# Properties of 6 Dimensional Spheres

## Main Question or Discussion Point

It occured to me that most unification theories propose at least 6 dimensions, and that the 6th dimension would look like a sphere or a doughnut. For the sake of greater simplicity, I will use a sphere for my question. My question is, if an object were travelling in the 6th dimension, would they be travelling 'in' or 'on' the the 6D sphere?

Also, is the 6D sphere a 2D sphere embedded in 3D space plus time, or is it a 6D sphere, as in 5D + Time? For instance, if you wanted to find the surface area of the sphere, would you use the regular equations of a 2D sphere, because it is a sphere embedded in 4D space-time, or would you use the equation for surface area of a 5D sphere and add time?

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It occured to me that most unification theories propose at least 6 dimensions, and that the 6th dimension would look like a sphere or a doughnut. For the sake of greater simplicity, I will use a sphere for my question. My question is, if an object were travelling in the 6th dimension, would they be travelling 'in' or 'on' the the 6D sphere?
A single dimension cannot look like something 2-dimensional, such as a sphere or donut - so I suppose you are either referring to 5+1-dimensional spacetime with two spatial dimensions compactified on a sphere, or to the 6 'extra' (compact) dimensions in superstring theory.
Anyway, in the first case, try to imagine ( 3+1d spacetime with a sphere at each point in spacetime. If an object were travelling on the sphere it is indeed restricted to those two dimensions.
In the http://en.wikipedia.org/wiki/Superstrings#Extra_dimensions" the situation is actually the same; supposing we compactified the extra dimensions on a (6-)sphere, you would be travelling on that sphere in any of those compact (periodic) directions.

Also, is the 6D sphere a 2D sphere embedded in 3D space plus time, or is it a 6D sphere, as in 5D + Time? For instance, if you wanted to find the surface area of the sphere, would you use the regular equations of a 2D sphere, because it is a sphere embedded in 4D space-time, or would you use the equation for surface area of a 5D sphere and add time?
Referring to the superstring case, your 6d sphere is actually embedded in 9+1-dimensional spacetime.

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Also, don't confuse spheres with balls (giggle).

A ball is a solid, spherical object, while a sphere is the surface (only!) of the ball.

So, for example, a 2-ball is a solid disk. The boundary of the disk is a one dimensional circle, or a 1-sphere. A 3-ball is a solid object, like a bowling ball. The boundary of this object is an infinitesimally thin shell, called a 2-sphere. And so on.

Also, don't confuse spheres with balls (giggle).

A ball is a solid, spherical object, while a sphere is the surface (only!) of the ball.

So, for example, a 2-ball is a solid disk. The boundary of the disk is a one dimensional circle, or a 1-sphere. A 3-ball is a solid object, like a bowling ball. The boundary of this object is an infinitesimally thin shell, called a 2-sphere. And so on.

So, going into an unrelated topic, if you were to establish that there is the sphere of 2 dimensions embedded in 4D Space-Time, what would the inside of the sphere be like, if there was one? From what I have concurred, the sphere is defined by having surface area, but if you were to travel through 'surface,' would you find that the area inside the boundaries of the sphere to be hyperspace?

A single dimension cannot look like something 2-dimensional, such as a sphere or donut - so I suppose you are either referring to 5+1-dimensional spacetime with two spatial dimensions compactified on a sphere, or to the 6 'extra' (compact) dimensions in superstring theory.
Anyway, in the first case, try to imagine ( 3+1d spacetime with a sphere at each point in spacetime. If an object were travelling on the sphere it is indeed restricted to those two dimensions.
In the http://en.wikipedia.org/wiki/Superstrings#Extra_dimensions" the situation is actually the same; supposing we compactified the extra dimensions on a (6-)sphere, you would be travelling on that sphere in any of those compact (periodic) directions.

Referring to the superstring case, your 6d sphere is actually embedded in 9+1-dimensional spacetime.
Now that you have taught me the right diction on this subject, can you answer my questions from my first post in a concrete way? Would one use the regular equation for a regular sphere to find the surface area of the higher dimensional sphere?

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I am interested in the language used here, as in phrases like 6-sphere and 5+1 D spacetime. Is there a guide to phrases of this kind? What branch of the math or physics tree do they come from?

