View Full Version : Saddle-shaped?

spaceprobe

2009-Jun-26, 02:29 PM

Hi all; hope this is the right place to ask a question.

I hear a lot about the "shape" of the Universe and really can't find a good explanation of what is meant.

I assume that if it were possible to look at it from the outside, the Universe would be a sphere radiating out and expanding from the big bang centre?

So, when astronomers say the Universe is saddle-shaped, what do they mean? I'm a graphic designer so need the answer described so as I can visualise it - if that's is even possible. How can a 3D space have a 2D shape?

Many thanks,

K.

hhEb09'1

2009-Jun-26, 02:41 PM

Welcome to BAUT, spaceprobe

I hear a lot about the "shape" of the Universe and really can't find a good explanation of what is meant.

I assume that if it were possible to look at it from the outside, the Universe would be a sphere radiating out and expanding from the big bang centre?

So, when astronomers say the Universe is saddle-shaped, what do they mean? I'm a graphic designer so need the answer described so as I can visualise it - if that's is even possible. How can a 3D space have a 2D shape?

It's an analogy. When they talk about the sphere, in the example you mention, the space is 2D--only the surface of the sphere, it doesn't contain the center. The examples are 2D analogies, since it is difficult, as you say, to visualize even 3D analogies.

EnigmaPower

2009-Jun-26, 03:25 PM

I assume that if it were possible to look at it from the outside, the Universe would be a sphere radiating out and expanding from the big bang centre?

There is no center to the universe. As far as the shape, when any structure is formed as long as it is large enough, gravity ensures it is spherical.

Ampatent

2009-Jun-26, 03:29 PM

There is no center to the universe. As far as the shape, when any structure is formed as long as it is large enough, gravity ensures it is spherical.

If it is spherical wouldn't it have a center?

EnigmaPower

2009-Jun-26, 03:34 PM

If it is spherical wouldn't it have a center?

Depends how you look at it. If you look at a balloon that is roughly spherical it has a center when you look at it from outside but in the case of the universe we are the surface of the balloon without the inside. What would the center be then?

Ampatent

2009-Jun-26, 03:38 PM

Depends how you look at it. If you look at a balloon that is roughly spherical it has a center when you look at it from outside but in the case of the universe we are the surface of the balloon without the inside. What would the center be then?

Point taken...

trinitree88

2009-Jun-26, 03:56 PM

spaceprobe. You will find your saddle shape here, with regards to the ratio of the energy density to the critical density...omega. It could have any value. Up until the distant type 1a Sne survey, most results favored a value near 1 (..oddly...:shifty:), since then there's a lot of arguing going on..(that's physics, marriage, politics, and umpiring...:shifty::lol:). There's a nice Wiki graphic, with an explanation...see:http://en.wikipedia.org/wiki/Shape_of_the_Universe

reminds me of Ali....sigh, who rides horses...miss her:doh::lol:

spaceprobe

2009-Jun-26, 03:57 PM

Thanks for the replies.

This is exactly what confuses me: What do you mean by...

we are the surface of the balloon without the inside.

Could you explain why "we are the surface"?

And how can the Universe be anything other than (roughly) spherical, if it expanded from a central point in all directions (if, as I said, you could look at it from the outside)?

Sorry if all this seems childish - I can normally grasp quite complex ideas and concepts, but this has me struggling.

EnigmaPower

2009-Jun-26, 04:07 PM

Thanks for the replies.

This is exactly what confuses me: What do you mean by...

Could you explain why "we are the surface"?

And how can the Universe be anything other than (roughly) spherical, if it expanded from a central point in all directions (if, as I said, you could look at it from the outside)?

Sorry if all this seems childish - I can normally grasp quite complex ideas and concepts, but this has me struggling.

Lets say we had a bunch of ants crawling on the surface of the balloon. To them there is no inside, just a surface. To us looking from outside there is an inside. The ants are constrained to their dimensions on the balloon's surface yet we as outside observers who do not have the same constraints, see the inside. This is one reason that the two common analogies of the universe (balloon and loaf of raisin bread) lead to nothing more than this type of confusion.

EnigmaPower

2009-Jun-26, 04:08 PM

And how can the Universe be anything other than (roughly) spherical, if it expanded from a central point in all directions (if, as I said, you could look at it from the outside)?I am pretty sure I addressed that in post #3.

spaceprobe

2009-Jun-26, 04:38 PM

Hmm, starting to get it now. Still not sure about the saddle thing. Ploughing through the wikki recommended earlier.

