coordinate (it is just 'a'), so when our 2d world gets reflected, we get (x,y,a) becomes (x,y,-a) So what has happened here? It is exactly the same as if we just moved the entire piece of paper in the z-direction, without actually changing anything that is on the paper. So we see that the '2d world' is not affected by a reflection in the 3rd dimension. And of course, if we also drew a room for the person on the piece of paper, that would not be changed either.

Now, if we reflected his '2d room', but left him the same (without reflecting him), then he would see a difference. The main point is that if everything gets reflected, then nothing is different.

Also, if we were somehow able to do a reflection of our 3d world, then we would expect that nothing would be different. But actually, the laws of physics in some cases don't obey this 'symmetry' (as jbriggs was saying). But this is very weird. In almost all applications, if everything is reflected, then nothing is changed.

well, in our 3d world, a general point is given by 3 coordinates (x,y,z) So if we say we have a 2d piece of paper, which is perpendicular to the 3rd coordinate, then all points on that piece of paper must have the same value for 'z', so let's call this same value 'a'. So then, a general point on the piece of paper is (x,y,a) Which makes sense, because in the 2d 'world', you can only move in 2 directions, since 'a' must remain the same.

Now, reflection in the z direction is given by the operation: (x,y,z) becomes (x,y,-z) And this happens for every point in 3d space. So for our 2d 'world', all the points have the same z coordinate (it is just 'a'), so when our 2d world gets reflected, we get (x,y,a) becomes (x,y,-a) So what has happened here? It is exactly the same as if we just moved the entire piece of paper in the z-direction, without actually changing anything that is on the paper. So we see that the '2d world' is not affected by a reflection in the 3rd dimension. And of course, if we also drew a room for the person on the piece of paper, that would not be changed either.

Now, if we reflected his '2d room', but left him the same (without reflecting him), then he would see a difference. The main point is that if everything gets reflected, then nothing is different.

Also, if we were somehow able to do a reflection of our 3d world, then we would expect that nothing would be different. But actually, the laws of physics in some cases don't obey this 'symmetry' (as jbriggs was saying). But this is very weird. In almost all applications, if everything is reflected, then nothing is changed.
But if something has changed, will it be him or the world around him?

BruceW
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I'm not sure what you mean. Under the mathematical definition of reflection, if the 2d person and his entire world is reflected in the 3rd dimension, then nothing has changed. (ignoring any effect due to the weak interaction).

Edit: and in fact, if we define his 2d plane to coincide with the origin of the 3d world, then even if we reflect some points through the 3rd dimension but not others, then still nothing changes.
Because in this case, a=0, so any general point in the plane will have a coordinate (x,y,0), so doing a reflection on any point on the plane (through the 3rd dimension) will mean (x,y,0) becomes (x,y,-0) and because zero = - zero, this means the point does not change. so even if we reflected some parts of his body, but not others, then he still wouldn't notice any difference.

jbriggs444
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But if something has changed, will it be him or the world around him?
[Ignoring the weak interaction] What difference does it make? What conceivable test could distinguish between the two descriptions of the situation:

"The world has been reflected and the ugly man is unchanged"
"The ugly man has been reflected and the world is unchanged"

Six of one, half dozen of the other.

[Ignoring the weak interaction] What difference does it make? What conceivable test could distinguish between the two descriptions of the situation:

"The world has been reflected and the ugly man is unchanged"
"The ugly man has been reflected and the world is unchanged"

Six of one, half dozen of the other.
Exactly my point!

Isn't it paradoxical, two different realities which are actually one and the same?

I'm not sure what you mean. Under the mathematical definition of reflection, if the 2d person and his entire world is reflected in the 3rd dimension, then nothing has changed. (ignoring any effect due to the weak interaction).

