Let's try something-let me know if you have difficulty

[marcus]

Let's try something--let me know if you have difficulty

Let's try something. Everybody imagine an expanding three-sphere.

When cosmologists give you the balloon analogy, that is what they are trying to illustrate. the balloon is just the 2D analog of the 3-sphere.

It is easy to imagine the 3-sphere (math'ians call it S³)
You just have to imagine the experience of living in it and moving around in it--what that would be like. It is hardly any different from the usual experience moving around in usual space---if the three-sphere is a large one---except for one slight detail.

I've noticed that some people probably newcomers at PF don't seem to "get" the balloon analogy. It is an example of a finite, boundaryless expanding space which can serve as a springboard to mentally comprehending the 3D version of the same thing.

A 3-sphere is also a finite boundaryless space---and the balloon surface is just the LOWER DIMENSIONAL ANALOG OF IT---if you can picture it at all then you will be able to imagine expansion: distances between points increasing at a percentage rate.

If everybody takes the (*easy*) first step of imagining S³ before they come into Cosmology forum, then maybe we won't have to repeat the same explanations over and over. If there is some reason why it is DIFFICULT for anybody, I would very much like to know, as it might help us improve explanations.

So let's try this. Anybody who wants to share their way of picturing the three-sphere please do!

I will sketch one or two ways I have of thinking of it and feel free to criticize if you find them not helpful or if you can come up with an approach that works better.

Before continuing with this, I'm going to wait a while to give anyone who wants a chance to respond, and say what is their preferred way of picturing S³, if they want.

 

[neutrino]

I "get" the balloon analogy, but I do find it difficult to "picture" its higher-dimensional analogue. But I am keen to see how this thread proceeds.

 

[marcus]

I "get" the balloon analogy, but I do find it difficult to "picture" its higher-dimensional analogue. But I am keen to see how this thread proceeds.

thanks for the reply, neutrino! I will give you a quick preview of the approach or approaches I have in mind. I know you from way back and I believe you are quick to understand analogies and also you've had some university math IIRC.
So I'll just say what I have in mind without (at this point) making a long story of it.

Maybe (I hope) other people will share their own approaches to thinking of this basic space.

do you know the "one point compactification" idea you get in a beginning topology class?

the one point comp'n of the real line R¹ is the circle S¹----you just add a point at infinity.

Is this familiar to you? (If not, don't be put off. I am skipping ahead because I think you know this stuff. I will start over later and go slow.)

So a 1D creature living in either R¹ or S¹ would not know which of them he was living in, unless he made a long enough trip and came around.

the one point comp'n of the euclidean plane R² is the two-sphere S²---- again you just add a point at infinity.

You've probably seen pictures mapping the plane onto a globe (minus one point, the "north pole"). You set the globe on the plane, and extend lines from the north pole thru some point of the sphere out to the corresponding point on the plane.

And again a 2D being wouldn't be able to tell which he was living in, unless he could do something like measure curvature, or traveled a long ways

so think of yourself as the analogous 3D being who can't tell the difference between the experience of living in R³ or the three-sphere S³.

there isn't any essential difference in what it feels like, except that one has an extra point at infinity. You don't discover which, unless you make long trips or are equipped to measure curvature.

That is not the only way to picture it! There is also thinking of it as the skin on a FOUR-BALL. But I will stop now for the time being. Hope others have some suggestions.

 

[neutrino]

Thanks, marcus. I have never had a course in topology nor have I got down to learning it myself. But I do hope to sit down with Munkres' text one of these days and learn the basics.

 

[marcus]

even an introductory book (like Munkre https://www.amazon.com/dp/B000OIXGYK/?tag=pfamazon01-20 )
could be over-kill. I don't say overkill is bad!
I just wish they'd establish a pre-topology course, or a two-week section of a basic calculus course, that was just about IMAGINING SPACES in several dimensions
without all the formality, abstract definitions, logic. BTW I used John Kelly's book and actually haven't looked at Munkre, but the first course in topology is pretty standard so I have a rough idea.

A person doesn't need to know the definition of a compact topological space in order to understand the one-point compactification of the ordinary 2D plane is a 2D sphere. that is visual.

imagining spaces and the relation between them---mappings from one to another---is fun and visual. I am not sure how much abstract logical proofs is needed. Of course proving theorems can be fun too. But we don't need that for cosmology forum.

