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Seeing into a 4D universe?

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What would the world look like to be in a 4D universe?

I’ve thought about this A LOT over the years - and while it's generally thought that we could never possibly imagine what it would be like - I disagree!

Eyes are not three dimensional cameras.

Your eyes are really TWO dimensional - like a camera. The retinas of our eyes are essentially flat surfaces (well, they curve a bit - but they are topologically 2D surfaces).

We mostly perceive the third dimension because we have TWO eyes - so we see the two 2D images and our brains infer distance from the small differences between them.

The way that three dimensions get squished into two is with “perspective”. Light travels into our eyes through a small hole and is focussed to make an image.

So - what happens if we were magically dropped into a 4 dimensional world?

Well, if we still have two-dimensional retinas and only two eyes - then you'd say that the fourth dimension would be “projected” into our 2D world in much the same way as our 3D world is - so it would look just like those 2D pictures you see if you google “tesseract” (a tesseract is a 4-dimensional cube):

 

A wire-frame terreract.

This is a tanslucent tesseract that's NOT changing shape - it's just rotating around some fourth dimensional axis.  It also has perspective in the 4th dimension as well as the 3rd.   This isn't some kind of bad visualization of what we'd see.  This is EXACTLY what we'd see.

Yeah - it’s a bit weird - but it hardly says “Four dimensional”.

But what about the two-eyes thing?

Well, our two eyes are separated by some distance and direction in 3D space - so the two 3D images that get squished into 2D and then our brains use the small differences in the images due to their positions to extrapolate a distance in the 3rd dimension.  But this distance is measured at right angles to the line between our eyes.

In an X,Y,Z world (Z is up!) then when we’re standing upright - our eyes differ in their X and Y coordinates - but they’re at roughly the same Z height - so the 3rd dimension distance is the distance in the direction we happen to be looking.

So if we’re dropped into a 4D (X,Y,Z,W) world - where do our eyes end up? Does the line between our two eyes remain strictly in three dimensions - with W constant? Or are we somehow able to turn our heads to look towards the extra dimension?  I presume not - we don't have the right W-direction muscles to do that...but perhaps some external device could turn us around in that extra direction.

It actually wouldn't make much difference to what we'd see - we'd just get the distance to the object at right angles to the line between our eyes - and it would still seem to us as if we were living in three dimensions...the two dimensions of our retinas - plus the third dimension we'd get from comparing the two images.

In fact, the only possible way for us to get a feel for the two distances that aren't on our retina surfaces would be if we had a third eye!  With a third eye - placed in the middle of our foreheads - we'd be able to turn our heads to place one pairing at right angles to the 3rd dimension axis and the other pairing in the 4th dimension - and what we'd see would be...well, none of us have three eyes - or the brain circuitry to decode them.   So all bets are off.  We can't tell what that would be like.

But with only two eyes - the answer is simple...nothing very special.

But there is a MUCH bigger problem...

Why did everything turn white?!?

We can see images clearly because light only enters our eyes through that little pupil hole and is focussed onto the retina with a 3D lens. But that hole is a three dimensional hole into a three dimensional spherical eyeball.

Instead of thinking how a 3D eyeball would work in a 4D world, let's do the 2D analogy here.  Let's imagine how a 2D eyeball would work in a 3D world!

Draw a circle with a small gap in it on a sheet of paper. That’s a 2D eyeball - it's a circle with a gap in it to let the light in.  In 2D, the light goes through the gap in the circle and is focussed onto the opposite side of the circle where the retina is.

But take that 2D eye out into our 3D world and there is a problem.  Light can shine from somewhere outside the plane of the paper onto the imaginary 2D retina!   So light from almost every direction in the 3D world floods into the eyeball without being focussed by the little 2D lens!

A 2D being would be blinded by unfocussed light coming from the 3D dimension - they’d be unable to focus an image at all.

The same exact thing would happen if our 3D eyes were transported into a 4D world - light would be able to flood into the interior of our eyeballs because a sphere isn't a fully enclosed to the 4th dimension - just as the 2D circle was flooded by light from our 3rd dimension.

We'd be completely blind...we'd probably get some kind of impression of whether it's dark or light - but that would be about it.   Unless the light was really dim, we'd be painfully dazzled with no way to close our eyes or shield them with our hands.  Very quickly, our vision would be destroyed forever.

Does this disprove String Theory?

String theory says that the world must have at least 11 dimensions (as many as 26 dimensions in some versions of it).   But we've just said that if there were more than 3 dimensions then we'd all be blinded.   So doesn't this disprove string theory?

Well, not really.   The string theorists propose that all but three of those dimensions are "small" so like an ant moving around on a donut (Eeuwww!") there is the "long" dimension around the outside of the donut, and the "short" dimension as you go through the hole in the middle and back around the outside.

If our three dimensions were all VASTLY long (much bigger than the visible universe) but the remaining 8 dimensions were less than a picometer across and 'wrapped around' - then there wouldn't be so much scope for light to travel diagonally in those directions and sneakily bypass our eyeballs to dazzle us.

That might still work.

CONCLUSION:

So there are really two possibilities:

If we (with our 3D eyes) were suddenly dropped into a 4D world, then all we’d see would be a uniform sea of the exact same color. We’d be more or less blind.

We’d see nothing whatever.

...OR...

If we were propelled into the 4D world with three proper hyper-spherical 4D eyes and 3D retinas - then our brains have not evolved or learned to see with those eyes. We’d have to have different brains - or at the very least to grow up from birth immersed in a 4D world.

