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When we see object around us in space, we can always interpret those in 2D, by considering them to pass through a plane, its only when we interact with those objects do we realise that it is 3D, is there any significant way of knowing this difference, using mathematics?

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    $\begingroup$ But we can tell the difference between 2 dimensional and 3 dimensional bodies without 'interacting' with them: both eyes see slightly different images, which allow humans to perceive depth. $\endgroup$ – user191954 Dec 31 '18 at 11:17
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    $\begingroup$ what if the human anatomy would not have allowed for us too perceive depth, but given us the power to think is there any logical or mathematical explanation for us to observe this difference? $\endgroup$ – Mundaplackal Dec 31 '18 at 11:20
  • $\begingroup$ What exactly do you want to know? $\endgroup$ – harshit54 Dec 31 '18 at 11:21
  • $\begingroup$ @Mundaplackal Do you mean that you wish to consider a case where a person is using only one eye? There're still an indefinitely large number of ways to observe that hidden dimension without physically touching it: shadows, off the top of my head, are one way. I don't understand how you would 'know this difference using mathematics'. The body either has a non-negligible third dimension, or it doesn't. Even casual observing can tell you the answer to that. $\endgroup$ – user191954 Dec 31 '18 at 11:25
  • $\begingroup$ Even if you have only 1 eye, or you're an animal with eyes on the sides of your head, so you can't perform stereo fusion, you can still obtain parallax depth information by moving your head. $\endgroup$ – PM 2Ring Dec 31 '18 at 11:28
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Using mathematics can mean different things. I will start explaining with a trivial example.

  1. I give you a two dimensional image(or even series of them ) and ask you to reconstruct a 3-D object from it. In this case, there is no inherent information in any two dimensional image that tells you what the object could be in three dimensional. No amount of mathematics with no additional information can take you from image on the right to the cube on the left in the image below.

    No amount of mathematics with no additional information can take you from image on the right to the cube on the left

  2. Luckily, in the real world we usually have some physical information about the object. For example if I just tell you that the two dimensional object(In the right image) is face of a cube. Then you can immediately reconstruct the cube from just one image !

Now this may seem too simplistic. But any reconstruction of a 3-D object from it's 2-D representation works in more or less a similar way. Our brain sees two 2D images and combining this with it's experience of the world it reconstructs a 3D representaion. Infact babies cannot percieve depth till the 5th month or so.

There are numerous ways out there of Depth reconstruction from 2D images, all of which heavily depend on mathematics. But there is no mathematical theory (and in my opinion there cannot be) that can just from a 2D representaion guess the 3D nature of the object.

Below, is a publication of depth reconstruction from a single image.

http://www.cs.cornell.edu/~asaxena/learningdepth/ijcv_monocular3dreconstruction.pdf

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