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Thread: Ep. 78: What is the Shape of the Universe?

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    Ep. 78: What is the Shape of the Universe?

    Some of the biggest questions in the universe depend on its shape. Is it curved? Is it flat? Is it open? Those may not make that much sense to you, but in fact itís very important for astronomers. So which is it? How do we know? How did we figure it out? Why does it matter?

    More...

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    I posted this question in the Q&A section, but perhaps this is the more appropriate place.

    If, due to the curvature of space as some theorize, any straight line in the universe will end up where it started. Or as Dr. Pamela put it, 'you shoot a laser beam into the universe it will eventually hit you in the back of the head.'

    And since the big bang everything is essentially moving in a straight line (more or less) outward.

    Will not all the stuff of the universe eventually re-converge on the same spot on the 'other side' of the universe?

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    Right before you got to the toroidal shape I was thinking of a rubber gasket (doughnut shape), and then Pamela said it.

    But then I had an idea about the possibilities.

    Of course I am not a mathemetician or a physics major, or even smart.

    But Please see if this thought can be entertained.

    If the toroid was like rubber and being stretched in every direction, then we would certainly see every point moving away faster and faster.... BUT....

    Here's my thought....

    Like a rubber gasket...the tighter you pull it some points begin to move closer together (squished), and some points move faster and farther away as it goes past the halfway point of its capabilities.

    Could OUR universe APPEAR to slow eventually, and then completely snap, breaking open the toroid?

    The implication of this is important because like that gasket... if it snaps, it is not only open, but very quickly will return to its original thickness.

    This would mean that even if the universe SNAPS apart... from any point in it, you would see everything rushing back AT you!

    (The universe would go from closed to open, but come back together in a balance)

    Maybe we are only right near the halfway point of the expansion.
    Last edited by EvilEye; 2008-Mar-04 at 02:16 AM. Reason: correction in context

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    You guys start to sound like a horror movie

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    OK, I can grasp the idea of space as a fabric. The analogy of a mass being like a weight on a trampoline, taken to 3D. I can get that.

    I can't get this geometry of the universe thing. I mean, I can imagine the universe being a ball or toroid.... but flat? Flat, beyond 2D, implied a thickness, which implies a 3D object. (Likewise saddle-shaped.)
    So in all these cases, there's an outer edge. Meaning, if one travel long enough you'll reach the edge. I don't buy that, and since I'm not an educated astrophysicist, the only answer is I have it wrong.

    OK, parallel lines forever. I get that concept on the surface of a sphere. But, and here's the crux of why I don't get this geometry topic--we're not on the surface of an object! We're, in a manner of speaking, INSIDE the object. So, if I were to imagine myself inside a sphere, and shoot two parallel lines, yeah, they'll remain parallel...until the reach the edge. Likewise if I were some raisin in the middle of a bagel, the lines would reach an edge.
    What then?

    Does this make any sense? (Sometimes naive questions from ignorant people can be just as baffling as educated answers from smart people.)

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    You almost have it, but then it fails when you jump up a dimention. The surface is every direction.

    By flat, they don't mean like a sheet of paper. They mean flat in that parallel lines never cross.

    You are at the center of the universe right now, and so is everything else. It is all there ever was, just spreading apart, (but not from a central point), but still the same thing with more room to do stuff.

    There is no outside or inside. The surface in 3D is all there is (plus time).
    Last edited by EvilEye; 2008-Mar-11 at 07:58 PM. Reason: more clarity

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    Quote Originally Posted by Mechphisto View Post
    OK, I can grasp the idea of space as a fabric. The analogy of a mass being like a weight on a trampoline, taken to 3D. I can get that.

    I can't get this geometry of the universe thing. I mean, I can imagine the universe being a ball or toroid.... but flat? Flat, beyond 2D, implied a thickness, which implies a 3D object. (Likewise saddle-shaped.)
    So in all these cases, there's an outer edge. Meaning, if one travel long enough you'll reach the edge. I don't buy that, and since I'm not an educated astrophysicist, the only answer is I have it wrong.