Because you and I are not those people - we can’t possibly imagine what the world would look like to them.

 

Reader Forum: Seeing into a 4D universe?

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From: Steve Baker  Date: 2017-11-01 17:45:35  

Oh...easy. Either there is no 4th spatial dimension...or...it's tightly rolled up as some string theorists claim and too small to measure or manouver within. I'm not calling time "the 4th dimension" here either. If there was a sizeable/unrolled 4th dimension there are any number of laws of physics that would be OBVIOUSLY different than they are.
From: Woelore  Date: 2017-11-01 16:13:40  

"* I don't think I understand your remark about NOT being blinded by 4D light rays shining around the edge of your 3D eyeball. We know that the inside of a 2D circle is fully illuminated by 3D light...you can see it! So the extra dimension analogy should hold."

According to your blinding premise, the fact that you and I currently aren't blinded proves either:

there is no 4th dimension;

or that it has no light;

or that we ARE blinded and the blindness manifests as a functional model of 3D space;

or that the premise is incorrect.

I would suggest there is plenty of radiation spectrum in our 3D world that the sensors in our eyes are insensitive to, and the lens of our eye does not refract in a way to focus onto the retina. This light IS flooding our eyes in an unfocused glare but does not "blind" us.

My response to your blinding premise meant that the 2D eye only sees as visible spectrum those waves/photons moving in its own 2D plane. That eye is not a flat slice of a 3D eye (which has z even if infinitesimal), it is only a two dimensional eye with no z dimension. The 3D world light flooding in from extra-planar dimensions is no more blinding to the 2D eye than the current flood of 4D light is to yours, or the 3D infra-red light you're not sensing, or the 3D x-ray particles you're not sensing. You see it from your 3D perspective only because it is bouncing back to your 3D eye in the third dimension, but it is entirely extra-dimensional to the 2D eye. These waves have no power to excite the 2D sensors, thus are invisible to it.

your turn...

From: Steve Baker  Date: 2017-10-26 12:11:29  

Wow! OK - well let me try to address some comments:

* Parallax due to head motion only works in 4D if you can voluntarily move your head in the 4th direction - otherwise it's the same argument as having two eyes only separated in dimensions 1,2 and 3. The other range-estimation methods are lens focal length (which fails because you have a 3D lens in a 4D world) and comparing known-sizes of things to perceived size...which would only help if there were lots of things in the 4D world that you're very familiar with. But it wouldn't make things look any DIFFERENT.

* It does help to think of lower dimensions as shadows of higher dimensions - but it doesn't allow you to understand what a higher dimension would look like from a lower one...which is the problem we're trying to address here.

* I don't think I understand your remark about NOT being blinded by 4D light rays shining around the edge of your 3D eyeball. We know that the inside of a 2D circle is fully illuminated by 3D light...you can see it! So the extra dimension analogy should hold.

* As for conciousness and self...I have no clue. It's not the question I was trying to answer here.

* Treating time as the 4th dimension is really only a mathematical convenience for some kinds of calculations...it's not saying that space and time are equivalent...they clearly aren't. For one thing, entropy increases in the "time" axis but not in the spatial ones. But even if you do count it - it doesn't change the answer here. If we consider there to be 4 spatial dimensions and one temporal one, nothing much changes.

(Now - if you REALLY want to futz with your brain - think about the string-theory concept of having multiple temporal dimensions too!)

From: Woelore  Date: 2017-10-25 15:11:59  

Is your consciousness, your "self", just a shadow of a higher dimensional consciousness?

Some people posit that time is the fourth dimension. If that were true, perhaps memory is the 2-D shadow of our consciousness, and intuition is our 3-D recognition that our consciousness is the shadow of a 4-D self.

Yea, let's run with that.

From: Woelore  Date: 2017-10-25 14:47:49  

Along those lines, you'll note that the zero-D point DOES exist in the 1-D space of the line, which in turn DOES exist in the 2-D plane, which in turn DOES exist in the 3-D world. It would be reasonable to exist that the 3-D world we inhabit DOES exist in the 4-D space, yet we are not universally blind nor is everything all white. The 3-D eye we see with is attuned only to the 3 dimensional aspects of the visible radiation spectrum.

I think your 2-D eye example would only be attuned to light radiation in its own plane, similar to a polarizing filter, and would not be blinded by radiation crossing its plane of existence from the hyper-dimension of 3-D space.

From: Woelore  Date: 2017-10-25 12:45:02  

One way to think about the various n-dimensional spaces is to view each as a shadow of its n+1 dimensional space. a zero dimensional dot is the shadow that a one dimensional line would cast in its one dimensional space. A line is the one dimensional shadow any two dimensional shape would cast in its 2D space. a 2D silhouette is the shadow a 3D object casts in 3D space. Similarly, perhaps the 3D objects we perceive are only 3D shadows of 4D entities.

You ARE only a shadow of your 4-mer self.

From: Woelore  Date: 2017-10-25 12:28:59  

Stereopsis (stereoscopic vision) isn't our only source of visual 3D data, and beyond a short distance probably isn't even the primary one. Relative motion in the field of vision changes significantly with distance (motion parallax) as can be demonstrated easily with a little experiment. Using only one eye, gaze out over a distance where you have objects at a variety of distances. Now, rock side to side and notice how they cross your filed of view at different rates, proportional to distance.

Within a few arm's length distance, the 6 or so centimeters between the eyes is a significant base for triangulation, but at a few hundred meters it becomes less precise. Motion parallax has been used to estimate distance to stars when taken at different points on the Earth's orbit about the sun.