# The universe's boundaries and origins

1. Jul 18, 2008

### thomasxc

If the entire universe expands from a single point in spacetime, and expands in a uniform manner, then the universe has a boundary, regardless how fast or whether or not it’s still expanding. If it has boundaries, then what’s on the other side? One might say that the big bang was/is only the expansion of matter into nothingness. But then the question arises: if, the universe has borders, how did something come to exist in the infinity of nothingness? This gives rise to another question: how do we differentiate nothingness from the near absolute vacuum of deep space? So by deduction, there must be more to empty space if it is to be different from nothingness-if there is nothingness on the outside. i cant make much sens of it. can anyone else?!

2. Jul 18, 2008

### mathman

Not so - you are implicitly assuming the universe geometry is Euclidean. It isn't.

3. Jul 18, 2008

### marcus

Mathman is quite right, Thomas. Your reasoning is incorrect from the word go, because of the pre-1850 way you imagine space.

People who can't get past 1850 hold up progress. Please get past that blockage---it really isn't hard---and start helping rather than hindering.

here's the story.

the best theory of gravity we have----the only one that predicts everything to high precision and really works---is vintage 1915 General Rel. People have tried to find better for generations and failed---and they have tried for 90 years to catch it out, in disagreement with some observational test, and it passes every test.

this theory of gravity requires the mathematics of manifolds developed around 1850 by Riemann.
Using that mathematical language one can (just to take an example) easily imagine and describe and calculate with a boundaryless 3D continuum analogous to the 2D surface of a sphere, and which is not immersed in any higher dimension surroundings.*

So here's the deal: your reasoning is wrong unless you take us back to the pre-1850 geometry of infinite 3D right-angle graph paper such as Newton or Descartes might have imagined----the Euclidean idea of space. If you want to eliminate the progress in mathematics made since 1850, and go back there, then it is incumbent on you to provide us with a theory of gravity which works as well for precision calculation as Einstein Gen Rel, but which uses the rigid foursquare Euclidean space instead of the continuum that mathematicians have been using since 1850 thanks to Riemann.

*that is just one of the many 3D geometries you get as a reward for switching over to the post-1850 continuum. I'm not saying it is the particular one the universe uses---it is just an example of a boundaryless manifold---a 3D continuum without edges. General Relativity is not dependent on any one particular geometric setup---it uses the formalism in a general way.
===================

Also look at this:
Thomas, as far as I know there is no astronomer who says this. Maybe I'm wrong. Can you give us an example of some professional writing where a real scientist says "the entire universe expands from a point in spacetime"? If you can, please give a link to click on. It would be interesting to see.

A common mainstream science view is that the universe IS all of spacetime together with whatever matter fields.
No surround need be assumed, no boundary needs to be imagined. The mathematical model allows space (in the sense of distances) to contract (in the sense of distances decreasing systematically) and likewise to re-expand.
So one version we are hearing a lot about recently has a socalled bounce. there is contraction to a point where the density becomes so high that quantum effects make gravity repel and then contraction reverses and the usual bigbang expansion story begins.

that contraction and expansion involves the whole of space----it is not something that happens IN space.
the bounce is something space and matter do together---as one organic unity.
it is not something that matter does, Newton-style, in the midst of a static empty space (that would have been the pre-1850 attempt at a description)

Also please keep in mind that there are several different pictures currently being studied besides the bounce picture. There is a book coming out next year with chapters by a dozen or more authors, laying out the different ideas. I wont waste time trying to list all of them that I can remember. The bounce picture has plenty of competition. But all the approaches are grounded in post-1850 kind of geometry. Keep an eye out for the book. Here is the amazon page

http://www.amazon.co.uk/Beyond-Big-Bang-Prospects-Collection/dp/3540714227

https://www.amazon.com/Beyond-Big-Bang-Prospects-Collection/dp/3540714227

Last edited: Jul 18, 2008
4. Jul 18, 2008

### Peter (IMC)

The way I understand it, thomas is that what you consider to be the whole universe now, is, in a way, just as big (infinite even) now as it was at the moment of the big bang. It never was a point in space that started to grow and became bigger from one side to the other. It's more like that everything that you consider to be space was the same place because time didn't exist in the exact moment of the big bang. And without time, our perception of distance doesn't exist. Then distances between matter started to grow, not because they were moving away from each other but because space it self was expanding and what ever is inside that space, moved with it. But there isn't an actual force pulling objects away from each other.

