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What Do You Mean, the Universe Is Flat?, Part 2: In Which We Actually Answer the Question

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walking a square

Stand up.
Walk 10 feet straight ahead.
Turn left by 90 degrees.
Walk another 10 feet.
Again, turn left by 90 degrees.
Do it for a third time: walk. then turn left.

Now the next time you walk 10 feet ahead, you’ll trace the fourth and last side of a square, and you’ll end up where you started. If you turn by 90 degrees for a fourth time, you’ll face in the original direction, too.

This seems intuitively obvious, even tautological—if you trace a square on the ground, well, you trace a square on the ground—but it is actually an empirical fact. And it’s important, so I’m gonna say it out loud:

There is no a priori reason why walking four equal sides and turning four right angles should take you exactly back to the same place. It is purely an empirical thing of our everyday experience.

As a matter of fact, it is not exactly true empirically, either. The failure to come back to the exact same spot—to precisely close a square—is not just true; it is one of the most important phenomena ever observed in the history of science. It is at the heart of everything. It is the way that gravity works the way that Einstein understood it. It tells us how black holes form and why they trap light. And it tells us if and how the universe should expand.

Our intuition tells us that every square should close. The world is far stranger than our intuition would have us believe.

In the previous part of this series, Part I, I promised that Part II would explain what it means for the universe to be flat. In this second part, I will talk about the concept—no, the phenomenon—of curved space, which is essentially when square paths fail to close, and about why flat space is where all square paths do close up.

Euclid Tried

So far I have intentionally emphasized the physical nature of this phenomenon called curvature of space. Most authors when they write about it follow a very different approach: they start with history.

You see, mathematicians came up with the idea of curvature—as a logically consistent but abstract concept—long time before anyone proved that it was relevant to reality. And measuring the curvature of space is actually very hard to do in practice, so it’s possible that no one would have tried if mathematicians had not told them that it was at least a possibility worth considering.

The mathematics required to fully make sense of curvature was invented in the mid-1800s by Georg Bernhard Riemann, and it is rather intricate. But curved space is a fact of life. In principle, you could discover it by walking around your room, without the need for mathematicians or physicists or philosophers to come up with abstract concepts first.

Euclid, the great geometer of Hellenistic Alexandria, was well aware of the fact that the closing of square paths is not a priori true. Euclid might have said it this way: the inner angles of a square (or of a rectangle or, for that matter, of a parallelogram) add up to 360 degrees. Going around a square means making four 90-degree turns.

diagonal of a rectangleAnother way that Euclid might have put it is by stating a related fact: that the inner angles of a triangle always add up to 180 degrees. Cut any rectangle into two triangles along its diagonal, and you’ll see why: your four right angles get divided into 6 angles, but the sum is still the same.

But geometry does not have to work that way. When it does, it is called Euclidean. But in the vast majority of cases when it does not, it is called non-Euclidean geometry.

Oftentimes, the way that authors introduce the idea of non-Euclidean geometry is by giving examples of what happens when instead of tracing triangles on a plane you trace them on a curved surface—say, on the surface of the Earth.

triangle on a sphereSo start at any point on the equator and head for the North Pole. Once you get there, you’ve covered one-fourth of the circumference of the globe, or about 10,000 kilometers. Now turn left by 90 degrees and start walking south. After 10,000 kilometers, you’ll reach the equator again. But you won’t be at the place where you started. Instead, you’ll be at a place 10,000 kilometers to the west of the starting point. Now turn left by 90 degrees so that you’re facing East, and walk another 10,000 kilometers: you’ll be back where you started.

You have traced a triangle on the surface of the Earth—and the inner angles are all right angles, so they add up to 270 degrees, not 180.

Notice that you have only done three legs of your trip. If you were to follow the instructions at the beginning of this post, you would still have another 90-degree turn and another full side to walk. In this case, the failure to close the square would be rather spectacular: instead of coming back to the original point on the equator, you would have ended up at the North Pole.

Tracing squares with sides that are 10,000 kilometers long is kind of extreme, of course. If you were to try a similar experiment with sides of, say, 1,000 kilometers instead, the error would be a lot smaller, but still conspicuous. And if you tried moving in 10-foot legs, you would notice nothing amiss: the world would look perfectly Euclidean to you. You could be forgiven for thinking that the Earth is flat.