    OK, parallel lines forever. I get that concept on the surface of a sphere. But, and here's the crux of why I don't get this geometry topic--we're not on the surface of an object! We're, in a manner of speaking, INSIDE the object. So, if I were to imagine myself inside a sphere, and shoot two parallel lines, yeah, they'll remain parallel...until the reach the edge. Likewise if I were some raisin in the middle of a bagel, the lines would reach an edge.
    What then?

    Does this make any sense? (Sometimes naive questions from ignorant people can be just as baffling as educated answers from smart people.)
    You could try thinking of it like this:

    Imagine 3d space. (Forget time for now and don't ask me how to visualise 4d spacetime - I haven't got a clue! )

    So you have your 3d space - you could be thinking of a cube, right, then divide it up in to a three dimensional grid so it looks like a stack of building blocks, or if you rather, like a 3d wire grid.

    Then just subtract one dimension. (I do it by taking away the building blocks and imagining the 'impression' they would have left on a sheet of paper, or rubber.)

    You end up with the rubber sheet 'grid with the bowling ball' familiar from the illustrations we see in books and on the 'net. And it isn't invalid because what happens in three dimensions also happens in two - they just take one away to make it easier to 'draw'.

    And then... the universe is 'flat' in the sense that the rubber sheet is flat, and the missing third dimension is also flat. (And of course the alternative geometries of the universe have the rubber sheet in the shape of saddles, spheres, donuts, soccer balls, etc... but in each visualisation the third dimension is 'missing'.)

    BTW I've seen 3d illustrations of the bowling ball scenario as well: if you imagine your 3d grid, when you add the bowling ball and everything curves due to gravity, your '3d wire grid' becomes pinched in around the bowling ball like the waist of a mannequin.

    Hope that helped? That's how I do it anyways...

  8. #8
    EvilEye:
    Thanks for the clarification; my puny brain just can't fully grok it. And maybe I never will--I'll just have to accept it.

    I can kind of "get" the idea of a surface of paper made 3D, just as I can get the dimple mass makes in the trampoline made 3D. But, I don't "get" the idea of a CURVED surface (e.g.: sphere, torus, saddle, etc) made 3D.

    I wish there were some example or analogy that could help...beyond what's already out there (like linked to the Astronomycast site) where all they show you is a sphere with a triangle on it. That just reinforces my inability to comprehend that surface, and not the solid content below the surface, as being 3D. *shrug*

    Oh well.

  9. #9
    Quote Originally Posted by Steve Limpus View Post
    You could try thinking of it like this:
    ([..snip..)
    Then just subtract one dimension. (I do it by taking away the building blocks and imagining the 'impression' they would have left on a sheet of paper, or rubber.)

    You end up with the rubber sheet 'grid with the bowling ball' familiar from the illustrations we see in books and on the 'net. And it isn't invalid because what happens in three dimensions also happens in two - they just take one away to make it easier to 'draw'.

    And then... the universe is 'flat' in the sense that the rubber sheet is flat, and the missing third dimension is also flat. (And of course the alternative geometries of the universe have the rubber sheet in the shape of saddles, spheres, donuts, soccer balls, etc... but in each visualisation the third dimension is 'missing'.)
    (..snip..)
    OK...I actually almost FELT a relay clacking in place in my brain. It's a really weird sensation.
    I got the grid of cubes within a cube, remove the cubes, left with grid thing.
    That's like why I can extrapolate the ball on the trampoline into 3D instead of just the 2D trampoline (which, yeah, the fact the trampoline is pitting I guess forces it into 3D already...but you know what I mean.)
    This is the point where the "flat" universe seems to click in place like a Leggo in my head...but then, replacing it with non-flat shapes shatters the near comprehension.