All I want in this context is that somebody has some idea what I'm talking when I say S³.

Neutrino, are you all right with the idea that S³ is boundaryless (on its own terms, ignore any higher dim. surrounding space, like a ring is boundaryless)?

 

[DaveC426913]

OK, so in a 3 sphere, you can move freely in any direction (X,Y,Y) and evntuially you will arrive at your starting point, right?



(One of the things I've always thought weird about this arrangement is that of what you would see. What you would see, failing any intervening dust or gas, is an inside-out version of your own head - filling your entire vision. And it doesn't matter if the 3-sphere is light yrears in diameter - the rays converge again. In every direction, the light rays impinge upon you. )

 

[marcus]

OK, so in a 3 sphere, you can move freely in any direction (X,Y,Y) and evntuially you will arrive at your starting point, right?

Right! In realistic situations there are other things to consider, like you can't travel more than speed of light. the thing might be expanding. we don't get light from beyond where the CMB originated because before that the universe was partly ionized gas (the CMB comes from the era when the universe first became transparent to light)

so there are a lot of things to think about. but for starters we can imagine a kind of kindergarten version where it is only as big as a football stadium and you can travel around in it.

(One of the things I've always thought weird about this arrangement is that of what you would see. What you would see, failing any intervening dust or gas, is an inside-out version of your own head - filling your entire vision. And it doesn't matter if the 3-sphere is light yrears in diameter - the rays converge again. In every direction, the light rays impinge upon you. )

this is good! this is exactly the way to go. Put yourself in the space and start thinking what you would see. I have to go do something, be back later.

 

[marcus]

I'm back now. It's probably a good idea to push ahead along two lines of imagining. Imagine the small three-sphere, where light can make the circuit almost instantaneously----there you might be seeing other people in the foreground but in the background you'd be seeing the back of your head.

Or if your head were transparent, you'd be seeing the back of your eye---as a background filling your whole field of vision.

But the more realistic picture is where the thing is expanding and light HASNT HAD TIME to make the circuit. In cosmology distances typically expand faster than the speed of light so if you picture yourself in a more realistic three-sphere you might very well only be seeing a modest, even a small, portion of it. And things might look rather much like they do in ordinary euclidean x,y,z space.

the thing is to be able to imagine both these versions, the small unrealistic version with instantaneous light and the more realistic expanding version with finite speed of light

 

[DaveC426913]

Say, if you were standing on a planet, you'd be looking at the antipodes of the planet. We should be able to calculate by diverging and converging angles how far away you would perceive yourself to be from that planet hovering over your head...

Following up on this, I realize it might not be as easy as I thought. (I thought the angle subtended by the distant planet's disc would make it appear to be D distance away, where D is the diameter of the 3 sphere. But in spherical geomrtry it's not as straightforward as this.)

So I attach a diagram, and ask for the help of someone who knows spherical geometry:

You have a sphere of diameter D, upon which is drawn an icoseles triangle with height D and base Length L. what is angle A?

 

[marcus]

Hi Dave,
I am glad you are following up on this and able to make diagrams. I haven't learned to do that so I rely on other people doing it where necessary. Thanks.

BTW as of the moment, you actually have something YOU said in a quotebox as if I said it

Posted by marcus:
Say, if you were standing on a planet, you'd be looking at the antipodes of the planet. We should be able to calculate by diverging and converging angles how far away you would perceive yourself to be from that planet hovering over your head...

antipodes is the word for it. the great circle rays emanating from one point converge at the other point----the way the lines of longitude do at the northpole and southpole, which are an antipodal pair.*

in a realistic situation the geometry would not be perfect and the curvature would only be average, not uniform------a "bumpy three-sphere". So even without the effect of expansion you would not get this kind of ideal optics.

but it is good to study the ideal (and nonexpanding) case anyway. even tho not realistic I think it is an invaluable tool to train one's imagination.

*if you think about this you will see why they reconverge at the source

 

[BoomBoom]

I can imagine the 2d balloon analogy, but I think it is difficult (if not impossible) to imagine the the same thing with a 3rd dimension. Mainly because there is nothing in our physical world that can describe that as all 3d objects have an outer boundary.