Thank you for this and many past kindnesses.

I am interested in the language used here, as in phrases like 6-sphere and 5+1 D spacetime. Is there a guide to phrases of this kind? What branch of the math or physics tree do they come from?
They're just fancy (short) names for '6-dimensional (hyper-)sphere' and '6-dimensional http://en.wikipedia.org/wiki/Minkowski_space" [Broken]', which treats the time-direction as something special (as opposed to 6-dimensional Euclidean spacetime, where all directions - including time - are on an equal footing).
I don't know of any guide to that kind of language but I'm sure the mathematics->geometry/topology section of Wikipedia can help you out.

Now that you have taught me the right diction on this subject, can you answer my questions from my first post in a concrete way? Would one use the regular equation for a regular sphere to find the surface area of the higher dimensional sphere?
At least in higher-dimensional Euclidean spacetime one would use the http://en.wikipedia.org/wiki/N-sphere#Volume_of_the_n-ball". To be honest though, I wouldn't know how this works out in a general spacetime - I guess it's just the same equation, but you'd have to be careful in stating what the radius of the sphere is, because the notion of distance can be different for each direction.

http://www.absoluteastronomy.com/topics/Sphere" [Broken] might be useful.

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Now, I'm pretty sure that if you had an object on a 5+1 D sphere, the object would travel on the sphere, and not inside of it, because the sphere by definition has no inside. My next question, which I've posted above already, is this:

If you were to establish that there is the sphere with 5 + 1D Space-Time, what would the inside of the sphere be like, if there was one? From what I have concurred, the sphere is defined by having surface area, but if you were to travel through 'surface,' would you find the area inside the boundaries of the sphere to be hyperspace?

I think it is probably incorrect to say that a sphere by definition has no inside. The definition of a sphere, including an n-sphere, starts with a defined origin point, from which every point on the sphere is equidistant. That point, as far as I can see, would be 'inside', for a 2-sphere anyway. None of the points of the sphere itself are 'inside', but there is still a definable ball, definably inside the sphere.

But perhaps you are saying the 5+1 sphere has no inside. I am still ignorant enough to imagine that as a possibility.

Hyperspace, if I understand correctly, is any space of higher dimension than 3+1 or 4. Or maybe it includes 4, but not ordinary Euclidian 3+1.

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I think it is probably incorrect to say that a sphere by definition has no inside. The definition of a sphere, including an n-sphere, starts with a defined origin point, from which every point on the sphere is equidistant. That point, as far as I can see, would be 'inside', for a 2-sphere anyway. None of the points of the sphere itself are 'inside', but there is still a definable ball, definably inside the sphere.

But perhaps you are saying the 5+1 sphere has no inside. I am still ignorant enough to imagine that as a possibility.

Hyperspace, if I understand correctly, is any space of higher dimension than 3+1 or 4. Or maybe it includes 4, but not ordinary Euclidian 3+1.

When I posted the last comment, I was looking off of what the others had responded. They have concluded that the sphere would have only surface area, and no volume. I actually can honestly say that I disagree with that. I think that the sphere has an inside, because it has an origin.

If you go further ahead, I have actually thought about the properties of these higher dimensional spheres, and if an object were to travel, is it traveling 'on' or in the 'inside' of these spheres embedded in 3+ 1 dimensional space. Is it possible to assume that there is an inside? From what I know, people say that objects would travel on the sphere, but what I have asked throughout this post is; can anything travel inside the sphere?

I would normally respond now, but I got married this afternoon and my bride is waiting.

I would normally respond now, but I got married this afternoon and my bride is waiting.

Congragulations!!!!!

Best wishes for you and your wife!

apeiron
Gold Member
The correct answer I believe would be that a sphere in the pure sense being discussed here has no physical origin point or interior. This would require the sphere to be embedded as an object in a higher-D space rather than simply being that space.

So think of it this way. The sphere surface has a curvature which at every location "points off" towards a virtual origin. There is no actual interior or origin, but there is an actual curvature which looked at in a certain light could be said to be angled off towards some imaginary reference point. That warpage might also be experienced by inhabitants of the surface as some other kind of physical constant. An "energy" perhaps.

