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Our retinas are two-dimensional surfaces. With two eyes we combine images to perceive 3-dimensions.

As a prelude to the main question below, there is the question of how can you describe mathematically the combining of two images like this ?

So, 2 x 2-dimensional sensing in a coplanar configuration to construct 3-dimensional objects.

  • Question: How does this generalize to higher-dimensions. i.e. how many "eyes" with m-dimensional sensing, and in what spatial configuration, is required to construct n-dimensional objects ?

There is the complication that retinas are not flat - I don't know how that affects things.

Some animals have more than two eyes, and it seems that two eyes only construct partial information about the 3D-ness of an object i.e. only info in one plane. Two more eyes on a vertical line could construct some up-down 3D-ness, and another set of eyes that are recessed compared with the others could allow sets of images from different distances to be combined so that the distance to an object can be perceived more accurately.

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More than "eyes" we need "mind" to perceive higher-dimensions :) –  user17762 May 25 '12 at 22:59
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It is not clear to me that light ought to behave in the same way in higher dimensions that it does in $3$ dimensions... –  Qiaochu Yuan May 25 '12 at 23:01
    
I think this is rather off topic here. You don't construct objects with eyes, but rather you process the light "input" and construct the images of the objects with your brian. I'm thinking it won't make any difference to have eyes in different directions. Horses have almost opposing eyes, which only enables them to have a more panoramic view, but that's all.. –  Pedro Tamaroff May 25 '12 at 23:26
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Perhaps the question should be edited to specify that the owner of the eyes should be able to make fine judgments of the position of viewed objects. Humans have much better depth perception than horses perhaps originally because they evolved from brachiating arboreal primates, and more recently because they have to throw rocks and spears while pursuing prey. Horses have to stand in one place and eat grass. –  MJD May 25 '12 at 23:36

1 Answer 1

up vote 12 down vote accepted

It is an interesting and non-obvious fact that geometric optics only works in an odd number of spatial dimensions. But let us gloss over that fact and assume that light travels as rays along straight lines no matter what.

Abstractly, assuming your eyes function as pinhole cameras, the information you get from a single eye can be thought of as a function from the sphere $S^{n-1}$ to some colour space $C$. In other words, for any given direction specified as a unit vector, you can tell what the colour of the object is that first meets the ray from your eye travelling in that direction. Given this distribution of incident light, you have some sort of internal image processing machinery that infers the outlines and identities of objects in your field of view, and has to deal with ambiguities and what not, but let's not worry about that. The key point is that you have angular information in terms of the direction that things are relative to your eye, but no radial information in terms of how far away they are. So if you identify a particular point source of light somewhere in your view, for example, you know it lies somewhere along a particular ray; there is exactly one degree of freedom left. All you need to determine its depth is to locate it in the other eye's view. And then, since the the separation between your eyes is fixed, you can find the intersection of the two rays and determine exactly the location of the point in the world, relative to your eyes. (This is really what happens subconsciously when you look at something with two eyes.)

All this is independent of the number of spatial dimensions you're living in. If your retina is an $(n-1)$-dimensional surface in $n$-dimensional space, depth is one leftover degree of freedom which you can determine using a second eye. Of course, your internal image processing machinery will have to be more sophisticated, but in principle two eyes are enough.

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Perfectly right, I think. When we reformulate the question in a more abstract geometric way, it just boils down to plane trigonometry. –  Lubin May 26 '12 at 0:20
    
Ah yes, that is the result I vaguely remembered. I wonder if there is a direct connection to the fact that the behavior of $\text{SO}(n)$ depends strongly on the parity of $n$... –  Qiaochu Yuan May 26 '12 at 1:25
    
@Qiaochu: Beats me! To be honest, I don't fully understand the origins of Huygens' principle myself. (Time to go read the answers on the page I linked to...) If you find a relationship with the special orthogonal groups, I'd be mighty curious to hear it. –  Rahul May 26 '12 at 2:29
    
What if one eye is only capable of 1 dimensional vision? That should be sufficient shouldn't it? –  Joel Cornett May 26 '12 at 2:40
    
@Joel: I don't know if I've misunderstood your comment, but here goes anyway. If the second eye can see the point of interest, then yes, it only needs to supply one dimension of information to determine the position of the point. But if the eye only sees a one-dimensional slice of the world, then most of the time it won't be able to see the point at all. –  Rahul May 26 '12 at 2:57

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