    A grid of doughnuts? Spheres? No, doesn't makes sense. I must have it wrong.
    Fraser (sp?) mentioned in yesterday's podcast that our intuition is left on the Savannah, it won't serve us here...and I'm afraid that's my sticking point. I just can't intuit a non-flat grid that uniform and universal.
    I mean, I can picture that 3D grid as if there's a finite mass in it pinching the lines...but it stops at some point. And a "flat" grid in that cube is easy ti imagine as stopping at the edges of the outer cube or going on forever, but any way in which the grid lines curve, has to end, and, be finite as you can't have (in my tiny mind) a 3D grid of curving lines without mass chaos in the lines.
    *sigh*
    Thanks anyway.

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    Quote Originally Posted by Mechphisto View Post
    OK...I actually almost FELT a relay clacking in place in my brain. It's a really weird sensation.
    I got the grid of cubes within a cube, remove the cubes, left with grid thing.
    That's like why I can extrapolate the ball on the trampoline into 3D instead of just the 2D trampoline (which, yeah, the fact the trampoline is pitting I guess forces it into 3D already...but you know what I mean.)
    This is the point where the "flat" universe seems to click in place like a Leggo in my head...but then, replacing it with non-flat shapes shatters the near comprehension.

    A grid of doughnuts? Spheres? No, doesn't makes sense. I must have it wrong.
    Fraser (sp?) mentioned in yesterday's podcast that our intuition is left on the Savannah, it won't serve us here...and I'm afraid that's my sticking point. I just can't intuit a non-flat grid that uniform and universal.
    I mean, I can picture that 3D grid as if there's a finite mass in it pinching the lines...but it stops at some point. And a "flat" grid in that cube is easy ti imagine as stopping at the edges of the outer cube or going on forever, but any way in which the grid lines curve, has to end, and, be finite as you can't have (in my tiny mind) a 3D grid of curving lines without mass chaos in the lines.
    *sigh*
    Thanks anyway.
    Stick with it, I'm sure the reason they draw 3d space in 2d is beacause we all see it better that way, I don't imagine the pictures in my mind are much better than yours!

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    The torroid's lines are still parallel no matter which way you go. (The doughnut)

    And the Saddle shape just represents the pinch.

    Our universe is flat in every direction.

    You are trying to make the universe a sphere, because we use a balloon analogy when explaining expansion.

    OK... lets take that balloon and make it as big as the universe. (forget about the inside and outside of the balloon... just the surface.)

    If you are standing on it, it appears flat no matter how far you go in any direction, just like a runway at an airport. But in reality, it follows the curvature of the earth. Now, when you jump up a dimention into 3, it is still flat but curved. And I think this is where you are getting stuck. Jump the surface of that sphere to 3 dimentions and any 2 parallel lines will stay parallel (flat) in ANY direction.... but going far enough in any direction will bring you back to your starting point.

    Close your eyes and point your finger in front of you, and you are essentially pointing at your back.

  12. #12
    Quote Originally Posted by EvilEye View Post
    Close your eyes and point your finger in front of you, and you are essentially pointing at your back.
    Gah! That makes no sense (to me). I still get it when we're talking surfaces...the runway, the line looking flat but coming around to the other side.
    But not when were not on the surface of anything. =P

    I dunno...maybe I'm making it harder than it is.
    I picture myself floating in space with two impossibly powerful laser lights. And they're wrapped together perfectly side-by-side as to (appear) perfectly parallel. They're 1 cm apart at the source.
    Now, in a geometrically spherical universe, once they get out far enough the two lines will be 2 cm apart, and 3, and eventually 1km apart?
    If that's right, the next obvious and naive question is...what's pulling them apart? And the answer would be: nothing 'cept the curvature of space. Yes?

    if I'm with it so far, the next question is...until what? If the universe is infinite, OK. But if it's finite, what's at the edge? A literal edge of space and then non-universe outside that?
    But what's this "back of your head" description? How's that possible?
    OK, I think I can get that in a spherical universe...I can imagine the laser lines separating more and more until they've curved back back to the source, and I can imagine that without there even needing to be an edge of the universe (ouch!!)