To get a deeper appreciation of the range of possibility here, think about what is the "opposite" of closed curvature. It is hyperbolic or open curvature.

So you can have 1) flat space (not "spherical" but infinite and straight in its directions).

You have 2) a self-closed space like a sphere where every point shares the same curvature that points towards a common connection. A single global value (looking liking the co-ordinates for a phantom origin) acts as a constant that describes an extra quality to be found at all the locations.

Then you could have 3) a space of hyperbolic curvature at every point, so with every point then pointing away from any common space. So not a very connected world but more a quantum foam, a dimensional roil.

So out of this, we have the alternatives of cohesive spaces and uncohesive spaces. With flat space being the exact balance point between the two.

None of these need to be embedded in a larger dimensional realm to have curvature, a positive or negative "tension" of some kind. Or flatness even.

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The correct answer I believe would be that a sphere in the pure sense being discussed here has no physical origin point or interior. This would require the sphere to be embedded as an object in a higher-D space rather than simply being that space.

So think of it this way. The sphere surface has a curvature which at every location "points off" towards a virtual origin. There is no actual interior or origin, but there is an actual curvature which looked at in a certain light could be said to be angled off towards some imaginary reference point. That warpage might also be experienced by inhabitants of the surface as some other kind of physical constant. An "energy" perhaps.

To get a deeper appreciation of the range of possibility here, think about what is the "opposite" of closed curvature. It is hyperbolic or open curvature.

So you can have 1) flat space (not "spherical" but infinite and straight in its directions).

You have 2) a self-closed space like a sphere where every point shares the same curvature that points towards a common connection. A single global value (looking liking the co-ordinates for a phantom origin) acts as a constant that describes an extra quality to be found at all the locations.

Then you could have 3) a space of hyperbolic curvature at every point, so with every point then pointing away from any common space. So not a very connected world but more a quantum foam, a dimensional roil.

So out of this, we have the alternatives of cohesive spaces and uncohesive spaces. With flat space being the exact balance point between the two.

None of these need to be embedded in a larger dimensional realm to have curvature, a positive or negative "tension" of some kind. Or flatness even.

I do see what you're saying. I have to wander though why, if one thought of the space being curved, why it would be curved in such a manner as to give the same angle of curvature to everyone at every reference point. I find this appearance of 'energy' a little hard to believe.

If what you're saying is correct, you are basically eliminating the sphere entirely, because one would have no need for the model of a sphere unless they were to take all measurements in very precise ways as to have a uniform curvature arise as the solution, which, from all vantage points, would reveal a sphere.

Even then, after all of the virtual origins, wouldn't the energy have to radiate from the 'center' of the sphere in order to make it warped in the precise and uniform manner you propose?

apeiron
Gold Member
When I say energy, I am trying in an abstract way to indicate how a tension at a location could be interpreted as a curvature towards something from a perspective outside - and then all that may actually exist is that tension!

And this is not far from general relativity is it? We can model gravity as a force (complete with particles) or a spacetime warp. And the warp can then be reduced to a tension, a potential acceleration, existing at some point.

I'm not actually thinking of an energy that can be radiated away from a surface. Though brane theories do sometimes venture into this territory with stories of unconfined gravitinos leaking away and making gravity so "unnaturally" weak.

Is the sphere eliminated? Yes, conventional ball sitting-within-a-space mental images have to be got past to talk about issues of naked dimensionality, naked topology. But some essential ball-like properties remain.

It would after all be meaningful to contrast a hyperspherical universe with a hypertorus or some other non-sphere geometry. The tensions across all the points in this case would not be so constant, so symmetrical, but would vary because not all would be pointing in the same common direction. Well, maybe the curvature of a torus is constant - a question others might be able to quickly answer.

When I say energy, I am trying in an abstract way to indicate how a tension at a location could be interpreted as a curvature towards something from a perspective outside - and then all that may actually exist is that tension!

And this is not far from general relativity is it? We can model gravity as a force (complete with particles) or a spacetime warp. And the warp can then be reduced to a tension, a potential acceleration, existing at some point.

