Thread: "How many eyes?"

Forgive me for asking a question that is only indirectly related to=20
higher dimensional cubing. I'm sure many of you can give me an=20
authoritative answer.

The question: how many eyes would a 4D creature, living in a 4D=20
world, need to see the world around (including, of course, his=20
Rubik's hypercube) in full 4D stereo vision?

My layman's answer, which I can't fully justify, is that two eyes=20
would suffice.

Ignoring any contextual clues to distance, in a 2D world, a single=20
eye reveals a closed 1D visual field (like the interior of a circle).=20
Adding a second eye allows the seer to see the plane in 2D. In a 3D=20
world, a single eye reveals a closed 2D visual field (like the=20
interior of a sphere). Adding a second eye brings depth to the=20
sphere. SO, the natural extension would be to say that in a 4D world=20
a single eye would reveal a closed 3D visual field (like the interior=20
of a hypersphere) and that a second eye would bring 4D depth to it.=20
Continuing the logic, two eyes would be enough for stereo vision in=20
any number of dimensions.

Is that right, or would more eyes be needed?

Guy

Guy,

Anything even remotely related to 4D cubing is fair game on this list so
don't worry about that. Your question is very interesting and appropriate.

It should be possible for a creature in N dimensions to fully perceive
the space with a single eye. The trick is that it would need to be able
to move that eye around. A brain can fuse multiple points of view into
a single mental perception of depth even if those points of view are
captured at different times. You can try this by closing one eye and
noticing how flat the world looks, and then moving your head around and
noticing how your ability to determine depth comes back. Once you stop
moving, it all goes flat again. Cats use this effect by moving their
heads around before making a big leap so that they can judge the
distance better than they can with just their normal eye separation.
You'll also notice the effect as a passenger in a moving car when you'll
find it much easier to judge long distances when looking out the side of
the car as opposed to out the front or back. At long distances, your
normal eye separation is useless but the motion parallax gives you
stereopsis.

Is this cheating or not really answering your question? Well yes and no.
"No" because stereopsis is the "perception" of depth (I.E. a purely
internal, mental phenomenon), but "yes" if you are asking a purely
geometric question. As to the geometric question, I've long thought that
the answer is N-1 but now I'm not so sure. In 3D for example, imagine
looking at a circular disk edge-on. If that edge is vertically aligned,
you'll be able to fully perceive its full 3D form but if its
horizontally aligned you will not. I don't think that suggests that you
really need 3 eyes to fully perceive a 3D space because, depending upon
how a scene is arranged, each additional eye can give you more depth
information. In order for the geometric question to make sense I think
we'll need to be a lot more specific and therefore more removed from
questions of actual physical worlds. For example, one reasonable
restriction might be to assume that the scenes to be viewed only contain
point objects. With that restriction it does seem that 2 eyes would
suffice in N dimensions. Except what about when trying to judge the
distance to a point that is perfectly in line with your 2 eyes? In that
case both eyes would see the point in the same position and you would
not be able to determine its distance. If you're allowed to turn your
head to face the point, then fine but that's a lot like the case of a
single eye that you're allowed to move around in time. Without the
ability to turn your head, the answer appears to be that you need N eyes
to perceive an N dimensional point space.

The geometric reduction above seems very much removed from your original
question of how many eyes an actual 4D creature in a real physical world
would need. Depending upon how you want to reduce it to a geometrical
question, it appears that you can defend any number from one to infinity.

-melinda

guy_padfield wrote:
> Forgive me for asking a question that is only indirectly related to
> higher dimensional cubing. I'm sure many of you can give me an
> authoritative answer.
>
> The question: how many eyes would a 4D creature, living in a 4D
> world, need to see the world around (including, of course, his
> Rubik's hypercube) in full 4D stereo vision?
>
> My layman's answer, which I can't fully justify, is that two eyes
> would suffice.
>
> Ignoring any contextual clues to distance, in a 2D world, a single
> eye reveals a closed 1D visual field (like the interior of a circle).
> Adding a second eye allows the seer to see the plane in 2D. In a 3D
> world, a single eye reveals a closed 2D visual field (like the
> interior of a sphere). Adding a second eye brings depth to the
> sphere. SO, the natural extension would be to say that in a 4D world
> a single eye would reveal a closed 3D visual field (like the interior
> of a hypersphere) and that a second eye would bring 4D depth to it.
> Continuing the logic, two eyes would be enough for stereo vision in
> any number of dimensions.
>
> Is that right, or would more eyes be needed?
>
> Guy

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Thank you very much for your answer, Melinda.

=20

The motion parallax point is interesting but yes, it is cheating as far as
answering my real question is concerned!! As you guessed, it was more the
geometry than the psychology of perception that was challenging me!