I have also heard you say (Marcus) that space may very well be infinite (which I can completely grasp and do believe), but if this were the case, how could you have an sphere of infinite volume or size? Or is this curvature in reference to all matter of the universe in an S3 sphere within (possibly) infinite space?

Is the evidence for this concept simply Omega is about 1.011 or is there more to it than that? If it were exactly 1, would that disprove this assumption? Or am I misunderstanding what this all means yet again?

The reality is that we can only observe a tiny fraction of our universe, so for people to say as a "matter of fact" that there is no beyond or boundary or anything like that is speculative by nature and akin to a discussion about the existence of God because it is most likely unkowable.

...just a hopeless skeptic at heart I guess.

 

[DaveC426913]

You have a sphere of diameter D, upon which is drawn an icoseles triangle with height D and base Length L. what is angle A?

Actually, while I can't do the math, I can say intuitively that angle A will be wider on this sphere than it will be in flat geometry making the distant object appear to be closer than it is (afterall tha'ts what convergent rays do!). In a universe of diameter D, there will be an Earth hovering over your head at some distance less than D. How much closer than D it is depends on the size of D, and that I can't calculate.


But what I want to is figure out what you'd see if you looked just off to the side of the Earth's disc. You should still see more Earth.

 

[cristo]

It's an interesting topic you bring up here, marcus. I have to admit that I've been following this since you posted it, and am still not closer to actually being able to envisage the 3-sphere. Of course, I can picture S², and picture it being boundaryless and expanding. I guess it also doesn't really matter that I can't picture S³, since I know it has the same topology as S², but just an extra dimension. I can also understand the fact that setting off from one point on S³, one would return to the same place after some amount of time. However, it would be nice to actually imagine it. So, marcus, I guess the question is, can you actually imagine S³ in your head?

 

[marcus]

It's an interesting topic you bring up here, marcus. I have to admit that I've been following this since you posted it, and am still not closer to actually being able to envisage the 3-sphere. Of course, I can picture S², and picture it being boundaryless and expanding. I guess it also doesn't really matter that I can't picture S³, since I know it has the same topology as S², but just an extra dimension. I can also understand the fact that setting off from one point on S³, one would return to the same place after some amount of time. However, it would be nice to actually imagine it. So, marcus, I guess the question is, can you actually imagine S³ in your head?

I do the same as you probably do. I imagine experiencing it---living in it---moving around---experiencing optically or various ways (and also by lower dimensional analog). I cannot see it from the outside.
As an exercise, I pass a plane through it and imagine the sections or "slices" intersecting that plane as they grow and then diminish
As long as you have spent a few minutes trying, you are probably on par with the rest of us--at least with me.

As someone who has taken Gen Rel and is in grad school, you must realize that what I am talking about here is very basic and takes only a few minutes to do all that one can reasonably expect to do.

plus there is the business of visualizing a 'bounce'-----imagine one of these things collapse and then re-inflate
and there is the business of the cosmological event horizon---the experience is qualitatively different if the thing is expanding----light may not EVER get all the way around.

In fact if we made a flash of green light, today, it could only reach stars which are at the present moment 16 billion LY distant. because if a star is 17 billion LY distant today, when we send off the light to it, the light will never be able to reach it in the whole future life of the universe-----I think you know this, perhaps you have written it in a post, but I say it in case somebody else is reading. the fact of expansion (especially accelerated kind) makes a big qualitative difference in the experience of living in a three-sphere (supposing that is the kind of space we live in)

BTW I congratulate you going grad school in math (with cosmology). Now is an exciting time to be doing that.

 

[marcus]

I have also heard you say (Marcus) that space may very well be infinite (which I can completely grasp and do believe), but if this were the case, how could you have an sphere of infinite volume or size?

That is right, it may well be infinite and I do often say this
There are competing models. One possible model is infinite flat, one is three-sphere.

Or is this curvature in reference to all matter of the universe in an S3 sphere within (possibly) infinite space?

No. Please keep the two cases separate. Don't try to merge them into a single story.