I'm not actually thinking of an energy that can be radiated away from a surface. Though brane theories do sometimes venture into this territory with stories of unconfined gravitinos leaking away and making gravity so "unnaturally" weak.

Is the sphere eliminated? Yes, conventional ball sitting-within-a-space mental images have to be got past to talk about issues of naked dimensionality, naked topology. But some essential ball-like properties remain.

It would after all be meaningful to contrast a hyperspherical universe with a hypertorus or some other non-sphere geometry. The tensions across all the points in this case would not be so constant, so symmetrical, but would vary because not all would be pointing in the same common direction. Well, maybe the curvature of a torus is constant - a question others might be able to quickly answer.
The idea of a tension causing the curvature is nice, but in simplistic general relativity, a mass causes curvature of space-time. From what I know, nothing can be simplified to 'just' tension. The curvature is caused by more than a potential acceleration existing at one point. It is caused by the mass, which creates the tension in space. So, if the tension is the curvature, then what causes the tension in space-time. I don't understand how a tension could be 'all there is.'

By the way, I'm just curious, is this idea of tensions a theory of publication or your own thoughts?

apeiron
Gold Member
What I was saying was not that one thing causes the other but what you might read off in one reference frame (as an an external observer) as a being a curvature might then be read off as something apparently quite different from another frame (as now an observer internal to the sphere surface).

Tension would be an example of an alternative reading, not necessarily the "right" one under GR, or indeed any other physical modelling. Although I feel it is certainly in the spirit of GR (tensor equations, etc).

So let's not get too caught up in the physical reality of "tension" (which may be a valuable idea or not) and just accept it as a way that you can understand why not to worry about the physical reality of origins to hyperspheres.

It is the reason I don't worry - and so it would be useful if someone else knew a flaw in this particular line of argument.

Now, I'm pretty sure that if you had an object on a 5+1 D sphere, the object would travel on the sphere, and not inside of it, because the sphere by definition has no inside. My next question, which I've posted above already, is this:

If you were to establish that there is the sphere with 5 + 1D Space-Time, what would the inside of the sphere be like, if there was one? From what I have concurred, the sphere is defined by having surface area, but if you were to travel through 'surface,' would you find the area inside the boundaries of the sphere to be hyperspace?
I'm sorry Benk, it must be me, but I still don't quite get your question. Let's for simplicity drop the whole hyperspace-thing for the moment, and just consider an 'ordinary' 2-sphere embedded in three spatial dimensions (no time).
Then, if one lived on the 'flatland' (or more perhaps more appropriately, 'curveland' - because of the curvature of the sphere) of the 2-sphere, one would indeed be confined to that surface of the sphere. However, due to the embedding in (i.e. the being-part-of) the higher dimensional space, it actually makes sense (at least for 'higher-dimensional creatures' living in the 3D (spatial) world) to talk about the inside of the sphere: it's a 3-ball.
You could of course wonder whether or not it makes sense for the 'curve-landers' to talk about the radius of the sphere, supposing they dont know about the existence of the additional (third) dimension that they cannot see. But at least they can calculate the curvature radius, and if they found it to be the same everywhere on the sphere, they could think of that as some kind of virtual origin, as was pointed out by Apeiron.

Now I think that by analogy this can simply be extended to the higher-dimensional case (still ignoring the time-direction for the moment, though). So creatures living on a 5-sphere could calculate the (for them invisible, virtual) origin in the 6-dimensional space.

benk99nenm312, thanks for your good wishes.

Apeiron, are you talking about tensors? Vector spaces? Covector spaces? Covariance and contravariance? I am studying these things but still not entirely comfortable with them. But maybe this conversation helps my understanding.

I think I understand that a torus has two directions of curvature. But I am running into language again. A point included in the set of points defined as a torus could also be said to be an element of a two dimensional surface, just as a point included in the definition of a 2 sphere is part of a two dimensional surface. The language difficulty seems to be centered on double meanings for the word "in": are we using 'in' to refer to a point being included in the set of surface points, or included in the ball, enclosed by the surface? It seems well-defined in a sphere, and we see the origin as not in or part of the surface, but as in or part of the ball. But look at the torus. Where is the origin? Is it even a point at all?