    But that doesn't make sense with a flat, finite universe. That implies an edge and an end, yes?
    Have I gotten closer?

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    You might like these images:

    http://plus.maths.org/latestnews/sep...stein/GPB1.jpg

    http://img506.imageshack.us/img506/5...acetimelu3.jpg

    http://www.ipod.org.uk/reality/reality_block_time2.jpg

    the third one is cool - a 3d image of the Earth and Moon in 4d spacetime!

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    Quote Originally Posted by Mechphisto View Post
    But that doesn't make sense with a flat, finite universe. That implies an edge and an end, yes?
    ...in the toroidal universe, the lasers will remain parallel, but will wrap around the 2d donut and smack you up the back of the head! But the 2d geometry is still Euclidean or 'flat'.

    I don't know how it works for the soccer ball... Bend It Like Beckham?

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    Here is your problem...

    Quote Originally Posted by Mechphisto View Post

    I picture myself floating in space with two impossibly powerful laser lights. And they're wrapped together perfectly side-by-side as to (appear) perfectly parallel. They're 1 cm apart at the source.
    Now, in a geometrically spherical universe, once they get out far enough the two lines will be 2 cm apart, and 3, and eventually 1km apart?

    This is where you are getting lost.

    It isn't a sphere. It is FLAT. The curvature is not like a ball. That's 2D. This is the surface being curved in 3 dimentions. Not 2.

    We aren't inside the ball. We are on it, and up, down and sideways is every direction depending on where you are.

    Are the people on the opposite side of the earth upside-down? Of course not. Just relative to you.

    You are not inside the universe. You are part of the whole. There is no center to the surface of a sphere.

    Your parallel lasers would stay parallel forever until they hit you in the back of the head.

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    Somehow I knew I shouldn't go through the entire trilogy in one sitting.
    Just finished part 2, and my head is smoking badly.

    Starting part 3 now ...

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    I warned you, take some ibuprofen.

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    I will offer answers to selected questions and responses to some of the comments appearing in this thread up to now. My responses start on new lines introduced by double dashes --.

    themank: Due to the curvature of space as some theorize, any straight line in the universe will end up where it started. Or as Dr. Pamela put it, 'you shoot a laser beam into the universe it will eventually hit you in the back of the head.'
    --Space curvature would cause any straight line in space to end up where it started only if space curvature is uniform through space, which is very unlikely. Curvature is affected by local mass. For example, light is bent around massive galaxies, producing "lensing".

    themank: And since the big bang everything is essentially moving in a straight line (more or less) outward,
    will not all the stuff of the universe eventually re-converge on the same spot on the 'other side' of the universe?
    --I visualize objects in the expanding universe not as traveling around the surface of a phere but as fixed on the three-dimensional "surface" of an expanding four-dimensional hypersphere,. See my thread entitled "The shape of the Universe".l

    EvilEye: If the toroid was like rubber and being stretched in every direction, then ---
    -- All of your comments are based on the idea that the Universe has a toroidal shape. Despite its popularity in some quarters, I find that idea to be so contrived and implausible as to be ludicrous. See my thread entitled "The shape of the Universe".

    Mechphisto: "The analogy of a mass being like a weight on a trampoline ---"
    -- I feel that the trampoline model is crude at best. It has a body pressed downward into the sheet by gravity, conveying an entirely erroneous idea as to the nature of gravitation. Time is a crucial component of the picture and is totally ignored in the trampoline model. See my thread entitled "The shape of the Universe".

    Mechphisto: "if it snaps,"
    -- The very idea of the analog of the trampoline snapping shows how far afield this analogy can take one

    Mechphisto: "Maybe we are only right near the halfway point of the expansion."
    -- Current evidence indicates that the expansion rate is increasing at an accelerating rate and will do so into the indefinite future.