In fact, the way I understand it, 2 objects are actually attracting each other, it's just that the further away they are from each other, the less strong that attraction is, so over big enough distances, they are moving away from each other with space, faster than they can attract each other. So locally matter atracted itself into big bubbles of matter and in between were created big voids of empty space, where the expansion of space is just making them go further appart.

5. Jul 19, 2008

### jonmtkisco

Hi Thomasxc,
Take a look at https://www.physicsforums.com/showthread.php?t=235035" thread where the model of a universe with a possible center and edges is discussed.

I have yet to read any serious explanation for why this model is ruled out by GR or current observations. Based on everything I've read, the model is not at all ruled out.

A scientific approach to this topic would benefit from more thoughtful analysis and less dismissiveness. Cosmology doesn't need more crackpots, but it also ought not to be treated as a popularity contest.

Morever, scientific demonstration of the physical reality of a 4th spacial dimension, which is crucial to the physical reality of GR mathematical prediction of spatial curvature, is utterly lacking. I find it distressing that mainstream cosmology has skipped over this huge missing link in the physical theory. There is a fundamental distinction to be drawn between the accurate mathematical predictions of GR, and the exotic physical reality popularly associated with that math by mainstream cosmology. The physical reality may eventually be demonstrated to be quite a bit more mundane. Like (prepare to be shocked!) the universe may be discovered one day to have only the 3 spatial dimensions we know and love (along with the time dimension if you want to think of it that way) and the underlying geometry might indeed be Euclidean. Even if that turns out to be so I expect the mathematical predictions GR makes about gravitational action will be reaffirmed.

Jon

Last edited by a moderator: Apr 23, 2017
6. Jul 20, 2008

### jonmtkisco

Here's a relevant excerpt from James B. Hartle's excellent textbook "Gravity - An Introduction to Einstein's General relativity" (2003) p. 368:

Emphasis added. So, a Euclidean universe with a center and an edge would not alter GR's observational predictions. Note the key bolded word "assume". It's a bit startling to see it stated that a centerless and edgeless universe is merely an assumption. Yes there's no evidence against it, but Hartle in effect admits there's no specific evidence for it either, other than the obvious point that if there is an edge, we are too far away from it to observe (at least so far) any phenomena that specifically reveal its existence to us.

Reading between the lines a bit, it seems that mainstream cosmology has opted for the mathematical elegance of a theory based on a "strong" cosmological principle that requires 4 spatial dimensions, over a supposedly messier (I question that) theory based on a "weak" cosmological principle that requires only our tried-and-true 3 spatial dimensions.

Jon

Last edited: Jul 20, 2008
7. Jul 20, 2008

### marcus

?

never heard anyone say that it requires a 4th spatial dimension in order for a 3D surface to be curved---or to mathematically define curvature

in GR an extra spatial dimension is not required----curvature is defined intrinsically just fine.

any differential geometry textbook or GR text should set you straight on this, Jon.

See if you can find an online GR text where an extra spatial direction is needed to define curvature and show us the reference.
============

Basic reasoning by analogy is that flatlanders have no need of an extra spatial dimension in order to discover if their world is curved.. They can find out just by measuring distances and angles. If their 2D world happens to be embedded in some 3D space, fine---the curvature will show up externally there. But if their flatland DOESN'T happen to be immersed in some surrounding space that is fine too. As long as the distances and angles are what they measured before, their flatland still has the same intrinsic geometry, same curvature etc. No physical need for an extra dimension exists.
=============

Sean Carroll has an online GR textbook. You might look there and see if an extra spatial dimension is ever invoked.
Good luck

Last edited: Jul 20, 2008
8. Jul 20, 2008

### Wallace

Good point Marcus, the balloon analogy often confuses people who wrongly assume that there must 'really' exist a 4th spatial dimension playing the analogous role of the 3rd dimension of the Balloon. It's an analogy, nothing more. Like all analogies, it is folly to try and push the analogy to ask any question that the analogy was not specifically designed to demonstrate. The balloon analogy just demonstrate how a homogeneous expansion looks at every point to be expansion away from that point. That is it. Nothing else should be attempted to be learned from that analogy and this kind of confusion it was follows from trying to do so.