In any event, the sphere a totally legit example of a non-Euclidean geometry, but can also be confusing. “Ok, the Earth is curved,” you say, “but what does that tell me about the curvature of space?”

“What if I had dug tunnels straight across the Earth, joining the two points on the equator and those two points with the North Pole? Together, the three tunnels would form an equilateral triangle. I could then imagine pointing lasers down the tunnels to join the three points with one another into a triangle of laser light. That triangle would surely have angles that add up to 180 degrees.”

Perhaps. But perhaps not.

Space in Outer Space

So here we come to the basic fact of life that I was referring to at the beginning of this post. The curvature of space itself.

To avoid any confusion caused by the Earth, take a trip to outer space. You could think of a  spacecraft tracing a triangle or a square by traveling in space. That would not be ideal, though, because it raises all sort of thorny issues about what exactly it means for a spacecraft to fly straight ahead or to turn by 90 degrees to the left.

Instead, you and two buddies each have a spaceship, and each of the three travels to some place in the near universe. Once you’re there, you point lasers at one another and form a triangle of beams.

three spaceships with lasers

Now each of you can measure the angle between the two beams that go in or out of the respective spaceship.

Fact: Those three angles won’t always add up to 180 degrees.

You could do the appropriate calculations and realize that this fact is a consequence of Einstein’s general theory of relativity. Or you could distrust math and physics and just go out to space to see for yourself.

Regardless, this is what it means for space to be curved. Whenever you can find three points in space, and join them with laser beams, and find that the triangle doesn’t have the expected 180 degrees, that means that space is curved.

And when no matter where the spacecraft are the angles add up to 180 degrees–that is what it means for space to be flat.

The mathematical machinery of Riemannian geometry goes much further and actually gives you a way to define and calculate numerical measures of curvature—not to just say if there is some or none.

There are two important special types of curved space. If in a certain region of space, no matter where you place your three spaceships the three angles they form always add up to more than 180 degrees, then the curvature is positive throughout the region. When they always add up to less than 180 degrees, that means the region has negative curvature. In the flat case, it’s precisely zero.

This post is part of a series on cosmology. Here are the previous posts: 

What Do You Mean, The Universe Is Flat? Part I
(On what I mean by “the universe”)

Being Mister Fantastic
(On visualizing a finite speed of light)

Under a Blood Red Sky
(On the afterglow of the big bang, and why the sky used to glow red)

Still to come: how do we know that the curvature of space is a fact of life; what would the world look like if space were very curved;  what is the curvature (and the size) of the observable universe; and what the heck does the observable universe have to do with Dante.

Footprint icon courtesy of palomaironique/Open Clip Art Library.
Spacecraft image courtesy of NASA. The artist’s impression represents the planned Laser Interferometry Space Antenna, or LISA, international space mission, which would in fact be unmanned. Also, LISA is not designed to measure the angles of the triangle but temporary changes in the distances of the space probes from one another due to the passage of gravitational waves–which are themselves perturbations in the curvature of space.

About the Author: Davide Castelvecchi is a freelance science writer based in Rome and a contributing editor for Scientific American magazine. Follow on Twitter @dcastelvecchi.

The views expressed are those of the author and are not necessarily those of Scientific American.

Comments 29 Comments

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  1. 1. MadScientist72 12:48 pm 08/1/2011

    There seems to be some inconsistency between the 2 exmples here. In the first, we’re tracing a 3-dimensional path along the surface of the earth but, in the second, we’re tracing a 2-dimensional path *through* space. If we were to this through the earth, our triangle’s angles would always add up to 180 degrees, but this doesn’t make the earth flat. We can’t make reliable statements about the 3D shape of the universe based on a 2D slice of its interior. To truly determine its shape, we’d have to somehow trace this path along the universe’s surface – if it has one.

    Besides, how can a flat (2D) universe contain 3D objects?

    Link to this
  2. 2. Torbjörn Larsson, OM 1:28 pm 08/1/2011

    It *is* a good story! Sort of like the mathematician’s fire and glüwein Yule story.

    I am probably jumping the gun by reminding that these ways of defining and measuring curvature is, as promised, valid on higher dimensions too. (Explaining why string theory 11D can, in some cases, look like 4D spacetime; “you say curvature, I say curvature”.)

    [I think; I never really got into this stuff for real. But I have gotten the impression that differential geometry et cetera is a *huge* subject.]

    Kudos for anyone trying to explain this!