    EvilEye: "You are at the center of the universe right now, "
    -- I believe that that statement, like the rest of your argument, is indefensible. See my thread entitled "The shape of the Universe".

    Steve Limpus: "you could be thinking of a cube --- "
    -- I feel that you derailed yourself as soon as you mentioned a cube. I dn't see any justification for seeing the Universe as having any sort of cubical symmetry. I feel that the geometry described in my thread entitled "
    The Shape of the Universe" is the only one in any way plausible.
    Last edited by dcl; 2008-Apr-25 at 02:21 AM.

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    I have a question about the parallel lines. Didn't Dr. Gay say that in a sphere, 2 parallel lines will not remain parallel, but in a cube or donut, they will? I don't get that. Seems to me that they remain parallel in a sphere

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    Quote Originally Posted by Steverino View Post
    I have a question about the parallel lines. Didn't Dr. Gay say that in a sphere, 2 parallel lines will not remain parallel, but in a cube or donut, they will? I don't get that. Seems to me that they remain parallel in a sphere
    I didn't hear the piece, so I'm not sure what was said.

    Generally, it depends on the lines. Like, on a globe, (vertical) lines of longitude meet at the poles, while (horizontal) lines of latitude are parallel.

    Geodesic lines are perhaps of most interest, as they define shortest paths. You've probably heard that in Euclidean space (flat, 3D space), the shortest path is a straight line. That's a geodesic. Longitude lines on a sphere are geodesics. Latitude lines aren't.

    See Wikipedia: Parallel (geometry) for descriptions of the different sorts of line parallelism in different sorts of geometries.
    0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 0 1 1 0 1 0 0 1 0 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0 ...
    Skepticism enables us to distinguish fancy from fact, to test our speculations. --Carl Sagan

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    I feel that thinking in this thread has gone wild with little thought given to what makes sense or is likely to reflect reality. This is a plea to put more serious thought into what seems plausible. For one thing, we need to avoid unnecessary complexity in our ideas. For example, a simple sphere seems more likely than a doughnut, cube, dodecahedron, or 3-torus to reflect the actual shape of the Universe. In the following, I describe what I see as the most plausible geometry of the actual Universe. I've already done that in a thread that I started some time ago called "The Shape of the Universe". In the following, I expand somewhat on that model.

    Observation has revealed that the Universe appears on the largest scales to be homogeneous and isotropic -- homogeneous in appearing to be statistically the same everywhere, and isotropic in the sense of being the same in all directions. We know from what we see that the Universe has at least three spatial dimensions -- for example up/down, left/right, backward/forward. In a universe of three spatial dimensions, that rules out all shapes except that of a sphere. In a universe of four spatial dimensions, that rules out all except the hypersphere. All exotic shapes such as cubes, 3-tori, and dodecahedrons, and doughnuts, all of which have been mentioned, are ruled out .on grounds of excess complexity -- why consider something more complex when a less complex one seems to meet all of the requirements. We don't know enough about the shape of the universe to rule out shapes of more than four dimensions, but I know of no plausible arguments for more than four basic dimensions. Warping of space by the presence of nonuniform distributions of mass may introduce at least one additional spatial dimension as a minor perturbation from one of the two basic shapes described above.

    My suspicion, based on the principle of Occam's razor, that the more complex an idea, the less likely it is to be correct, is the four-dimensional hypersphere. We need a fourth dimension in modeling an expanding Universe to avoid having to consider the possibility that there is a unique point in it away from which the rest of it expanded. In a four-dimensional hypersphere, all points on the three-dimensional "surface" are equidistant from its center.

    If the three-dimensional "surface" of the four-dimensional hypersphere were perfectly uniform, its optics would have properties similar to those of great circles on the surface of a three-dimensional sphere: Lines starting out parallel would cross at points one quarter of he way around it, reach maximum separations and parallelism half way around it, We should not expect this to happen in the actual Universe because of slight deviations from sphericity because of slight randomness in the distribution of matter as exhibited, for example, by cosmological optical lensing..