The closed, finite, FRW universe does not imply the existence of any more than the regular 3 spatial and one temporal dimensions.

9. Jul 20, 2008

### jonmtkisco

Hi Wallace,

Of course the FRW metric itself does not imply the existence of more than the regular 3 spatial dimensions -- for example, the case where the universe is expanding from a single origin point is easy to portray as a solid FRW model in 3 dimensions.

But 4 spatial dimensions are indeed required in order for an FLRW universe to expand "from every individual point in the universe, in all directions simultaneously," where, by definition, there is no single origin or center point. Try to build a solid model of that... Better textbooks feel obliged to mention that it can't be done in 3 dimensions.

Hi Marcus,

Sorry to hear that you've never read that it is impossible to construct a 3 dimensional solid object which exhibits 3 dimensional surface curvature. CompuChip and I recently wrote about this very subject on https://www.physicsforums.com/showthread.php?t=243442".

Please explain how you would go about constructing a 3 dimensional solid object which exhibits 3 dimensional surface curvature. Sure, that's a rhetorical invitation, because we all know it's impossible. The fact that a 3 dimensional surface is capable of being embedded mathematically into a 3 dimensional manifold does not mean it is capable of physical embodiment in 3 dimensions. All of us should be frustrated that the advanced geometry textbooks you perhaps have in mind aren't more clear about that distinction.

It's pretty obvious to anyone who is thinking rather than reciting, that mathematical/geometrical theory and tangible, locally touchable physical reality are two very separate things, which may or may not coincide on any give topic.

Jon

Last edited by a moderator: Apr 23, 2017
10. Jul 21, 2008

### Wallace

GR is a 4D theory, 3 spatial and one temporal. That's it. Nothing that comes from GR can possibly suggest that any more dimensions exist (that's not to say they don't, just that GR does not describe there effects if they do).

In Euclidean space you need extra dimensions to define curvature, because the space itself is not curved. In general, metric spaces, such as those found in GR, do have curvature. A curved N dimensional manifold can either be described as a curved N dimensional manifold, or a curved surface within a higher dimensional space. What you can do is shift the intrinsic curvature of the space into curvature of the surface in higher dimension. The maths looks different but the reality is the same. Don't mistake co-ordinates for reality!

Have a look at section 3.1 of Peacock (page 69 in my copy, maybe different in other editions?). It describes how a constant time slice of an isotropic, homogeneous, closed Universe can be defined as equivalent to a Euclidean 4 sphere by a suitable transformation. This is interesting, but it does not really mean that if the Universe is finite it has 5 dimensions and 4 if not.

People have played around with non 4D GR like theories, the conservation laws and then all different and you get strange results. For instance in 2D, gravity is repulsive, and for other dimensions gravity no longer goes to 1/r^2 for the Newtonian limit. Therefore the physics, which is what is real, is intimately connected to the number of true spatial co-ordinates. You can't just add extra ones in and think it won't change everything.

11. Jul 21, 2008

### jonmtkisco

Hi Wallace,

I have no objection to most of what you say. But it is quite straightforward to demonstrate that a model of 3-dimensionally curved space or "centerless space" cannot be physically constructed in 3 dimensions.

An isotropically expanding sphere with a unique center is easy to model in 3 dimensions -- just start with a sphere of arbitrary size and add on nesting shells of ever increasing size - like Russian dolls. The existence of a unique "center" or coordinate origin is never in doubt, at least until infinite size is attained.

Now try constructing a solid model (in incremental pieces) of an isotropically expanding sphere with no center. It can't be built. It certainly isn't modeled by an expanding loaf with raisins, which has an obvious center in 3 dimensions. It's extremely difficult to even imagine it in 3 dimensions let alone to construct it. At best it can be constructed only as a 2-dimensional model of such a surface on a 3-dimensional manifold, e.g the surface of an expanding balloon.

It's just not physically real, in 3 dimensions. It's real only in people's heads -- as a mathematical/geometrical construct. And it can be physically real, but only if there are 4 spatial dimensions.

Jon

Last edited: Jul 21, 2008
12. Jul 21, 2008

### Wallace

Your reasoning is correct, if all spaces are Euclidean. General Relativity is founded on the idea that in general the space-time of our Universe is non-Euclidean. You can take the view that only Euclidean spaces can be 'real' but then you are forced to use a theory of gravity other than GR. What you cannot do is use GR to make a prediction and then violate an assumption of the theory that made that prediction in order to derive a physical interpretation! You can't use the GR result for an closed Universe, then pretend that aspects of GR don't exist when making an interpretation!