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  3. 3. @dcastelvecchi 4:02 pm 08/1/2011


    I may have gone too quickly in that part of the post. What I meant was the following: I am showing an example of a curved surface. But what I am really aiming at is explaining how 3-D space can be curved. The paths on the surface of the Earth do not tell us anything about the curvature of space because they are forced to move along the surface. If you want to see what those three points tell you about 3-D space you have to trace the shortest paths between them in space, i.e., shoot lasers in tunnels.

    Incidentally, even that triangle of lasers, which seems to be a normal flat triangle, has angles that don’t add up exactly to 180 degrees, because the Earth’s own gravitational field makes space curved.

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  4. 4. JoeBussey 8:06 pm 08/1/2011

    So… I thought I was kinda smart, but after reading this article. I’m going to go with not dumb. (maybe)

    I think its great that the author posts in the comments section. I don’t think I’ve ever seen that before.

    I’m gonna go watch TV and try to feel smart again.

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  5. 5. dthoang 8:24 pm 08/1/2011

    There is an (unstated) assumption that light travels in a straight line. Given the fact is that light can be bent by gravity, does it still make sense to define a straight line by the path a light beam takes? Is there another definition of straight line that is not affected by gravity?

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  6. 6. Bill Crofut 8:35 pm 08/1/2011


    Re: “If in a certain region of space, no matter where you place your three spaceships the three angles they form always add up to more than 180 degrees, then the curvature is positive throughout the region. When they always add up to less than 180 degrees, that means the region has negative curvature. In the flat case, it’s precisely zero.”

    Has anyone tested this hypothesis by actually using three spaceships?

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  7. 7. JoeBussey 10:19 pm 08/1/2011

    We’d have to rent the spaceships from Russia to find out. :)

    Link to this
  8. 8. @dcastelvecchi 1:00 am 08/2/2011


    Very poignant remark. No, I did not make any assumption about light traveling in a straight line. The reason I didn’t is that if you give up the rock-solid foundations of Euclidean geometry you also give up the very notion of “straight”.

    In Riemannian geometry the correct concept to replace straight lines is that of geodesics. (This is a very subtle point over which even professional mathematicians often stumble, and one that will certainly be the subject of a future post.)

    The reason why I avoided talking about spaceships moving “straight ahead” and instead I talked about laser beams is that in empty space (as opposed to in space-time) laser beams follow geodesics, which are roughly speaking the shortest paths between two points.

    Yes, geodesics in space are indeed affected by gravity. However there is an additional complication: to really understand the propagation of light you have to look at the path of photons in space-time, which has a more fundamental meaning than their path in space. There you also have a notion of geodesics but it is even weirder in that it is not simply defined as a shortest path.

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  9. 9. @dcastelvecchi 1:09 am 08/2/2011

    JoeBussey: bloggers are supposed to respond to comments, aren’t they? A lot of bloggers I know do it.

    I would be interested in knowing which parts you found confusing so perhaps I can be clearer next time.

    Bill Crofut:

    I don’t know if anyone has directly measured the internal angles of a triangle in space, but I suspect that it would be exceedingly difficult because the effects are tiny. Gravity Probe B however has measured the geodetic effect, which is another way of essentially measuring the same phenomenon. I plan to talk about it in the blog soon. Here is their web site:

    Link to this
  10. 10. vanzilar 7:23 am 08/2/2011

    Good article.

    But yes the curvature of space is due to gravity for the most part (I like qualifying everything because more than gravity, its the assumption that there are more than 3 dimensions but each of the extra ones if just a tad bit more that leads to GR), and light does bend when it travels through space due to the curvature of space and NOT due to gravity since photons are (at least to a insanely large degree, and greatly debatable) massless. So the laser example in a way would be a bit confusing to someone who does not understand general relativity, but has heard of gravitational lensing. Basically, the laser would not travel a straight line geometrically and thus it would not close the square properly and that would show the curvature of space! So good article!

    (Also NASA has plenty of unmanned spaceships zipping about, a spaceship does not require a human).

    Link to this
  11. 11. Bill Crofut 10:59 am 08/2/2011


    Thank you for the website reference; it’s my plan to follow your development.

    Link to this
  12. 12. @dcastelvecchi 9:39 pm 08/2/2011

    Thank you for your nice words. In a non-flat space, light does not travel in a straight line, but not because it deviates from going straight: it’s because there just is no notion of “straight”. The correct concept to replace the notion of straight line is that of a geodesic. So in a sense you could say that light still goes “straight” in the sense that it follows a geodesic.