K.

Sam5

2009-Jun-26, 04:38 PM

Thanks for the replies.

Could you explain why "we are the surface"?

It’s nonsense. We are not the “surface” and we are not on the “surface”. We are somewhere inside a 3-D universe, looking “out” in all directions. However, the universe is too large for us to see any outer “edge”. So, we don’t know how large the universe is or where we are within it.

Visually, it appears that we are in the “center” of a spherical universe, but many people don’t want us to be in the center, for philosophical reasons. If all is random chance, then we should be at some random place in the universe and certainly not in its center.

But of course we are not on the “surface” of the universe, and our galaxy is not on the “surface” of a giant expanding balloon-universe. We are somewhere inside a very large 3-D universe which appears, visually, from our viewing position, to be spherical and expanding.

Here’s Eddington’s 1933 version of the balloon nonsense, which has been repeated many times since then:

http://i31.tinypic.com/2e194cn.jpg

EnigmaPower

2009-Jun-26, 04:48 PM

Visually, it appears that we are in the “center” of a spherical universe, but many people don’t want us to be in the center, for philosophical reasons. If all is random chance, then we should be at some random place in the universe and certainly not in its center.]

It does? Oddly enough that would mean we really are not 2/3s of the way from the center of the Milky Way and are really in the center and residents of Sagittarius A :)

Sam5

2009-Jun-26, 05:00 PM

It does? Oddly enough that would mean we really are not 2/3s of the way from the center of the Milky Way.....

No it wouldn’t. We now know the extent of our galaxy and our position within it.

We do NOT yet know the extent of the entire universe or our position within it.

trinitree88

2009-Jun-26, 06:10 PM

Actually we're pretty special. It appears that the CMB's axis of polarization lines up nicely with the symmetry of the plane of the solar system, with a chance alignment being about 1/10,000....nice. see:http://www.newscientist.com/article/mg19425994.000

hhEb09'1

2009-Jun-26, 07:20 PM

Actually we're pretty special.Well, Sol is. :)

It appears that the CMB's axis of polarization lines up nicely with the symmetry of the plane of the solar system, with a chance alignment being about 1/10,000....nice. see:http://www.newscientist.com/article/mg19425994.000

BlueCoyote

2009-Jun-26, 11:39 PM

I get all of these things about saddles , donuts, circles etc and I understand if you have a 2d projection of a surface of a sphere you end up with a map projection somewhat like your standard world maps where exiting top leads to entrance bottom.

What people really struggle with, myself included, is I think the idea that, in the universe, if you travel towards the pole star there is a suggestion that you might eventually pass the pole travel on and on and on and on for an extremely large distance and eventually hit the south pole of the very planet you had left.

Normally we are used to feeling that straight lines are straight but the suggestion is that this is not the case. It would suggest to my limited understanding that a dimension itself is bent round upon itself.

In our every day lives the x y and z axis are never bent and time continues in a straight line.

I guess thinking about it Einstein proved that time can vary according to your point of reference so I guess something similar will need to exist for the three standard dimensions for all of this less than intuitive geometry to exist.

But I also take Sam5's point about the surface of a balloon although I haven't had time to read through the linked article. There is no shape on the earth or object on the earth within which or on which straight travel in any of the 3 differning dimensions within it can be continued infinitely albeit repeatedly revisiting a single location.

Therefore to my thinking if this exists as a characteristic of our universe, at the universal scale, then saddle donut or any other shape would simply have to be a 3d projection of a universe that must in someway have more than 3 dimensions. Thus the saddle is a 3D projection of something with more than 3 dimensions just as a map is a 2d projection of a 3 dimensional object, in my asteroids example. (That's my current thinking based on what I've learnt from those that know more)

I also recognise that those that know more haven't completly agreed between themselves yet.

StupendousMan

2009-Jun-27, 12:00 AM

Hmm, starting to get it now. Still not sure about the saddle thing. Ploughing through the wikki recommended earlier.

K.

People have trouble visualizing four-dimensional spaces, as you have discovered. Therefore, many scientists have made analogies to three-dimensional spaces, with which we have much more experience. Instead of speaking of a three-dimensional universe curving through a four-dimensional space, we simplify to a two-dimensional universe curving through a three-dimensional space.

