How to imagine higher dimensions?

[Saffat Rafsan]

In the links below Carl Sagan and TED-Ed described about higher dimension:




and here's a description of Brian Greene:



Carl Sagan and TED-Ed explained, we can not see the higher dimensions because we are limited to perceive only three dimensions. They didn't say a dimension can be small or big. These explanations completely makes sense.

But Brian Greene explains, higher dimensions can be tiny and curled up.

Isn't every dimension perpendicular to each of the other dimensions? If so, then how can a dimension be tiny or big? I want to know, which is the right way to imagine higher dimensions?

 

[wabbit]

Think of a hair. Does it look one-dimensional or three-dimensional ? Or a piece of cloth - two or three dimensions ? It all depends if you can look closely enough to see the extent of the extra dimensions.

 

[Saffat Rafsan]

Think of a hair. Does it look one-dimensional or three-dimensional ? Or a piece of cloth - two or three dimensions ? It all depends if you can look closely enough to see the extent of the extra dimensions.

please explain. Brian Greene said the same thing i didn't understand.

...and what about the explanation of Carl Sagan and TED-Ed?

 

[rootone]

Mathematical arguments can be made in which other dimensions exist, and in fact must exist to formulate a coherent description of reality.
String theories are the well known example.
However since these dimensions, if they exist, are at a scale which makes no difference to humans, they will have played no part in human evolution.
We just are not wired to perceive those dimensions directly, so probably a mathematical description is as close as we can to comprehending.

 

[wabbit]

please explain. Brian Greene said the same thing i didn't understand.

...and what about the explanation of Carl Sagan and TED-Ed?

Sorry I first replied without wartching those videos...

Just looked at the TedTalk - it is good but I feel one part is misleading : the part about the square suddenly discovering the third dimension as he is lifted out of the plane. His senses aren't able to perceive three dimensions, so he would see weird things for sure, shapes going in and out of nowhere, but these would still look two dimensional for him.

Regarding the hair I'm not sure how to explain it better. But staying with the flatland analogy: imagine these surfaces as really three dimensional but just one atom thick. How would they perceive the world? Would they talk of that third dimension? All their experiences can be explained without it. Their sensory organs are all layed out on a plane. To them the world would still be two dimensional - until one day some flatland physicist makes subatomic experiences and discovers that there is a tiny third dimension that no one can see. And that physicist would probably sound a little crazy to them.

Does that help a little?

By the way there is no tiny dimension as large as an atom in our world. Extra dimensions are strictly speculative, and if they do exist they must be much smaller than that. For physicists here have already probed scales much smaller than an atom and found nothing like a fourth dimension.

 

[wabbit]

please explain. Brian Greene said the same thing i didn't understand.

...and what about the explanation of Carl Sagan and TED-Ed?

I tried to watch that Brian Greene video. I don't find it any more enlightening than you do.

 

[Saffat Rafsan]

Just looked at the TedTalk - it is good but I feel one part is misleading : the part about the square suddenly discovering the third dimension as he is lifted out of the plane. His senses aren't able to perceive three dimensions, so he would see weird things for sure, shapes going in and out of nowhere, but these would still look two dimensional for him.

I completely agree with you in this case. That part is misleading indeed. I can imagine myself in a four dimensional space seeing weird things.

Regarding the hair I'm not sure how to explain it better. But staying with the flatland analogy: imagine these surfaces as really three dimensional but just one atom thick. How would they perceive the world? Would they talk of that third dimension? All their experiences can be explained without it. Their sensory organs are all layed out on a plane. To them the world would still be two dimensional - until one day some flatland physicist makes subatomic experiences and discovers that there is a tiny third dimension that no one can see. And that physicist would probably sound a little crazy to them.

Thank you so much. Now this completely makes sense. I was dying to get this concept. You made it all clear.

Just one more thing. Here I get the concept of tiny dimension. But I'm not getting the concept of curled up dimension. Can you please explain this like the 'one atom thick' universe?

 

[lavinia]

I found this article by Yau a mathematician who seems to have worked with Physicists in developing the mathematics of String Theory.

http://www.ams.org/notices/201108/rtx110801067p.pdf

 

[Matterwave]

Extra dimensions are hard (impossible maybe?) to visualize because we've never perceived them. It would be like asking you to visualize what the Sun looks like in infra-red. You can make a picture that sort of explains it in terms of what you are able to perceive (i.e. you can make a picture of the infrared waves by using different colors to indicate different infrared wavelengths or intensities) but it will only be a picture.