Edit: and in fact, if we define his 2d plane to coincide with the origin of the 3d world, then even if we reflect some points through the 3rd dimension but not others, then still nothing changes.
Because in this case, a=0, so any general point in the plane will have a coordinate (x,y,0), so doing a reflection on any point on the plane (through the 3rd dimension) will mean (x,y,0) becomes (x,y,-0) and because zero = - zero, this means the point does not change. so even if we reflected some parts of his body, but not others, then he still wouldn't notice any difference.
Weak interaction? There's no "standard" physics involved.

But not every point is zero right?

Weak interaction? There's no "standard" physics involved.

But not every point is zero right?
but you have not described the situation clearly

Exactly my point!

Isn't it paradoxical, two different realities which are actually one and the same?

BruceW
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Weak interaction? There's no "standard" physics involved.

But not every point is zero right?
Right. not every point is zero, but all the points in the 2d world have the same z component, and we are doing the reflection through the z-direction. This is why the 2d man and his 2d world remain unchanged.

I have to mention the Weak interaction, because in reality, the parity-transformed world is not identical to the real world; parity is not conserved. http://physics.nist.gov/GenInt/Parity/parity.html [Broken] Essentially, if we do a parity-violating experiment with some particles, then do the same experiment, but with everything turned round so it is the 'mirror image', then the outcome of the experiment does not happen in the way we would expect if parity was conserved. You could say that this is just a property of the particles, not due to actual parity violation. But since these are fundamental particles, physicists simply say that it is parity itself that is violated.

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Right. not every point is zero, but all the points in the 2d world have the same z component, and we are doing the reflection through the z-direction. This is why the 2d man and his 2d world remain unchanged.

I have to mention the Weak interaction, because in reality, the parity-transformed world is not identical to the real world; parity is not conserved. http://physics.nist.gov/GenInt/Parity/parity.html [Broken] Essentially, if we do a parity-violating experiment with some particles, then do the same experiment, but with everything turned round so it is the 'mirror image', then the outcome of the experiment does not happen in the way we would expect if parity was conserved. You could say that this is just a property of the particles, not due to actual parity violation. But since these are fundamental particles, physicists simply say that it is parity itself that is violated.
Not sure what direction z-axis points in. It'll be good if you can provide a diagram.

I'm discussing Burger's scenario, nothing to do with real world physics.

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BruceW
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well, lets say that we have a piece of paper, which is perpendicular to the z-direction. So a line going in the z-direction would pass through the paper at right angles. So then if we say the piece of paper is the '2d world', and we can draw the 2d person and his 2d house on the piece of paper. Then this means that every point in the 2d world has the same z coordinate.

well, lets say that we have a piece of paper, which is perpendicular to the z-direction. So a line going in the z-direction would pass through the paper at right angles. So then if we say the piece of paper is the '2d world', and we can draw the 2d person and his 2d house on the piece of paper. Then this means that every point in the 2d world has the same z coordinate.
Correct.

Imagine the 2D person is like Pacman, or Pancake man.

We pick him up like a pancake, and flip him over.

But don't forget, as I mentioned in #1, he is ugly and asymmetrical.

BruceW
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I see. So we effectively pick up the piece of paper, and turn it around (as if we were reading a book, except he is still visible even from the other side of the page). This isn't a reflection in the 3rd dimension. That's probably where some of the confusion came from.

In this case, if the entire page is flipped, then I would say nothing changes. But if some of the page was flipped, but other parts not flipped, then something does change. For example, if we cut out his head, and flip the rest of his body, and place his head back on, then all of his body except his head will have flipped, so from his perspective, his right hand would have changed places with his left hand. Of course, it might be difficult to do such a feat of surgery on a real person :)

And about the house thing, if we cut out the person, and flipped him, but didn't flip the house, then from his perspective his house will have been flipped around. The key thing to keep in mind is what is being flipped and what is not. If everything is flipped, he won't be able to tell the difference.

So let's move up by one dimension of Euclidean space.

Now you are analogous to Pancake Man. A 4-D creature picks you up and flips you. But not your room.

From your POV, your room has been flipped. It now looks like a mirror image of it's original self.

But from the perspective of your family, you are the one who has been flipped. Weird right?