When I post it is usually with some conventional model in mind---one that fits the data fairly well. I am talking about this or that model.

Is the evidence for this concept simply Omega is about 1.011 or is there more to it than that? If it were exactly 1, would that disprove this assumption? Or am I misunderstanding what this all means yet again?

I think you are NOT misunderstanding.

If you like to picture the universe as spatially flat and infinite, then I am happy for you to picture it that way. You are in good company. A lot of professional cosmologists assume this.

there is uncertainty in the measurement of Omega. It is not decided yet.

However some people are not happy merely having the flat infinite model, with infinite energy, and big bang having infinite extent etc etc. and they prefer to picture the universe beginning with something finite----so the model for them to picture is this threesphere.

I like both. I will only decide which to prefer when more data is in.
However some things are easier to define and talk about in the finite case.

 

[cristo]

I do the same as you probably do. I imagine experiencing it---living in it---moving around---experiencing optically or various ways (and also by lower dimensional analog). I cannot see it from the outside.
As an exercise, I pass a plane through it and imagine the sections or "slices" intersecting that plane as they grow and then diminish
As long as you have spent a few minutes trying, you are probably on par with the rest of us--at least with me.

As someone who has taken Gen Rel and is in grad school, you must realize that what I am talking about here is very basic and takes only a few minutes to do all that one can reasonably expect to do.

Yes, I realize now that we are pretty much visualising this in the same way. To start with I thought you meant you could picture this 3 sphere from the outside (something which is nigh on impossible!) and so thought I was missing something

 

[BoomBoom]

OK, so I think I get it in the sense that there is no way to picture such a thing from the outside since there is no outside. I understand the concept of traveling in a straight line would eventually bring you back to where you started, but what if you traveled in a line that countered the curvature?

 

[marcus]

OK, so I think I get it in the sense that there is no way to picture such a thing from the outside since there is no outside. I understand the concept of traveling in a straight line would eventually bring you back to where you started, but what if you traveled in a line that countered the curvature?

I think you have figured it out already. If you don't insist on the line being straight---that is a GEODESIC, the shortest distance between any two points along it----then it is just a random wandering wiggly line that could go anywhere. what we are talking about is the geodesics, the STRAIGHT lines

the detail that Cristo did not mention and I didnt mention, but which everybody is aware of I think, is how do you define distances? the "metric" function defines the geometry. if you don't have a distance measure you can say "shortest distance" so you can't define what is geodesic alias straight. So we need to get back to 3-sphere and think about defining distance.

the picture of a big room with a point at infinity you can pass thru and come back from the other direction is OK as far as it goes, but it does not tell you about distances. The distance function that we want for the 3-sphere is JUST AS IF IT WERE THE SKIN OF A FOUR-BALL.

there is something called the "euclidean metric" formula that works in any kind of graphpaper setup whether it is xy, or xyz, or wxyz or whatever.
it is the most obvious thing----squareroot of sum of squares. think about it:
sqrt(x² + y²) or sqrt(x²+y²+z²) or
sqrt(w²+x²+y²+z²)

it works in any dimension and it is a nice straightforward thing to try and in this case it works. you track your way along the skin of the 4-ball little step by little step using the plain vanilla 4D euclidean metric, everything is just as you would expect from how it works in 3D on the skin of the 3-ball!

so that defines what DISTANCES are in the 3-sphere
and now we can throw away that "point at infinity" idea if you want.
the experience of moving around in the 3-sphere is ALMOST like moving around in 3D space except there is this very slight curvature that you almost don't notice. You really only realize it is there when you make very careful measurments of the angles in large triangles or do other high-precision work

like you seldom need to allow for the curvature of the Earth's surface in your ordinary dailies

You don't have to REMEMBER that euclid distance formula (which is as old as Pythagoras anyway) all you need to do is recognize that there is a distance function in our 3-sphere----and so we CAN define geodesics, that is straight lines----and they correspond to what we normal consider to be straight---light goes along, so if you look along the edge it looks straight---and those are the ones where it is the shortest path between any two points on it----and those are the straight lines which if you SEND ONE OFF IN ONE DIRECTION IT COMES BACK TO YOU FROM THE OTHER.

thanks for asking BoomBoom. I think that completes the idea. Anyone with any other question please ask!