Or should we say the origin of the torus is a circle, with the torus defined as a set of points equidistant from the circle? This last works for me except I need to add the constraint that the distance from the circle which generates the torus must be less than the radius of the circle. Otherwise I get confused, and lose the idea of the torus shape.

I suspect the language used here, such as 2-sphere and 5+1 space, must have been defined so as to eliminate the double meanings. That's great, but I need to understand the grammar to understand the conversation. A phrase book might help.

If I am working in the right direction, I think I understand that the tangent space of a point on a 2-sphere is a disk or plane. The point is the origin of the disk, or the (0,0) of the plane. Each point on the surface of the 2-sphere has its own unique tangent plane. The line normal to the plane or disk through the origin of the disk also passes through the origin of the 2-sphere.

Not so for the torus. The tangent space is not always a plane or disk. It ( the tangent space) may sometimes be constrained by the curvature at some points to be a single line rather than a plane. The lines normal to the tangent spaces of a torus do not converge to a single point as they do in the 2-sphere.

Sorry to go on and on but I feel my understanding has moved forward during this conversation and I hope for confirmation or correction. I also hope my thoughts are useful to the original poster.

Thanks!

I'm sorry Benk, it must be me, but I still don't quite get your question. Let's for simplicity drop the whole hyperspace-thing for the moment, and just consider an 'ordinary' 2-sphere embedded in three spatial dimensions (no time).
Then, if one lived on the 'flatland' (or more perhaps more appropriately, 'curveland' - because of the curvature of the sphere) of the 2-sphere, one would indeed be confined to that surface of the sphere. However, due to the embedding in (i.e. the being-part-of) the higher dimensional space, it actually makes sense (at least for 'higher-dimensional creatures' living in the 3D (spatial) world) to talk about the inside of the sphere: it's a 3-ball.
You could of course wonder whether or not it makes sense for the 'curve-landers' to talk about the radius of the sphere, supposing they dont know about the existence of the additional (third) dimension that they cannot see. But at least they can calculate the curvature radius, and if they found it to be the same everywhere on the sphere, they could think of that as some kind of virtual origin, as was pointed out by Apeiron.

Now I think that by analogy this can simply be extended to the higher-dimensional case (still ignoring the time-direction for the moment, though). So creatures living on a 5-sphere could calculate the (for them invisible, virtual) origin in the 6-dimensional space.
It actully answers all of my question. When I posted the quote that you quoted on your post, I was bluntly stating what the others before me had answered. I actually agree with your statement, that it would be more of a 3-ball.

Thanks for the answer. This really clears things up.

apeiron
Gold Member
Apeiron, are you talking about tensors? Vector spaces? Covector spaces? Covariance and contravariance? I am studying these things but still not entirely comfortable with them. But maybe this conversation helps my understanding.
I'm trying to avoid getting into specifics of GR modelling as conversations can become religious. It is the intuitive content (nice phrase that) which is at a level above - a meta-level - that interest me here.

Does "curved" space need to be oriented to something beyond itself to be curved. Intuition suggests yes - until you perhaps adopt another point of view with different intuitive content, one based on the notion of a tension.

But tensors are an approach to GR modelling for a good reason.

I think I understand that a torus has two directions of curvature.
Not my subject area as I said. But that seems likely - and so extending the meta-discussion, every point would be found to have a stress tension oriented in two directions.

And as you say, instead of having a phantom point as the origins, a torus has a phantom circle - so a 1D origin rather than a 0D one, and thus has double the directions.

The relative size of the donut's hole would be irrelevant I believe to this principle. Or it would be set by a symmetry constraint.

I think I see that one could assign a stress value to every point in a plane, or more generally on a manifold, and that the stress value could be independent of any geometric distortion of the surface. Think of pulling on a rope that does not stretch. You can pull harder, so long as there is a counterbalance pull. The stress at the point measured, which point would not have to change physical coordinates, would be a variable, hence a dimension, albeit not one of position. I guess this is not unlike the idea replacing the curvature with a heat distribution, as I have seen suggested in texts, and in another thread in this forum, although I have not been following that one closely.

In short, one can assign a variable quantity to every point in the manifold, and that can act just like projecting to an origin that is not related to any physical coordinates of the surface.