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  13. 13. GoodLibran 5:38 am 08/3/2011


    If one has to understand what you are trying to imply by the first example, they have to come to terms that the surface on which we would walk to form a square
    is not flat, and hence your assertion. If you just add this line, it would be much simple for a layman to understand the concept.

    Link to this
  14. 14. verdai 6:49 pm 08/4/2011

    The use of words is more advanced than this author.
    One of you clue it.

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  15. 15. Postman1 8:00 pm 08/4/2011

    I think space is more than curved, warped and twisted is more like it. Thanks for a good article.

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  16. 16. jtdwyer 8:14 pm 08/4/2011

    I’ve never really been interested enough to investigate the details, but it seems to me that GR defines ‘the curvature of space’ as a set of coordinates in empty space. Personally, I could never accept that imaginary abstractions alter the motion of objects. It seem to me that space must have some form of physicality to produce physical effects.

    If the gravitational effect (for ideal spherically symmetrical distributions of mass) is considered to be the radial contraction of external kinetic space-energy, producing a field which imparts increasing accelerating velocity to objects of potential mass-energy, generally directed to the local center of mass (each of which produce their own external field), the motion of objects is influenced by directional acceleration produced by the directional flow of kinetic space-energy.

    This kinetic space energy may be simply the residual energy wave produced by the big bang that imparted the initial expansion of the universe.

    In this scenario, spacetime contains the locally directed flow of kinetic energy that determines the motion of objects (in conjunction with external ‘peculiar’ velocities).

    To me this sort of physical description of spacetime is not only possible (admittedly I may have missed something) but should be necessary for any aspect of spacetime (curvature of dimensional coordinates) to impart motion.

    In this scenario, light curves when traversing a significant gravitational field because the field’s radially directed velocity is proportionately imparted to the path of light in conjunction to light’s own independently directed linear path of self propagation. The vectored net velocity of the light wave is redirected within the gravitational field to propagate in a curved path.

    Sorry, but light following an imaginary ‘straight line in curved space’ is too abstract to satisfy me, and I think unnecessary. I prefer an equivalent space containing kinetic energy that imparts physical effects.

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  17. 17. Wilhelmus de Wilde 11:11 am 08/5/2011

    Space/time is a unit that makes it possible for us to be aware of our geometric places and history, we are using for that Cartesian coordinates and clocks, both have to be agreed upon by humans.

    The essence of space/time is that the one (for a human living being) cannot exist without the other, different places mean different coordinates , mean distance , mean time to go from one place to another.

    Einstein added to that the mass, and used the speed of light as the maximum speed, so the limit. This limit included that at that speed time did not pass, so we arrived at a “frozen” universe. The mass influenced the curvature of space, so the road that that light takes by passing through space. If the mass were too massive, light could not “escape” its influence and stood still, does this mean that time evolves at the maximum speed ? (compared to the speed of light wher time stands still) and what the hell is the maximum speed of time ? Are we touching here INFINITY (black hole), is it the same INFINITY as we are touching it at the speed of light?

    second problem :

    During Inflation it is stated that space on itself expanded, and time did not have any influence on that, they take time away from space to arrange that space could expand to the volume needed by now. I think that is impossible because :

    if you extract time from space space has no meaning any more.

    I will come back in alater post on this item.
    (in the meantime if you have “time” available read my essay on FQXi:

    keep on thinking free


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  18. 18. kebil 8:56 pm 08/5/2011

    JT, what are you talking about when you say kinetic space energy? Fields imparting radially directed space energy? This does not make any sense to me.

    GR is more than imposing coordinates on spacetime, it is a succinct, if complex description on how matter curves alters the flow of time and the curvature of space. In fact, it is more the slowing of time in the presence of massive objects that gives rise to gravitational effects, and the curvature of light passing nearby, than the warping of space itself.

    And I don’t know what you can mean when you insist that their must be a “physicality” to space. Space is the arena in which physical phenomena occur, just as time is the dimension through which objects travel as change occurs. Space is very much a part of the physical universe, even if their is no “material” structure. Indeed, to insist on a physical substrate upon which space rests is to insist on a preferred frame of reference, which absolutely makes all of astrophysics impossible to make sense of unless one could determine this frame of reference and why it is preferred.

