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I hope this question has some sort of meaningfulness. Suppose you are on the phone with an alien which is on his planet. For some reason he know which are our UP and DOWN and our FRONT and BACK. It's not difficult to explain him where is the UP or where is the BACK in a "physical way".

Today I found out that there is a way to explain the alien the LEFT and RIGHT too, by means of some decaying process, but now I'm interested in a more mathematical way to do that.

I suppose that it can't be done because there is an "unbreakable symmetry" between the two direction.

How to prove the impossibility of solving the problem? How could the problem be modeled in a more abstract one?

I make this reasoning: i tell to the alien to think of the floor as a plane and to draw a line on it which have the direction of his nose; then, maintaining his own orientation draw another line perpendicular to the first and chose an arbitrary direction for it. Let's call this direction A and the opposite would be B. Now, if a can make "operations" on A and B that converge to my RIGHT (but there has not to be reference to my left/right in these operations) I would have done. But this can't be done, I think.

For operations I mean, for instance, to pick a vector in the plane, rotating it, etc..

user3621272
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  • Does he know which is the north pole of a magnet? Because if he does, you can tell him to align a magnetic field vertically, in some specified direction, and have an electrical conductor sending current forwards or backwards. Then the conductor will be moved either left or right depending on the orientations you've chosen. – Arthur May 07 '15 at 16:42
  • You can explain up and down using gravity, I guess, but suppose the alien is rotationally symmetric (e.g. a sphere) and sees in all directions at once. How do you go about explaining front and back? – Qiaochu Yuan May 07 '15 at 16:46

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Whether this is possible depends on the shape of the universe, in the following sense.

Suppose the alien has guessed some answer. How could you confirm with the alien that it was correct? Well, the alien could travel to where you are, and you could compare. But it turns out that depending on the shape of the universe, whether you and the alien agree can depend on what path it takes to get to you.

More precisely, the issue is that the universe might fail to be orientable. This is easiest to visualize on a Möbius strip. If you and an alien are on opposite sides of a Möbius strip, then whatever you both think left, right, up, and down mean, whether your answers agree or not depends on whether the alien approaches you from one direction or the other. (Draw a picture! Or watch this lovely video by Vi Hart.)

Qiaochu Yuan
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  • The OP said the alien is on earth. Are you suggesting there are orientation-switching paths on the face of the earth? – A. Donda May 07 '15 at 16:48
  • @A. Donda: the OP said the alien is on another planet. In order to travel to our planet the alien has to travel through space, and it might be the case that space is non-orientable. – Qiaochu Yuan May 07 '15 at 16:49
  • Sorry, misread "his planet" for "this planet". Objection withdrawn. :-) – A. Donda May 07 '15 at 16:52
  • I need to explain to the alien which is my right on the phone. I'd like he put his right hand up for example. I don't need him to come here on Earth. I began to read a book about flatland, by Jeffry Weeks. Maybe my question is more complicated than I tought and I can't deny the physical concept of the problem? – user3621272 May 07 '15 at 16:53
  • @user3621272: suppose the alien has no hands, or thirty hands arranged in a circle. Then what do you do? In any case, whatever you end up saying to the alien on the phone, you need some way of checking whether you did it correctly, and I'm saying that whether this is meaningfully possible depends on the shape of the universe. If the universe is non-orientable, then there is no such thing as a globally meaningful notion of left vs. right (given notions of up vs. down and front vs. back). – Qiaochu Yuan May 07 '15 at 16:55
  • @user3621272: let's even cut out the alien. Suppose you met someone who did not speak your language, and you wanted to communicate to them which of their hands was their right hand or their left hand. How would you do it? I would go up to them and grab their right or their left hand, and shake it. But that requires that I go to where they are, and again, in a non-orientable universe it can happen that depending on how I do this I get different answers. In other words, it can happen that I take a trip around the universe and come back as a mirror image of myself, with left and right switched. – Qiaochu Yuan May 07 '15 at 16:59
  • Ok, I see. But I have formulated the problem in a different way than my friends proposed to me. Suppose he is on a room with a trasparent wall (the mirror in the room of police interrogations for example): you can see him, but he can't see you. He has two hands, and I think it's not too important how the space is... (o maybe is) let's say, euclidean space. You can communicate with him: explain him where is the right. But without cheating saying "an hand up". Ok, that's the left. I want math here. – user3621272 May 07 '15 at 17:01
  • @user3621272: that's a different problem; an interrogation room is orientable. – Qiaochu Yuan May 07 '15 at 17:02
  • Maybe I'm not being able to explain what type of argument I'm looking for. – user3621272 May 07 '15 at 17:03
  • @user3621272: this is not a math problem; it is a physics problem, and its answer depends on the physical features of the universe. For example, it might be the case that 1) the universe is orientable, and 2) the laws of physics fix an orientation of it (e.g. I think that some descriptions of magnetism might do this, but I'm not sure). Then you could solve the problem using whichever laws of physics fix the orientation (e.g. tell the alien to do some experiment involving magnets). – Qiaochu Yuan May 07 '15 at 17:07