This is interesting to me because of the gradual development of the idea of dimensions which are not position coordinates, yet can be treated as if they were geometrical coordinates, with the usual addition and multiplication rules. So a two dimensional surface can be given a kind of geometry which does not have anything to do with deforming the surface.

Then I wonder if this idea generalizes to give us a means of talking of higher dimensional objects in ordinary three dimensional space. Every point in a ball can have its own temperature. Maybe the temperature is an even distribution from hot in the middle to cool on the surface, like a star or planet. So the ball, seen as a simple filled sphere, can be given another dimension (a temperature distribution) and be called a four dimensional object. Temperature is an added independent dimension, as if it were another basis line.

Then I suppose one could challenge this idea, by arguing that what we really want is a fourth spatial dimension, not merely a column in a database. Then we have to ask what is the difference between the two ideas. In what way is our perception of space different from a database? Are the three spatial dimensions special somehow? Intuition tells me the space we live in is somehow different. Somehow it exists on its own, independent of how hot it is in one place compared to another, independent of how fast or how far you travel.

This then leads to information theory, and suggestions from the black hole theorists that all that really exists is entropy.

I have an appointment and have to go, but I look forward to hearing any thoughts you have on this.

Thanks,

r

apeiron
Gold Member
Sounds like you are re-inventing the notion of the field :-0

There are so many different angles into this subject of 'what is a dimension' that perhaps any further discussion should go into the dimension creation thread...

But even if we just stick to geometry, consider what happens to the "constant" of pi as we move from flat space to curved space, or alternatively to hyperbolic space.

So pi could be said to stand for a tension - the value at every point of space when it is flat. From memory, pi falls to 2 in a hypersphere, and rises to infinity in a hyperbolic space.

And this is a way to measure what kind of space we are in. Set up a big enough triangle and see what the angles add to.

I am not reinventing it, Apeiron, merely trying to get a better understanding. Mathematically a field is just a collection of numbers that obey certain rules of addition and multiplication. Physically we have gravitational fields, electromagnetic fields, and corn fields, as well as others. I would like to have a full understanding of how the physical fields are related to the mathematical fields, but as of yet it still looks like fog to me.

I see in the Wikipedia article on n-spheres (http://en.wikipedia.org/wiki/N-sphere) under the heading "Volume of the n-ball" that pi is treated as a constant. The volume of the 6-sphere is listed as [(pi ^3)/6]R^6= 5.16771....R^6. (Sorry, my latex is too rusty to use here now). Pi raised to the third power, divided by six, and multiplied by the radius of the ball. I think we don't lose anything by setting R at unity, so it drops out of the formula. Anyway, you see here they have evidently intended pi to be held constant across dimensions.

Of course you are free to change pi to an independent variable and re-calculate R as the dependent variable. I am sure a consistent reformulation could be made that way. But since this is the best exposition of higher dimensional math I have yet found, I am tempted for now to stick with the author and call pi a constant.

The Wiki author later says "It is meaningless to compare the unit-sphere volumes in different n's, just as it is meaningless to compare a length to an area to a volume." They suggest comparing a dimensionless measure, such as the ratio of the unit n-sphere to its superscribed hypercube volume.

The case we are familiar with is the 2-sphere of a 3-ball, where the ratio is found by comparing the volume of an 'ordinary' sphere (we can use the schoolroom globe as an example) with the volume of an 'ordinary' cube (like the cardboard shipping box the globe just fits into).

apeiron
Gold Member
You are talking about extending a "flat space" geometry into n-dimensions (so of course pi would be the same), and I was talking about what happens to a constant like pi in a dimensionality allowed a constant curvature (either closed and open).

See for example - http://mathcentral.uregina.ca/QQ/database/QQ.09.01/alex1.html

I agree I probably am confusing matters because the thread was about flat space geometry - the most highly symmetric case where there is the symmetry due to "flatness" and then the further symmetry due to "sphericity".

My own interest is in modelling dimensionality itself, so I am talking about the naked curvature that comes even before flatness. Where flat space constants like pi break down - where they "unwind" - along with other symmetries we tend to take for granted.