Anyways, with regards to extra dimensions, you can think of it this way. You are in a 3-spatial dimension world because you can move in 3 orthogonal directions: forward and back, up and down, left and right. A 4th spatial dimension would mean being able to move in 4 orthogonal directions. Meaning instead of forward and back, up and down, left and right, you would also be able to move in and out of this 4th dimension. Obviously we can't actually do this in real life, which begs the question "if reality is made up of more than 3 spatial dimensions, why can't I ever move in those other dimensions?" This is where compactification comes in. A dimension is said to be "tiny" if after you've moved a certain very small distance in that extra dimension you end up back where you are. This periodicity of a dimension is not intuitive to us because our 3 dimensions don't appear to be periodic.

Imagine being restricted to a circular loop. If you are a 1-dimensional being and you have no perception of this circular loop in extra dimensions all that you would notice is that if you star moving forward (and in 1-d you can only move forward or back, those are your only 2 options) you will eventually end up where you started. Now make that loop really really small. That is the nature of these extra compactified dimensions.

 

[OmCheeto]

...
Isn't every dimension perpendicular to each of the other dimensions? ...

Only if you are stuck in the cartesian coordinate system.
How would you describe the world, if such a system didn't exist, and everything was measured with angles and distances?
Would there be a limit to the number of dimensions?

Take a look at this simulation: New Project: Interactive Celestial Map with D3
published: Thursday, April 9, 2015

You can click and drag the image and get all sorts of weird, non-right angle shapes.
Once the image is moved, it's almost completely void of right angles!
I would imagine it takes on such weird distortions, as it is attempting to display 3 dimensions, on a 2 dimensional display.

And I would imagine the higher dimensions are screwy like that.
And philosophical scientists, might argue that there's no such thing as a right angle, due to gravity warping space, and hence, there's no such thing as "perpendicular", except in textbooks.

I want to know, which is the right way to imagine higher dimensions?

I have no idea.

 

[Ibix]

A 1d universe is just a line. Everything is lined up and nothing can pass anything else without moving through it.

A 2d universe is a plane - it looks like a sketch on a piece of paper. The interesting thing about this universe is that, although you can line things up in a line, you can also step out of the line and see other entities from an angle.

You could take that piece of paper and bend it so that two of the edges touch, forming a tube. This is still a 2d universe - you can still line things up in a line, then step out of the line and look at them from an angle. This works whatever size the paper is. If you do this with an incredibly long, incredibly narrow piece of paper, you still get a tube. But if the piece of paper is so narrow that the atoms drawn on it are billions of times longer than the paper is wide, entities on the surface could be forgiven for mistaking it for the 1d case I discussed above. Although they can step out of line, they cannot move even one billionth the length of an atom before they are back in line again. They can never see other entities from an angle because there simply isn't room to do so practically - although it remains a possibility, mathematically speaking.

That's the image I'm getting from Matterwave's post, anyway. String Theory is rather beyond my level of expertise. Hopefully @Matterwave will correct me if I'm way off beam.

This kind of thing is harder to imagine in higher dimensions because you need extra dimensions to "embed" this kind of model, and you rapidly run out of extra dimensions.

 

[wabbit]

Just to add one more perspective to the explanations provided above by others about what a curled up dimension means.

First le'ts forget about tiny dimensions - just think of ordinary dimensions, a few meter large here.

Imagine you are sitting in a room, facing a wall. There's another wall behind you of course. Now imagine these two walls are really the same one. If you were to poke a hole in the front wall, looking through it you'd be peeking through the back wall and seeing yourself from the back.

Well, let's enlarge the hole and tear down those walls. You can now walk freely towards the front, as long as you want. But walking say 4 meters you find yourself were you started. You see yourself and the objects surroundind you 4 meters ahead - and also, behind that and a bit smaller, 8 meters ahead. And 12, and 16, and so on ad infinitum. This is what it must feel like to live in a world with a curled up dimension.

Now the next step is harder and to be honest I don't fully picture it, it's too hard - but still I find it suggestive.