    Sorry JT, but on this issue, I think I will have to disagree, unless I have terribly misunderstood you. Of course, if you can explain it more precisely, preferably concisely, I can’t really explain what I think is wrong with your hypothesis. A more coherent understanding of the GR explanation for gravity may help you to frame your argument better.

    I am always willing to listen to alternative explanations

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  19. 19. kebil 9:29 pm 08/5/2011

    Willhelmus, the expansion of space in the early universe was no more outside the measure of time than the expansion of space at any other time. If there was no measure of time at this period, it would have been instantaneous, which obviously makes no sense.

    The speed of light being the maximum speed has no such effect as freezing the universe. It is merely the fastest speed anything with no mass can travel. Anything with mass must travel at a slower speed. Anything travelling at the speed of light does not experience time, that is true, but their is good reason for that.

    When you add the speed an object travels through time to the the speed it travels through space, you come up with the speed of light, that is just an observed fact of the universe. This may not be the simple mathematical sum, but involves some higher forms of mathematics, but it is true. An object at rest in one frame of reference moves through time at the speed of light. If you use a different frame of reference, it may be moving at some velocity, but it is travelling through time at a lower.
    speed (hence the contraction in time, in that frame of reference). Einstein did not just arbitrarily pick the speed of light to be the maximum, he had good reason for it, albeit it reasons that I cannot easily explain.

    Light does not stand still in a black hole, but falls towards it at the speed of light, as always. The infinity in a black hole is it`s density, although what happens to time I am not sure, but it does not become infinite anymore than time is infinite in direction everywhere (at least in many cosmologies.).
    Of course if you extract time from space, it loses much of it`s meaning. Without time to travel through, nothing can change, as change occurs through time. Without time, change is meaningless and does not occur. This is why photons do not change, they are timeless in a sense. Light moves at the same speed to all observers. This is what keeps a universe with no preferred frame of reference from become absurd and contradictory.

    It is not life that cannot exist without time, it is change.

    Einstein did not really add mass to space time so much as he fleshed out what space-time was. Mass alters space-time more in that it alters the flow of time, and only secondarily in that it curves space. Objects fall towards massive objects because time slows in their presence primarily, and only secondarily because they curve space.

    Space and time are inextricably linked, we live in a four dimensional (at least at the macroscopic level, 10 or eleven possibly at the sub quantum level) universe.

    At least this is what my reading of general and special relativity, string theory, quantum mechanics, and the geometry of the universe tell me. As always, I have been wrong once or twice in my lifetime, and there could be a third time (but only as long as I don`t move at the speed of light).

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  20. 20. kebil 9:45 pm 08/5/2011

    MadScientist72 – the surface of the earth is a curved 2 dimensional surface. While one can think of the surface as curving into a third dimension, the geometrical meaning of a curved surface does not require this. This is why a three dimensional surface (this is using the geometer`s definition of a surface, it is not a volume in our usually understanding) can be curved without the need of a fourth or more dimension into which it can be curved. This is what I get from my reading of Shing_Tung Yau, one of the mathematicians/geometers who brought us Calabi-Yau spaces, the spaces of the string physicist’s 10 or 11 dimensional worlds.

    The two dimensional path traced by the triangle on the surface of the earth would represent a three dimensional curved space, while the 2-dimensional surface through the middle of the earth would represent flat space (ignoring the curvature of space that earth produces, which is generally to small for us to notice without precise observations). They are both two dimensional as you only need two dimensions to indicate where on the surface they lie.

    In general, the universe is thought to be flat on the largest scales, but on smaller scales, where massive objects like the sun and the earth produce local curvatures of three dimensional space (and four dimensional space-time) on these “smaller scales” (small scales being those in which the effects of suns, planets, galaxies, and black holes are noticable).

    Experiments are being done on intergalactic scales to see if the inner angles of huge triangles between galaxies in space add up to 180 degrees or not. This could tell us if the universe if flat or curved. Their are other ways to detect the curvature of space as well, but I won`t attempt to describe them now. So far, all results seem to indicate that on the largest scales, the universe is flat, or extremely close to being flat. However, if the universe is infinite, which it probably is, even a very small curvature of space could make the universe a closed system (like a circle, with no ends, yet not infinite), or curved the other way.