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    Quote Originally Posted by Steverino View Post
    I have a question about the parallel lines. Didn't Dr. Gay say that in a sphere, 2 parallel lines will not remain parallel, but in a cube or donut, they will? I don't get that. Seems to me that they remain parallel in a sphere
    Lines can be LOCALLY parallel on the SURFACE of a sphere. Thus, close-spaced lines of constant longitude are parallel at the equator, but they intersect at the poles.

    As for Dr. Gay's assertions regarding parallel lines inside various solids such as spheres, cubes, and doughnuts, I find them thoroughly incomprehensible. For any such remarks to make sense to me, they need to be made with regard to objects in spaces with specified dimensionalitires. Lines can certainly be parallel throughout the interiors of all sorts of solid figures in three-dimensional space, and they will also start and terminate on faces of such figures, anything Dr. Gay says to the contrary notwithstanding. I have been unable to make any sense whatever of her assertion that the shape of the Universe in any way resembles that of a doughnut.

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    me neither

    i dont really even see why it needs 4d

    lets just have and edge a glorious energy crackling terrifying edge rushing away from us at ls or more

    it would be a sight !

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    Is this correct?
    When cosmologists say Flat universe, it means (among other things) angles of a triangle adds up to 180 deg. So according to cosmologist point of view, you can have a "flat" AND spherical universe. Here, Flat refers to the curvature of space(-time). Which is a dependent on the density of the universe.

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    Quote Originally Posted by suyuti View Post
    Is this correct?
    When cosmologists say Flat universe, it means (among other things) angles of a triangle adds up to 180 deg. So according to cosmologist point of view, you can have a "flat" AND spherical universe. Here, Flat refers to the curvature of space(-time). Which is a dependent on the density of the universe.
    In a "flat" Universe, Euclidean geometry is applicable. This means, for example, parallel lines are parallel along their entire lengths, the angles in triangles sum to 180 degrees, those of rectangles to 360 degrees, etc. It does NOT assume that space is three-dimensional; it can also be, for example, four-dimensional. In discussing space, only spatial dimensions are involved. That excludes time. In relativity, time is also a dimension, but not a spatial one and should not be confused with spatial dimensions. Flat does NOT refer to curvature of spacetime. Density of the universe refers only to density of matter in space and causes curvature not only of space but also of spacetime.

    I'm not aware that any cosmologists would talk of a "spherical universe", and I cannot imagine what they would mean by other than a three-dimensional space with a spherical boundary, which would imply that it was bounded. I ca not imagine any competent cosmologist as regarding such a geometry as plausible.

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    Quote Originally Posted by dcl View Post
    In a "flat" Universe, Euclidean geometry is applicable. This means, for example, parallel lines are parallel along their entire lengths, the angles in triangles sum to 180 degrees, those of rectangles to 360 degrees, etc. It does NOT assume that space is three-dimensional; it can also be, for example, four-dimensional. In discussing space, only spatial dimensions are involved. That excludes time. In relativity, time is also a dimension, but not a spatial one and should not be confused with spatial dimensions. Flat does NOT refer to curvature of spacetime. Density of the universe refers only to density of matter in space and causes curvature not only of space but also of spacetime.

    I'm not aware that any cosmologists would talk of a "spherical universe", and I cannot imagine what they would mean by other than a three-dimensional space with a spherical boundary, which would imply that it was bounded. I ca not imagine any competent cosmologist as regarding such a geometry as plausible.
    Cheers. I had agreed with you before but then another person mentioned that and i got confused!

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    The universe is either finite or infinite.

    If it's infinite then there are considerable problems with the idea that it developed from a singularity through a very condensed form.

    If it's finite then, either it has a boundary or it is some manifold that is locally three dimensional - various possibilities proposed include a hypertorus or a hypersphere.

    There is, of course, no reason at all to suppose that the large scale nature of the universe (or anything else about it) is explicable to me. I envy those whose intuitions and mathematics enable them to rule out any of these cases or to imagine alternatives.