The FRW metric can describe a closed, finite, homogenous and isotropic universe and that metric contains only 3 spatial dimensions, and was derived from a theory relying on there being only 3 spatial dimensions.

I didn't make the Universe so I can't apologise for relativity being difficult to understand, but that's just how it is.

13. Jul 21, 2008

### Haelfix

Jon distinguish between the topology of spacetime and the metric of spacetime. The two are related but not necessarily restrictive.

So for instance you CAN have a big bang occuring at one point, just like you can have a big bang occuring everywhere.

For instance in the closed FRW universe, from the point of view of the metric, you can just as easily make the topology a flat plane(s), as well as a sphere as well as various other multiple connected topologies. In the case of a sphere, all homotopic curves contract to a point, so in that case you have a unique origin point.

But for the case of a flat plane (identify the boundaries), you have no such thing, all points feel the big bang, even though the distance between them (even arbitrarily far apart) shrinks to zero. All this is the same exact metric, but what occurs at the boundaries differs physically (and presumably its observable in principle).

Other than that, spacetime does not need to be embedded in a higher space. In general, you actually need twice the amount of dimensions to embed an arbitrarily complicated geometry (Nash embedding theorem).

14. Jul 21, 2008

### jonmtkisco

Hi Wallace and Haelfix,

Any geometry can be "physically real" in 3 dimensions if, and only if, we can build a solid 3 dimensional model of it in our real world. That is a very straightforward criterion of what "physically real" means; no subjective judgment is involved. If space can be curved in 3 dimensions, then we should be able to construct a solid physical model of curved space in 3 dimensions. Alas, we can't. And we certainly can't build a solid model of a "centerless" 3-dimensional manifold. If we had 4 spatial dimensions to play with, we could do both. GR may specify only 3-dimensional manifolds, but they simply can't be physically constructed unless we have 4 spatial dimensions available to work in.

The precise mathematical predictions of GR can be physically interpreted in more than one way. It is widely recognized in GR technical literature that the physical interpretation of the individual components of the Einstein Field Equations are ambiguous.

I submit that a physical interpretation is possible, consistent with the precise mathematical predictions of GR, which explains the so-called spatial curvature as substantively representing an integration across an infinite series of local SR reference frames immersed in a homogeneous cosmic gravitational field, over which the length of our physical rulers is affected by a series of Lorentz Transformations. Eliminate those Lorentz Transformations, and the manifestation of spatial curvature vanishes.

Jon

Last edited: Jul 21, 2008
15. Jul 21, 2008

### Haelfix

"Any geometry can be "physically real" in 3 dimensions if, and only if, we can build a solid 3 dimensional model of it in our real world."

Last I checked, GR is (3+1) dimensional. And I suppose that depends what you mean by physically real. Physically real to me, means it gives values for all experiments in its domain that I can measure and that are accurate.

"And we certainly can't build a solid model of a "centerless" 3-dimensional manifold"

Umm, sure you can. A 3 dimensional flat plane (R^3) is a manifold and centerless..

Also you cannot embed GR by adding an extra spatial coordinate. As I mentioned you actually need 8 dimensions (6 + 2) for the most general (torsionless) geometry.

16. Jul 21, 2008

### thomasxc

excuse me for being an idiot. im only a 10th grader trying to understand this stuff or the first time.

17. Jul 21, 2008

### marcus

Heh heh. You're fine. You are asking questions. No excuse needed! There is a problem when people start declaring unfounded stuff as if it were God's truth. But that's not you.

I wish more highschool students would get some exposure to cosmology (and not just by reading some popularization). What kind of library resources do you have?

18. Jul 21, 2008

### marcus

Interesting conversation. I think the nub of it is that Jon wished to persuade Thomas (a 10th grader new to the forum) that General Relativity is physically unreal because we cannot build a solid model of curved 3D space in our normal Euclidean 3D laboratory space.

This is an unusual viewpoint to say the least (and may involve private non-standard definitions of terminology). I replied that I had never heard of GR needing a 4th spatial dimension and invited Jon to find a passage in an online GR text, e.g. Carroll's, that we could look at. Wallace also responded to Jon, very helpfully I think, as follows:

Jon responded by giving his what seems to be his own private definition of what it means to be physically real---namely that you can build a solid model of it in ordinary 3D space.