So, instead of humans, imagine a civilization of teeny little ants. They live in the middle of a large region which appears, on casual inspection, to be flat. Therefore, the ants imagine that their world is just a large, flat, rectangular table extending out in all directions. We humans, looking at this world from far above it, would say, "The ants live on the surface of a flat table."

Now, ant-scientists make more precise measurements over large areas, and discover some peculiar things; for example, when they measure the properties of very large triangles, they find that the sum of the angles inside triangles is not 180 degrees, but more than 180 degrees. How is this possible? One ant-scientist postulates that the surface on which the ants live is "curved". We humans, again looking at the world of the ants from far above, understand at once: the ants are actually living on the surface of a very large sphere. If you examine only a tiny portion of the sphere, it appears NEARLY flat ... but if you make measurements over a large area, it is obviously curved in three dimensions.

The sphere is an example of a flat surface which is curved in three-dimensional space. Geometers describe the curvature of a sphere to be positive.

However, it is also possible for a surface to be curved in another way -- such that the angles of a triangle add up to LESS than 180 degrees. If you look at the surface of a saddle, you'll see that it curves one way, upward, from front to back, but a different way, downward, from the left side to the right. Geometers describe this sort of curvature as negative.

The saddle is just a somewhat common object in the everyday world which has negative curvature. There's nothing special about it -- one could use other objects, like donuts, if one wished.

People can usually follow this sort of argument when it is presented in this simplified manner, because we can look at ordinary objects in the real world and imagine tiny beings living on their surfaces. We tend to have trouble when we try to imagine the equivalent three-dimensional surface curving through a four-dimensional world.

Amber Robot

2009-Jun-27, 12:06 AM

Therefore to my thinking if this exists as a characteristic of our universe, at the universal scale, then saddle donut or any other shape would simply have to be a 3d projection of a universe that must in someway have more than 3 dimensions.

The thing is, mathematically, you don't need those extra dimensions to have a non-Euclidean 3d space.

mugaliens

2009-Jun-27, 01:53 AM

If it is spherical wouldn't it have a center?

The balloon analogy is a 2D representation whereby 3D space is represented by (flattened onto) a 2D surface. There is nothing on the inside or the outside of the balloon analogy - the entirety of the universe is analogously represented on it's 2D surface.

The entire point of the analogy is to help explain expansion. It is not to be any sort of a replacement for, or representation of a 3D universe.

The thing is, mathematically, you don't need those extra dimensions to have a non-Euclidean 3d space.

Exactly. The three spatial dimensions, along with the component of time, fully describe our macroscopic universe. The other dimenions are extremely small.

Jeff Root

2009-Jun-27, 03:52 AM

Amber Robot said that additional dimensions aren't needed in order for

space to be able to curve. That's sort of correct. No additional dimension

of space is needed. Space and time are somewhat interchangeable,

though, so what happens is, in effect, that the three dimensions of space

curve in the time dimension. Gravity curves space that way. Time passes

a very tiny bit more slowly on Earth than it does on the Moon, because

Earth's gravity curves the space near it into a little bit of additional time.

-- Jeff, in Minneapolis

Jeff Root

2009-Jun-27, 04:06 AM

I think of the balloon analogy like this: The spherical surface of the balloon

represents a plane slice through the Universe. Since three points define a

plane, let's say that it is the plane which contains the center of the Milky

Way galaxy, the center of the Sombrero galaxy, and the center of M81 in

Ursa Major, just to pick three semi-random points. What appears to us to

be a flat Euclidean plane would actually be a small portion of a spherical

surface on the scale of the entire Universe.

As far as I know, this analogy only applies to the Universe if it has positive

overall curvature. There is no evidence that it has such curvature.

And the analogy can be misleading in many ways. My "slice" version of

the analogy has the particular misleading feature that you can imagine

other slices parallel to the given slice as being like concentric balloons.

It doesn't work like that. Every slice through the Universe is about the

same as every other slice. You can only look at one slice at a time.

It is a limitation of our ability to visualise spacetime curvature and a

limitation of the analogy. All analogies are flawed, but they can still be

helpful.

-- Jeff, in Minneapolis

mugaliens

2009-Jun-27, 04:41 AM

A few weeks ago, I recommend using the sponge analogy, as a sponge (the universe) can expand rapidly, while the microbes (galaxies) crawling through the sponge travel much more slowly.

BlueCoyote

2009-Jun-27, 06:05 AM

The thing is, mathematically, you don't need those extra dimensions to have a non-Euclidean 3d space.