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    Quote Originally Posted by agingjb View Post
    The universe is either finite or infinite.

    If it's infinite then there are considerable problems with the idea that it developed from a singularity through a very condensed form.

    If it's finite then, either it has a boundary or it is some manifold that is locally three dimensional - various possibilities proposed include a hypertorus or a hypersphere.

    There is, of course, no reason at all to suppose that the large scale nature of the universe (or anything else about it) is explicable to me. I envy those whose intuitions and mathematics enable them to rule out any of these cases or to imagine alternatives.
    I endorse your statement with one exception: I feel that the hypertoroid model is so contrived as to be preposterous. Dr. Gay defined it as follows:

    "You can do this in different ways. One of the weirder ways is to start with a cube Ė nice, friendly
    normal cube. Wrap it around so that the two ends of the cube, the left side and right side come
    together and touch. What youíve just made is a doughnut, a toroid."

    In spite of her admitting the weirdness of his model, Dr. Gay appears to endorse it as a plausible model for the actual shape of the Universe.

    I regard the four-dimensional hypersphere that I described in my thread entitled "The Shape of the Universe" is highly plausible. It describes the universe as basically homogeneous, isotropic, finite, unbounded, and expanding from a point equidistant from all points in our perceived three-dimensional space..

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    So let me get this straight.
    According to the flat universe, the shape is like paper(as NASA said) or as a table or a book one can say or is the comparison wrong? Thats the closest we can get to imagining it right? Or like how the people in the past believed in the flat Earth?
    I agree though that the universe must be finite, otherwise it can't be expanding! After all, nothing can be added to infinite.

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    Quote Originally Posted by suyuti View Post
    So let me get this straight.
    According to the flat universe, the shape is like paper(as NASA said) or as a table or a book one can say or is the comparison wrong? Thats the closest we can get to imagining it right? Or like how the people in the past believed in the flat Earth?
    I agree though that the universe must be finite, otherwise it can't be expanding! After all, nothing can be added to infinite.
    That comparison is not valid. All of the examples you cited, the paper, the table, and the book cover, are flat, but they are only two-dimensional, so citing them is oversimplifying the picture of what we suspect our Unvierse to be. In geometry, "spaces" can have any numerical dimensionalities and can be either flat or curved. In this sense, flatness refers to properties that determine, for example, what the angles of triangles add up to. In a flat Universe, these angles add up to precisely 180 degrees, and those of rectangles, including squares, add up to precisely 360 degrees. Three- and four-sided figures that we still call triangles and rectangles can be drawn on surfaces of spheres, but their angles add up to more than 180 and 360 degrees, respectively.

    Conceptually, there are spaces with these properties, and in fact, many people, including me, suspect that we have our Universe exists in such a space. One of the purposes for which the Wilkinson microwave anisotropy probe (WMAP) was created and put into orbit at enormous expense was to determine whether the three-dimensional space in which the Universe that we perceive is "flat", meaning not curved in that sense. WMAP data have thus far not been able to detect any curvature. This has led many people to conclude that space is flat. Actually, that conclusion is premature. Space may still possess such a curvature so slight that WMAP has thus far not yet yielded data precise enough to enable it to detect any curvature that may exist. I believe there is plausible reason to expect that WMAP will detect some slight curvature after it has accumulated data with small enough uncertainty to enable it to do so. My belief is based on the fact that the integral of the product of two finite variables must be finite, those variables in the present case being the age of the Universe, found by WMAP to be about 13.71 billion years, and the expansion rate, believed to have always been finite, even during the inflation era when it is believed to have been many times the speed of light but always finite.

    Incidentally, an infinite Universe CAN expand.

    I hope the above dispels your confusion regarding the meaning of "flatness" when applied to the geometry of space. If it doesn't, I hope you will ask more questions. Hopefully, one of us can resolve your uncertainty.

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