Haelfix took issue with Jon's idea of what is physically real (which indeed does seem peculiar in the context of physics---although it might seem reasonable to general audience.)

It is an interesting stand-off. Let's stay tuned and see what we can learn, if the discussion continues.

19. Jul 21, 2008

### marcus

Jon, I see you have given your own coinage of two other technical-sounding terms:

strong cosmological principle

and

weak cosmological principle

These sound like they might be accepted cosmology terms. But in fact I don't believe they are. Here is how you introduce and use the terms:

In fact mainstream cosmology is based on Gen Rel which does not require a 4th spatial dimension. Haelfix and Wallace have spoken to that.

I don't know where the quote-marks come from, around "strong" and "weak". Who, if anyone, is being quoted? I tend to be dubious of posts with a lot of quote-marks unless there is a link that allows me to easily locate the source.

Anyway, my question is this: Jon, what exactly is the strong cosmological principle and what exactly is the weak cosmological principle?

The cosmological principle I am used to has to do with assuming uniformity---the premise that the world looks pretty much the same from different standpoints and in all directions.

Last edited: Jul 21, 2008
20. Jul 21, 2008

### shamrock5585

you should understand the entire universe by tenth grade thomas... shame on you... study up haha

21. Jul 21, 2008

### jonmtkisco

Hi Marcus,
It is obvious in context that I put the words "strong" and "weak" in quote marks because they are not defined terms. Generally speaking I wouldn't put defined terms in quote marks.

I tried to use the notion of a "strong" cosmological principle to mean an admitted ASSUMPTION by popular mainstream cosmology that the expansion of the universe has no unique center or origin point. In other words, that we prefer not to have a center because then it would make the single random place in space where the big bang started too "unique" from a certain purity of perspective.

I tried to use the notion of a "weak" cosmological principle to mean an ASSUMPTION popular mainstream cosmology is currently unwilling to make, which is that the expansion of the universe could have a unique center or origin point. That such a center point is not "unique" in any fundamental sense, because the underlying physics of space are the same at that point as at every other point in space. It's just the random place where a certain event occurred. In fact, if such a center point actually exists, it would be impossible to distinguish it from any other point in the universe today, other than by measuring the circumferance of the universe's outer edge (if there is one) and using that to geometrically estimate the center point's location.

I've tried to be very clear about what mainsteam cosmology agrees with and disagrees with on this subject. Even in 10th grade I don't think it's too early to learn that some of the most basic teachings of cosmology theory are admittedly based on assumptions which are preferred less because of specific observational evidence than because of the their philosophical and mathematical elegance. My advice is, study voraciously but try to think independently. When you ask "but why not?", think about to what extent the answer you get back is based on observational fact, mathematical theory, or convention. Then try to peel it back a layer or two.

Jon

Last edited: Jul 21, 2008
22. Jul 21, 2008

### jonmtkisco

Hi Haelfix,
Sometimes the physical interpretation of a single mathematical algorithm can be ambiguous, i.e. more than one physical embodiment can be interpreted. Saying that any one of such possible interpretations is the only correct one, merely because it generates accurate results, is not persuasive logic if an alternative interpretation generates equally accurate results.

ALL of the alternative interpretations can't be physically real at the same time, even if they are all consistent with the same mathematical predictions. One of them is actually physically real and all the others are unreal.
I'm not following you. Of course any finite 3 dimensional flat plane has a physical centerpoint. Keep in mind that the term "center point" is subject to varying meanings depending on context: The weighted-average spatial center, or the center of mass, etc. If a 3D plane is self-expanding from a singularity, then it has an historical origin point as well, which may or may not coincide with the spatial centerpoint at any moment in time.

Conversely I never intended to suggest that an object of infinite extent can meaningfully have a spatial center point. But if it is self-expanding from a singularity, it can meaningfully have a single historical origin point.

