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If a four-dimensional world were to exist, how many eyes would a creature minimally need to see it (in three dimensions)?

Three? Four?

(Bonus question: how should these eyes be spatially configured?)

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closed as off-topic by Aaron Stevens, Bill N, Kyle Kanos, user191954, stafusa Nov 9 '18 at 7:49

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    $\begingroup$ Just think about how many "2D" eyes are needed to see 3D space. $\endgroup$ – Aaron Stevens Nov 7 '18 at 18:20
  • $\begingroup$ If the fourth dimension is time you might need a memory and or some predictive cabability to be aware of it. $\endgroup$ – JMLCarter Nov 7 '18 at 18:48
  • $\begingroup$ More on vision in other dimensions. $\endgroup$ – Qmechanic Nov 8 '18 at 5:14
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    $\begingroup$ I'm voting to close this question as off-topic because it's about the vision capabilities of a hypothetical alien species and not physics. $\endgroup$ – Kyle Kanos Nov 8 '18 at 11:05
  • $\begingroup$ one is enough to see $\endgroup$ – Wolphram jonny Nov 9 '18 at 1:54
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One eye would be enough. (Try closing one eye and you can experience the result for a 3-D world).

But I guess you had in mind what is the value of $N$ such that $N$ eyes give geometric info not available with $N-1$ eyes.

First suppose that in the 4D world, an eye has a 3D volume into which it projects whatever light arrives along a given direction in the 4D space, and the brain has access to the result at each point in this 3D volume. Then 2 such eyes enable the brain to do the triangulation to get distances as well as directions, so 2 is a sufficient number of eyes to get a complete 4D picture.

Now suppose that the eyes are more primitive. Say they project incoming light onto a plane, and the brain scans this plane. (You can compare this with what our eyes would be like if they could determine left from right but not up from down.) Label the 4 directions $w,x,y,z$ and suppose there are some coloured dots to look at, all of different colours or brightness. One eye can determine the $w,x$ direction of a given dot. In principle a single further eye could determine the $y,z$ direction of the dot of the same colour, but such an eye would have to be a long way from the first one. If Instead we suppose the eyes are not far from one another, then I suppose the second eye would be oriented so as to determine the $w,y$ direction, if we take it that $z$ is the direction extending away from the location of the eyes, in the direction they are looking. Thus for any given dot the brain can now figure out $w,x,y$. It can then also get $z$ by using triangulation since it has the $w$ information twice. So once again two eyes are sufficient.

Having said that, I must admit that I wrote the above without drawing any diagrams and I am not absolutely confident whether I have missed something.

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  • $\begingroup$ A very good answer. It can be noted that 2 eyes suffice for any number of dimensions as each defines a line which intersect in a point in a flat space of any dimension. $\endgroup$ – Paul Childs Nov 8 '18 at 5:51
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    $\begingroup$ Triangulation is only one example of depth perception (why 3D movies are still lacking). Focus, shading and motion parallax all contribute. For focus it would depend on the physiology of the hyperdimensional eye, but all the same, the one-eyed is not going to miss out. $\endgroup$ – Paul Childs Nov 8 '18 at 5:55
  • $\begingroup$ An interesting problem is to determine what kind of receptors work for 4D vision. As shown in fma.if.usp.br/~amsilva/Livros/Zwiebach/chapter3.pdf the 4D counterpart to our electromagnetism is tensor-valued rather than vector-valued. While simple eyes might just detect power and hence measure a scalar across the retinal volume, more advanced ones may pick up more components of the field tensors. There is also a faster ($1/r^3$) attenuation of light over distance maybe suggesting a need for more light-gathering capacity. $\endgroup$ – Anders Sandberg Nov 8 '18 at 10:59
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You can already see four dimensions with the two eyes you've got. It's just that two of those dimensions happen to be the same, so you don't distinguish them. But it could be otherwise.

In addition to distances up/down and left/right, you can perceive distances ahead of you by the fact that the light rays reaching your two eyes from a single object form an angle, and from that angle you can infer the distance to the object.

Or you can perceive distances ahead of you from the fact that your lenses need to adjust their focal length in order to bring things into focus, and from the amount of adjustment you can again infer the distance to the object.

The info you get from the angle and the info you get from the focal length is, in ordinary circumstances, redundant, so it adds a total of one dimension to what you can perceive. But in principle, if you were in an artificial environment where you got one piece of information from the angle and an independent piece of information from the focal length, you'd have all the information you need to perceive an additional dimension.

Whether you brain would learn to make use of this information in a way that feels subjectively like seeing a fourth dimension is an open question --- but of course any non-standard configuration of eyes leads to the same open question.

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