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  21. 21. kebil 9:47 pm 08/5/2011

    vanzilar – the curving of spacetime by mass is the same thing as gravity, at least according to Einstein. So yes, it is gravity that is bending the path of light, just as yes, it is the curvature of space that is bending the path of light. Even massless particles are affected by gravity, but through this effect of mass on space. Another way of thinking about it is that, in a sense, light does have mass, because mass and energy are equivalent (remember, E=MCsquared)

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  22. 22. jtdwyer 7:51 am 08/6/2011


    I think that my conception of kinetic space energy can be considered an approximation of Vacuum Energy. Please see:

    Fields imparting radially directed space energy are related to the classical conception of vectored gravitational fields – that produce the ‘curving’ effects imparted to spacetime as described by GR. Please see:

    Newtons’ attractive force between two objects can be approximated by two objects, each locally contracting spacetime, producing a local field of radially directed kinetic energy that imparts accelerating velocity to material objects.

    The net effect of the interaction between the two fields is the accelertion imparted to each object, effectively producing a virtual attraction between the two objects.

    In GR, what is the physical medium being “warped” or “curved”? As I understand their is no physical medium or force involved, only the abstract effect on the relative motions of objects is described. GR is a very useful analytical model describing the effects of gravitation, but as I understand does not describe any physical mechanism that produces the described effects.

    Likewise, Newton admittedly had to employ an imaginary attractive force to describe gravitation’s effects.

    Sorry that I am not satisfied without physical effects being described as the products of physical processes involving physical entities. I hope this helps my inadequate explanations outside the context of common conceptions.

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  23. 23. kebil 6:42 pm 08/19/2011

    Jtdwyer – I cannot really explain to you in simple terms what is meant by space being warped. This comes down to a description of geometry rather than physics. I understand the desire to want to have a “something” that is warped, rather than the “emptiness” or “nothingness” of space and time. What would it mean for space to be a “physical” thing? Are not all “physical” things merely interactions of fields, giving rise to such “physical” things as quarks, electrons, etc. Space is the canvas against which these forces are played out. I would recommend to you Yau’s book “In The Shape of Inner Space” in which he explains in detail the geometry of space, Calabi-Yau manifolds, and other such things, giving a greater appreciation for the “thingness” of space without it become a physical entity such as the “ether” was once thought to be. I would love to explain it better to you, but Yau does a much better job

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  24. 24. jtdwyer 10:05 am 08/24/2011

    kebil – Exactly: the effect of gravitation is described by general relativity in terms of a geometric abstraction rather than physical process effecting real elements. It is defined by Newton as an imaginary force. But then, I think I’m repeating myself.

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  25. 25. jtdwyer 12:46 pm 08/24/2011

    Kebil – I almost forgot: you keep wanting to dismiss any thought that space might have physical attributes as a ‘throwback’ to the antiquated concept of “ether”, but I apparently have to ask you once again to please refer to:

    The idea that space is an unphysical idealized vacuum is invalid.

    Link to this
  26. 26. mpv55 10:46 am 04/28/2012

    Why is the Universe flat and not spherical?
    To understand it let us consider that there are only two mass bodies (or particles) in the universe. Initially, they are moving or stationary, or may be very far apart. There will be the gravitational force of attraction between them and it will merge then into a single body after some time. Thus one has to rotate around the other body in order to sustain them apart. Like earth moves around the sun.
    Now, if there are many bodies moving around a single body, initially they may in any orientation around the centre body. But slowly they will be in a plane as we know the formation of Saturn ring.
    Above two observations clearly suggest that the universe is also rotating around some body. We have to be outside the universe to see this rotation of universe. Similarly, it is expanding since it is getting flatter day by day.

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  27. 27. rohna 8:39 pm 01/12/2013

    Anyone else noticed that the triangle formed by cutting tunnels through earth would not be equilateral due to the fact that the earth has a larger equatorial girth than it’s height?

    :) Great Article for a lover of Physics.

    Link to this
  28. 28. rohna 8:43 pm 01/12/2013

    sorry for double post but if the universe was rotating around another body would the gravitational pull from the other body not stop the expansion of the universe at least in some directions?

    Link to this
  29. 29. erica1222 1:58 am 02/5/2014

    I don’t understand the last paragraph:
    “…if the three angles they form always add up to more than 180 degrees, then the curvature is positive throughout the region. When they always add up to less than 180 degrees, that means the region has negative curvature. In the flat case, it’s precisely zero.”

    Is it a typo and supposed to say “it’s precisely 180 degrees”?

    I can’t comprehend 3 closed connected angles adding up to zero degrees.

    Link to this

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