OK I think I get it - so there's not necessarily more than 4 dimensions but what is being bent , in some fashion, is both time and space as predicted through special relativity. In some way this allows for the strangeness of the dimension aspect. So the saddles and donuts are 3D physical analogies to our present 4 dimensions the important point being that this physical analogy includes time.

Right I'll try and remember it but not sure that I really understand it...

Time to go off and study non-Euclidean mathematics!

These last points are good examples of why Maths is important in astronomy.

spaceprobe

2009-Jun-28, 12:20 AM

I think I get it now. I'll have to mull it all over for a few days. If I get time I'll make a diagram and you all can tell me if I'm wrong!

Cheers,

K.

robross

2009-Jun-30, 11:31 PM

I'll just add a few comments to the already many useful ones above.

We humans appear to be "trapped", both physically, and perceptually, in 3D space. We understand up-down, left-right, forward-backward intuitively. It appears our brains have evolved this way of thinking about reality, so it is core to the way we visually process everything we can see.

Mathematically, it's easy to construct a model of less or more than 3 dimensions, and have all the math be self-consistent and solve problems; but our human brains don't understand what these other spaces would "look like." I imagine it is the same kind of problem someone who is blind from birth would have with the concept of "color", as described by a sighted person.

So the best that we can do is use analogies and examples of things we *can* visualize, and try to apply them to things we cannot, in order to get a "taste" of what that might be like.

We deal with surfaces of objects all the time, so they're easy to describe and visualize. We can talk about the surface of a sheet of paper as a plane of 2 dimensions. We know the paper actually does have a 3-dimensional extent; it might be very thin but the thickness is non-zero. Yet, we can imagine a zero-thickness piece of paper and think about what that would be like, even if we cannot perfectly visualize it. And we can think about how that 2D plane would behave in our 3D world. This is how we try to comprehend how our 3D world behaves in higher dimensional space.

If you have not read "Flatland" by Edwin Abbott Abbott I would recommend it. (It's a very quick read.) The author had many purposes in mind when writing it, but the main vehicle of the story is a 2-dimensional world populated by 2-dimensional beings, and what that would "look like" to we 3-dimensional beings, and how the 2-dimensional beings would perceive us.

Rob

DrRocket

2009-Jun-30, 11:53 PM

Hi all; hope this is the right place to ask a question.

I hear a lot about the "shape" of the Universe and really can't find a good explanation of what is meant.

I assume that if it were possible to look at it from the outside, the Universe would be a sphere radiating out and expanding from the big bang centre?

So, when astronomers say the Universe is saddle-shaped, what do they mean? I'm a graphic designer so need the answer described so as I can visualise it - if that's is even possible. How can a 3D space have a 2D shape?

Many thanks,

K.

There are a lot of assumptions approximations and analogies that result in the situation about which your question pertains.

General relativity models space-time as a 4-dimensinal Lorentzian manifold. The curvature of that manifold is determined by the distribution of mass and energy. A priori the curvature tensor is unknown, and is determined by solving the Einstein field equations that result from mass-energy distribution (mass-energy includes potential energy in the form of stress and pressure as well as things you are more used to thinking about). Because of the curvature, one is forced to use the language of manifolds and of Riemannian geometry. This in turn results in an inability to talk sensibly about the distinction between space and time in a global way in general.

Cosmologists make the assumption that the universe is homogeneous and isotropic (which it seems to be on the largest scales although there is some debate on the implications of recent observational data). Under the assumption of homogeneity and isotropy, space-time decomposes into what mathematicians call a one-parameter "foliation" of space-like 3-dimensional hypersurfaces. It is these hypersurfaces that play the role of "space" when cosmologists talk about space. The parameter, plays the role of time, but it is not really time.

It turns out that the space-like hypersurfaces are of constant curvature, and therefore there is a classification of such hypersurfaces that comes from differential geometry. They are classified by their sectional curvature, positive, negative or zero.

The idea of a "saddle" (a hyperbolid paraboloid in the language of elementary calculus) is used to provide an example of a 2-dimensional surface that has negative curvature. The usual example for positive curvature is a sphere and for zero curvature the plane. However, there are other more exotic examples, and it is possible, for instance, to have a torus with zero curvature, or a cylinder with zero curvature.

