# Human perception of distance

When we see things around us, distant objects look smaller to our eyes than nearby objects do.

Is there any physics-related reason why our eyes or brain perceive things like this?

Or if this is purely biology issues (<-but nothing is purely biology or something like that..), can anyone briefly show me the processes behind those things?

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Things at a distance look smaller because they subtend a smaller angle. It's simple geometry. Is this what you're asking? – Mike Dunlavey May 12 '12 at 2:09

Yes, it's very much physics related: The perceived smallness of distant objects is a direct function of how many space dimensions we live in.

Here's an example: For a one-dimensional or "string land" creature, what would be the apparent difference in size between a dot nearby and a dot many miles away?

If you think about it a bit, the answer is "none" -- they both look like dots, since light in string land can only travel one way and never changes angles or intensity. It may take longer to arrive, but that's about it.

This is why optical fibers are so great for communications, incidentally, since they are one-dimensional worlds where the light just keeps on doing the exact same thing no matter how far it travels, and looks just the same when it arrives.

It's also why they don't let explosives trucks into long tunnels. An explosion in a tunnel is channeled entirely into just two directions, and only very slowly loses force through friction as it moves. The explosion "looks the same" at the mouth of the tunnel as it does a mile in.

Mathematically, that comes out to a factor of 1 for any length $s$. That is, it doesn't matter how far away the object is, it's size (and impact) will still be multiplied by exactly one. You had better be on good terms with all of your neighbors in string land, no matter how far away they are!

How about two dimensions, which was called Flatland in a famous 1884 book?

To figure that one out, draw two circles, one an inch from a center, and the second one two inches away. If you use standard geometry or just measure it, you will find that the second circle is twice as long as the first one. That means that an object on that circle must look have the size as the same object on the inner circle, because it will take up only half as much room on the doubled circle length.

Mathematically that comes out to $1/s$, where $s$ is the distance to the object on a more distant circle if the first circle is at $s=1$.

Now what about us? We live in a three-dimensional land, or $n=3$ if you use n to give the number of dimensions. The circle trick works exactly as before, so an object twice as far away will again look half as wide. But since we see distant images as two dimensional, like images on movie screen, the total image size for something twice as far away as $s=1$ is also cut in half for its height, as well as its width. So, like a piece of cloth cut in half both vertically and then horizontally, you end up with a total image size of only $1/(2*2)$, where each $2$ is the distance $s$ again.

Put that all together and you get a more general rule. The size of an object, as measured by its total "area" in whatever form you see it (a dot for string land, a line for flatland, and an area or flat image for our world), will always be $1/s^{n-1}$, where $n$ is the number of dimensions of that world, and $s$ is the distance to the object.

All of this has some rather practical consequences in terms of the number of dimensions we actually live in. One dimension is very dicey indeed due to everything colliding at full strength, as well as for other reasons such as just not being rich and diverse enough. Two dimension is more interesting and plausible, but you would still want to be farther away from anything hot, since temperatures from a fireplace would fall off a lot more slowly, at just $1/s$ to be precise. There are still lack-of-richness problems in flatland, also. For example, if you have a full alimentary canal in two dimensions, you end up being divided into two separate pieces!

Our world of three dimensions is, well, sort of a nice balance. Objects can get very complicated in three dimensions, as anyone who has tried to untangle a messed-up chord can testify. That's good if you want the complexity of living things! Also, radiation falls off a lot faster, but not too fast. So the sun gives us plenty of heat, and that heat neither incinerates us (as it would in two dimensions) nor fades away too fast.

And on that last point I'll close with a thought about higher dimensions: You would probably not want to be a 20-dimensional creature. The reason why is probably not what you would expect, however: You would have a very hard time not freezing to death!

The reason is that equation again: $1/s^{n-1}$. For twenty dimensions, that comes out to $1/s^{19}$... and that is a number that gets very, very small in a hurry. A sun in such a world would be a tiny (from your $n=20$ perspective) object that would provide essentially no heat at all, since all the heat would be streaming off into all 20 of those dimensions.

So if you ever find yourself wishing you could be some kind of magical four dimensional creature who could flit in and out of our ordinary 3D world at will, be careful what you wish for, since getting your wish might prove a lot more dangerous than you might think. While 3D is a bit boring at time, it's also a nice comfortable kind of boring: not too hot, and not too cold. In fact, it's just about right!

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"The perceived smallness of distant objects is a direct function of how many space dimensions we live in." And the particular geometry of the world we live in. While in Euclidean space is the angle (which distant objects subtend) is proportional to the distance, that is not so trivial in a 2d sherical space, for example. – Leos Ondra May 12 '12 at 10:11

Well, it's all because we have two eyes

First off, a home-experiment: Close one eye. Make your index fingers point out, the rest should be folded in. Now, hold your hands out, but have them bent 45° at your elbow. Your fingers ought to still be far away. Now, with your eyes closed, try to make them touch. More often than not, you'll miss. THis is not the case when both your eyes are open, since your eyes can judge depth.

There are multiple ways that our eyes can judge depth, but most of them do with the fact that we have two eyes, and they compare the sizes of the image/location of the image projected onto our retina. Since our pupils are small, our brain can backtrace the rays falling on your retina, and two eyes-->two rays, so we can pinpoint the object:

(yes, the ovals are eys and the yellow splotches are where the image is on the retina).

Where the rays meet, your brain mentally perceives the object.

Incidentally, this is how 3D movies work--they pass different images to each of your eyes, tricking your brain into seeing a 3D image.

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I don't think the OP is asking how binocular vision works. – Mike Dunlavey May 12 '12 at 2:08
@Mike whoops. I'll add the angle thingy later.. – Manishearth May 12 '12 at 3:16