So imagine these 4m become one, then 50cm... Your nose is touching the back of your head. Already there's not much of a third dimension to speak of - the rest of the world is to your right and left, up and down, but there's not much of anything back and front.

To continue that process, you must now get compressed, flatter and flatter, until what's left of the third dimension is so tiny it's negligible. Your only contact with the world is now through the contact you feel on your right, left, above and below.

You are now a flatlander, in a flatland with a tiny curled up extra dimension.

This isn't the same thing as being 3D plus a tiny curled up 4th dimension, but perhaps the analogy can still be suggestive.

 

[jerromyjon]

I prefer to stick with 3 spatial dimensions and leave the mathematical dimensionality as its own "loophole" in reality. The way my logic works a fourth spatial dimension would mean a presumably infinite slew of alternate universes, like the many worlds interpretation. Every possible outcome of every interaction occurs in one of these alternate universes and the results we observe determine which universe we inhabit. Just more food for thought...

 

[FactChecker]

Something else to consider:

There can also be dimensions that are so "large" that we have never detected them. The only way humans can detect a dimension is if something in our observed world changes in that dimension. What if there are things that we think never change, but their variation is so small that we have just not detected it. There are a lot of constants that we use in physics. Are they really constants? Suppose we do an experiment with such precision that the only explanation of the results are that the 20'th decimal place of a "constant" must vary. Normally we would attribute that to experimental error, to the uncertainty principle, or to some other disturbance. But it might be a dimension that we should consider. The uncertainty principle might be a consequence of dimensions that we have ignored.

 

[stevendaryl]

please explain. Brian Greene said the same thing i didn't understand.

...and what about the explanation of Carl Sagan and TED-Ed?

The number of dimensions is roughly speaking the number of coordinates you would need to specify a location.

If you look at a drinking straw, it's two-dimensional. To specify a point on the straw, you have to say how far you are from one end, say one inch away from the end. That's one dimension. But that doesn't uniquely specify a location on the straw, because there is a circle of points that are all one inch away from the end. So to uniquely specify a location on the straw, you have to give two numbers: D, \theta. D tells how far from the end, and \theta tells how far around the circle (\theta goes from -180^o to +180^o). So it's two-dimensional, but the dimensions are not the same. One dimension, with coordinate D, measures distance along a straight line, and it keeps increasing. The other dimension, \theta, measures distance around a circle, and it doesn't keep increasing; once you go through 360^o, you're back where you started from. That's what Brian Greene means by a "small" dimension.

If you have a very long, very narrow drinking straw, you might not realize that it's two dimensional. From a distance, it looks one-dimensional. You have to look very closely to see it as a two-dimensional space.

 

[Saffat Rafsan]

Just to add one more perspective to the explanations provided above by others about what a curled up dimension means.

First le'ts forget about tiny dimensions - just think of ordinary dimensions, a few meter large here.

Imagine you are sitting in a room, facing a wall. There's another wall behind you of course. Now imagine these two walls are really the same one. If you were to poke a hole in the front wall, looking through it you'd be peeking through the back wall and seeing yourself from the back.

Well, let's enlarge the hole and tear down those walls. You can now walk freely towards the front, as long as you want. But walking say 4 meters you find yourself were you started. You see yourself and the objects surroundind you 4 meters ahead - and also, behind that and a bit smaller, 8 meters ahead. And 12, and 16, and so on ad infinitum. This is what it must feel like to live in a world with a curled up dimension.

Now the next step is harder and to be honest I don't fully picture it, it's too hard - but still I find it suggestive.

So imagine these 4m become one, then 50cm... Your nose is touching the back of your head. Already there's not much of a third dimension to speak of - the rest of the world is to your right and left, up and down, but there's not much of anything back and front.

To continue that process, you must now get compressed, flatter and flatter, until what's left of the third dimension is so tiny it's negligible. Your only contact with the world is now through the contact you feel on your right, left, above and below.

You are now a flatlander, in a flatland with a tiny curled up extra dimension.

This isn't the same thing as being 3D plus a tiny curled up 4th dimension, but perhaps the analogy can still be suggestive.

Your explanation totally makes sense. Now i can imagine tiny and curled up dimensions. Thanks for the detail explanation.