 

[chrisina]

Hi Marcus,

I am going to try to explain how I visualize the 3 sphere.
Let me know what you think. It's a purely mental exercise, impossible to draw. Coulours are essential.

I start first with the 2 sphere : example the earth.
We all know what is the equator, ok ? In your mind, paint that circle in red.
Now, move a little bit north, paint that 10º lat. circle in orange. It's going to be a bit smaller.
Continue doing this as you are moving north. Use the colours of the rainbow. ok ? You get the picture ?
Now look from outside (in your head) : what do you see ? We all get the same picture, a couloured hemisphere with the gradual colours of the rainbow. No problemo.

Now let's do the same thing with a 3 sphere.
Instead of the equator, we got a 3-quator. Now that's actually a 2 sphere, (like the surface of the earth) we can picture that. Colour it in red in your mind.
Now move away from the 3-quator, up north (I mean perpendicularly to that 3-quator sphere). You are going to arrive on a smaller sphere, that is somehow contained in the previous one (in the sense of russian dolls I mean). Colour that one in orange.
Continue the mental exploration, with the colours of the rainbow, until the sphere you are painting is almost a point.
Now, think of that thing you just drawn in your mind, try to "feel" what its like inside there. That's of course only half of the 3 sphere.

 

[marcus]

Now let's do the same thing with a 3 sphere.
Instead of the equator, we got a 3-quator. Now that's actually a 2 sphere, (like the surface of the earth) we can picture that. Colour it in red in your mind.
Now move away from the 3-quator, up north (I mean perpendicularly to that 3-quator sphere). You are going to arrive on a smaller sphere, that is somehow contained in the previous one (in the sense of russian dolls I mean). Colour that one in orange.
Continue the mental exploration, with the colours of the rainbow, until the sphere you are painting is almost a point.
Now, think of that thing you just drawn in your mind, try to "feel" what its like inside there. That's of course only half of the 3 sphere.

That's a nice way to to it!

that means that you have deformed half a 3-sphere into a 3D BALL!
(and the surface of the ball is the "equator" of the 3-sphere)

you can continue the same train of thought and deform the OTHER HALF of the 3-sphere into a SECOND 3D ball.

Then you can get a very nice complete model of thw whole 3-sphere by "gluing" the two outer surfaces together-----which I guess you can imagine as a process of identifying points on the surface of one with points on the surface of the other and connecting them with almost invisible threads so that a person who comes out of one ball is instantaneously transported to the corresponding point on the surface of the other.

Or maybe by some mental trick you can picture turning one ball inside out and actually gluing it onto the surface of the other (the center of the ball you turn inside out has to be treated with special care, perhaps temporarily removed, it corresponds to the "point at infinity" in another construction).

But even without imagining physically joining the two balls, you have a pretty good image of a 3-sphere just as the two separate balls with their surface points identified "by fiat" or instantaneous transporter threads.
================

Good.

BTW yesterday I was listening to the audio of Jan Ambjorn's Loops '07 talk and he was describing Monte Carlo universes that they have been running at Utrecht. Spatially they are S³.
The Utrecht group runs literally millions of universes in order to study (quantum) averages.

Because of computer limitations, each universe can only last a finite time. It expands from some minimal stem and evolves according to its internal laws in some kind of shape or history and then collapses back to nearly nothing.

They run these things and pick random ones to study and make averages. All the universes are spatially 3-sphere TOPOLOGICALLY but they can be very lumpy and bumpy. Maybe analogous to the quantum fluctuations imprinted in the CMB map---so spatially not perfectly uniform curvature or perfect in any way. It might have a simple overall topology but realized with realistic puckers warts and wrinkles.

Ambjorn said that at one point they averaged up a huge number of these spacetime histories and found it was an approximately smooth uniform 4-sphere.

 

[Coolphreak]

I think it's somewhat difficult for people to imagine 3 spheres, since it is essentially a 4 dimensional object with a 3 dimensional surface. At some point though...there's really no reason to visualize iti guess. But here's a 3sphere for anyone who is interested.
http://upload.wikimedia.org/wikipedia/commons/d/db/Hypersphere_coord.gif