Jon

Last edited: Jul 21, 2008
23. Jul 21, 2008

### jonmtkisco

Hi Wallace,
I'm trying to contain this discussion to spatial curvature (and centerless manifolds) rather than to get into the related but somewhat broader concept of spacetime curvature. I think it's more accurate to say that the FRW metric (not GR itself) was devised as a theoretical model of universes characterized by either flat or curved spatial geometry. The physical reality of cosmic spatial curvature has pretty much been assumed since the FRW metric was adopted in the 1930's, despite the fact that there has been no observational evidence since which makes in highly likely that our universe has anything but zero cosmic spatial curvature. As you know better than most, current observations by WMAP constrain any spatial curvature to being exceedingly close to zero, if it is non-zero at all. In the FRW metric of course, cosmic spatial curvature is zero in any homogeneous region whose radius is expanding at the escape velocity of its contents. If we eventually are able to measure with reasonable confidence that the cosmic spatial curvature is vanishingly close to zero in our current universe, we should remain open to the plausible possibility that cosmic spatial curvature simply isn't a physically real phenomenon at all. Such a measurement of course wouldn't categorically rule out the possibility that cosmic spatial curvature is physically possible in our 3D universe, but for practical purposes it would relegate the phenomonon to the realm of pure theory. In any event, we could safely continue using GR's math and the FRW metric to make observational predictions in a flat universe.

I recall you stating in the past that spatial curvature in GR metrics can be eliminated by a suitable coordinate change. This tells us that spatial curvature is not a universal, covariant phenomena. It might simply be a manifestation of using a particular coordinate system.

For example, one might think of a black hole as being a strong candidate for generating extreme local spatial curvature in the Schwarzschild metric. This is the case when curvature is calculated in certain coordinate systems. But in the rest frame of an observer who is freefalling radially toward the black hole at exactly its local escape velocity, the local space is observed to be entirely flat and Euclidian. In this context "escape velocity" means the speed to which an object initially at infinite distance from the black hole and at rest relative to the black hole would be progressively accelerated by the black hole's gravity at each point along its plunging radial path into the black hole. The local observation of Euclidian space theoretically holds true even inside the Event Horizon where it is no longer actually possible to "escape" from the black hole's gravity at any attainable speed.

Jon

Last edited: Jul 21, 2008
24. Jul 22, 2008

### Haelfix

"Of course any finite 3 dimensional flat plane has a physical centerpoint."

Well then it wouldn't be a manifold would it. Go back to the definition of a manifold.

You will run into pretty much a major mathematical problem with any definition of a 'center' that you can think off in differential geometry, the language of GR. The best you can do are find priviledged points that are poles, or say that are fixed points (in a topological or homotopy sense), or simply work locally.

I know what you want to do, that is try to imagine physical concepts like the center of mass or somesuch. But thats an entirely coordinate dependant construction from the point of view of a metric and there are no unique ways to construct such an object. Again, the best you can do is think topologically.

25. Jul 22, 2008

### jonmtkisco

Hi Haelfix,
OK, I see where this dialogue went off-track. When I use a simple term like "center" you are applying concepts and terminology from differential geometry that are inconsistent with my intended meaning.

For the purpose of this discussion, the term "center" means nothing more than the unique point determined simply by measuring the intersection point of any two different (non-identical and non-opposed) vectors which are inwardly normal to the line (e.g. circle) or surface (e.g. sphere) constituting the outer edge of the expanding manifold.

If a flat 2D plane in Euclidian space originates from a single point and expands outward isometrically and homogeneously through all of the dimensions available to it, say the x and y dimensions, then the "center" as defined above will be both the physical center and the "historical origin". As long as its spatial extent remains finite, the meaning of the center will never be lost; it can always be located.

Even if a big bang starts with a pre-existing microscopic 2D plane which encompasses more than a single point, and the expansion energy is felt "everywhere on that plane", no matter how far the plane expands, as long as it remains finite it will have a meaningful center, because it will have an outer edge. Just use the same measuring technique with inward normal vectors.

A 3D "originless expansion" also can be imagined, for example, as a whole bunch of protons (hydrogen nuclei) compressed through external pressure into a small ball. At T=0 the external pressure is released, and each proton repels each other proton electrostatically due to their like positive charges. Despite the fact that the "expansion force" did not literally originate at a single point, and is felt "everywhere" in the compressed ball of protons, the expanding ball of protons will have an outer "edge" (the expanding spherical surface of the ball), and inward normal vectors can be taken from that to measure the center point of the expanding ball. Thus in Euclidean space, even an "originless expansion" leaves behind a meaningful center.

Jon

Last edited: Jul 22, 2008