BlueCoyote

2009-Jul-01, 09:35 PM

Professor Richard Pogge of Ohio State University goes through the various points as described by DR R in lecture 36 "the Big Bang" of his 2006 Course Astronomy 162 which is free through Itunes by the way.

He spends about 4 lectures talking to Einsteins Universe which I think you might find interesting....

Really though this stuff is really hard to understand.

.....listen mull over

.....listen mull over again.

.....consider learning the mathematics then realise I ain't got time.

.....accept.

.....that's where I'm at

Sam5

2009-Jul-01, 11:41 PM

If you have not read "Flatland" by Edwin Abbott Abbott I would recommend it. (It's a very quick read.) The author had many purposes in mind when writing it, but the main vehicle of the story is a 2-dimensional world populated by 2-dimensional beings, and what that would "look like" to we 3-dimensional beings, and how the 2-dimensional beings would perceive us.

Rob

I think Edwin Abbott wrote “Flatland” in 1884 as a humorous fairy tale booklet, much like Alice in Wonderland, but designed for educated adults. It was a joke, a satire about Arthur Cayley’s address to the British Association in 1883 and about Cayley’s remarks about flat little beings in a 2-dimensional world. These are the flat being referred to in Chapter XXXI of Einstein’s book “Relativity, the Special and General Theory”:

http://www.bartleby.com/173/31.html

Cayley’s remarks at the 1883 meeting were re-published in The Living Age in October 1883:

First page of article, lower right side, page 177:

http://digital.library.cornell.edu/cgi/t/text/pageviewer-idx?c=livn;cc=livn;xc=1;idno=livn0159-3;g=moagrp;q1=Arthur;q2=Cayley;node=livn0159-3:1;size=s;frm=frameset;seq=187

See his remarks about the flat beings are on page 183, left column, about halfway down the page:

http://digital.library.cornell.edu/cgi/t/text/pageviewer-idx?c=livn;cc=livn;xc=1;idno=livn0159-3;g=moagrp;q1=Arthur;q2=Cayley;node=livn0159-3:1;size=s;frm=frameset;seq=193

See:

”I have just come to speak of four-dimensional space. What meaning do we attach to it? or can we attach to it any meaning? It may be at once admitted that we cannot conceive of a fourth dimension of space; that space as we conceive of it, and the physical space of our experience, are alike three-dimensional; but we can, I think, conceive of space as being two or even one-dimensional; we can imagine rational beings living in a one-dimensional space (a line) or in a two-dimensional space (a surface), and conceiving of space accordingly, and to whom, therefore, a two-dimensional space, or (as the case may be) a three-dimensional space, would be as inconceivable as a four-dimensional space is to us.”

Also see Richard Proctor’s response to the article in his own article titled “Dream Space”, published in The Living Age in January of 1884 (see the beginning at the bottom left side of page 228):

http://digital.library.cornell.edu/cgi/t/text/pageviewer-idx?c=livn;cc=livn;xc=1;idno=livn0160-4;g=moagrp;q1=Richard;q2=Proctor;q3=%C3%83%C2%A2%C 3%82%C2%80%C3%82%C2%9CDream-Space%C3%83%C2%A2%C3%82%C2%80%C3%82%C2%9D;node=liv n0160-4:1;size=s;frm=frameset;seq=238

Sam5

2009-Jul-02, 01:10 AM

The concept of flat two-dimensional beings goes back at least as far as Helmholtz in 1870. Einstein mentions Helmholtz as being one of the sources of the concept in Chap. XXXI of his own book. There are several internet sources for the Helmholtz “two-dimensional beings” lecture of 1870:

“ON THE ORIGIN AND SIGNIFICANCE OF GEOMETEICAL AXIOMS”

(English translation of a) Lecture delivered in the Docenten Verein in Heidelberg, in the year 1870.

Go here and do a search for the word “beings”:

http://www.archive.org/stream/lecturespopularo00helmrich/lecturespopularo00helmrich_djvu.txt

Also go here and see page 650 or use the search term “beings”:

http://books.google.com/books?id=UQqLHyd8K0IC&pg=PA647&lpg=PA647&dq=helmholtz+%22Origin+and+Significance+of+Geometr ical+Axioms%22&source=bl&ots=smYlxQCSb4&sig=qgyUcDUzv2r9ixVcsEh5ZpHb4is&hl=en&ei=7gFMStOEKofkNdDL3f8P&sa=X&oi=book_result&ct=result&resnum=1

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