=20

I think you have confirmed (tell me if I=92ve misunderstood you) that two
stationary eyes would suffice to see at least a part of N-dimensional space
in full N-D stereo, just as in 3D reality our two eyes allow us to see dept=
h
in a part of our visual field. Many predators remain stock still while they
observe their prey. Herons, for example, stand motionless, bill poised over
the water, waiting for a fish to move into spearing range (they have to dea=
l
with refractive depth effects too when they strike, of course!). If I
understand you aright, a 4D heron would indeed only need two eyes for there
to be a part of its visual field in full 4D, enabling it to strike with
accuracy within that range.

=20

Most animals with good binocular vision are hunters =96 the hunted tend to
have widely separated eyes pointing in different directions, like rabbits.
Hunters only need really accurate depth perception in a limited field =96
namely, in the direction of the prey, for the final attack. So maybe
predatory animals in N dimensions would be able to get away with just two
eyes.

=20

Thanks again,

=20

Guy

=20

_____=20=20

De : 4D_Cubing@yahoogroups.com [mailto:4D_Cubing@yahoogroups.com] De la par=
t
de Melinda Green
Envoy=E9 : samedi, 13. septembre 2008 22:23
=C0 : 4D_Cubing@yahoogroups.com
Objet : Re: [MC4D] How many eyes?

=20

Guy,

Anything even remotely related to 4D cubing is fair game on this list so=20
don't worry about that. Your question is very interesting and appropriate.

It should be possible for a creature in N dimensions to fully perceive=20
the space with a single eye. The trick is that it would need to be able=20
to move that eye around. A brain can fuse multiple points of view into=20
a single mental perception of depth even if those points of view are=20
captured at different times. You can try this by closing one eye and=20
noticing how flat the world looks, and then moving your head around and=20
noticing how your ability to determine depth comes back. Once you stop=20
moving, it all goes flat again. Cats use this effect by moving their=20
heads around before making a big leap so that they can judge the=20
distance better than they can with just their normal eye separation.=20
You'll also notice the effect as a passenger in a moving car when you'll=20
find it much easier to judge long distances when looking out the side of=20
the car as opposed to out the front or back. At long distances, your=20
normal eye separation is useless but the motion parallax gives you=20
stereopsis.

Is this cheating or not really answering your question? Well yes and no.=20
"No" because stereopsis is the "perception" of depth (I.E. a purely=20
internal, mental phenomenon), but "yes" if you are asking a purely=20
geometric question. As to the geometric question, I've long thought that=20
the answer is N-1 but now I'm not so sure. In 3D for example, imagine=20
looking at a circular disk edge-on. If that edge is vertically aligned,=20
you'll be able to fully perceive its full 3D form but if its=20
horizontally aligned you will not. I don't think that suggests that you=20
really need 3 eyes to fully perceive a 3D space because, depending upon=20
how a scene is arranged, each additional eye can give you more depth=20
information. In order for the geometric question to make sense I think=20
we'll need to be a lot more specific and therefore more removed from=20
questions of actual physical worlds. For example, one reasonable=20
restriction might be to assume that the scenes to be viewed only contain=20
point objects. With that restriction it does seem that 2 eyes would=20
suffice in N dimensions. Except what about when trying to judge the=20
distance to a point that is perfectly in line with your 2 eyes? In that=20
case both eyes would see the point in the same position and you would=20
not be able to determine its distance. If you're allowed to turn your=20
head to face the point, then fine but that's a lot like the case of a=20
single eye that you're allowed to move around in time. Without the=20
ability to turn your head, the answer appears to be that you need N eyes=20
to perceive an N dimensional point space.

The geometric reduction above seems very much removed from your original=20
question of how many eyes an actual 4D creature in a real physical world=20
would need. Depending upon how you want to reduce it to a geometrical=20
question, it appears that you can defend any number from one to infinity.

-melinda

guy_padfield wrote:
> Forgive me for asking a question that is only indirectly related to=20
> higher dimensional cubing. I'm sure many of you can give me an=20
> authoritative answer.
>
> The question: how many eyes would a 4D creature, living in a 4D=20
> world, need to see the world around (including, of course, his=20
> Rubik's hypercube) in full 4D stereo vision?
>
> My layman's answer, which I can't fully justify, is that two eyes=20
> would suffice.
>
> Ignoring any contextual clues to distance, in a 2D world, a single=20
> eye reveals a closed 1D visual field (like the interior of a circle).=20
> Adding a second eye allows the seer to see the plane in 2D. In a 3D=20
> world, a single eye reveals a closed 2D visual field (like the=20
> interior of a sphere). Adding a second eye brings depth to the=20
> sphere. SO, the natural extension would be to say that in a 4D world=20
> a single eye would reveal a closed 3D visual field (like the interior=20
> of a hypersphere) and that a second eye would bring 4D depth to it.=20
> Continuing the logic, two eyes would be enough for stereo vision in=20
> any number of dimensions.
>
> Is that right, or would more eyes be needed?
>
> Guy

=20


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