Thinking in 4 dimensions of space
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Re: Thinking in 4 dimensions of space
@Paul Richard Martin wrote:Jon,
if you know the formula for the area of surface of a sphere, which is A = 4 pi r^2
...which shows that the surface increases to the square.
As the force of gravity decreases to the same proportion,
it thus decreases to the inverse of the square.
Precisely my point!
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Jonathan Ainsley Bain forum physicist
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Re: Thinking in 4 dimensions of space
ok.
You are correct, there is a small error in the diagram,
which would be more accurately described with a sphere
instead of a flat surface.
Perhaps if you spent some time in google sketchup you could make it up?
However, in this diagram:
An effect is spread out in a similar manner.
So if it was gravity, it would be
g=m/r
for a flat 2d universe.
(yes if the edge was curved it would be more accurate)
(but it would also be more difficult to easily measure!)
so we can see that in a 2d universe g=m/r
in a 3d universe g=m/r^2
in a 4d universe g=m/r^3
5d is g=m/r^4
but not just for gravity
any effect (like light for eg) that spreads out does so
in the same proportions
in our 3d universe the intensity of light decreases by the inverse of the square
so i looked through every denominator in several thousand pages of textbooks
searching and searching for a denominator that decreases to the inverse of the cube
and i found it
You are correct, there is a small error in the diagram,
which would be more accurately described with a sphere
instead of a flat surface.
Perhaps if you spent some time in google sketchup you could make it up?
However, in this diagram:
An effect is spread out in a similar manner.
So if it was gravity, it would be
g=m/r
for a flat 2d universe.
(yes if the edge was curved it would be more accurate)
(but it would also be more difficult to easily measure!)
so we can see that in a 2d universe g=m/r
in a 3d universe g=m/r^2
in a 4d universe g=m/r^3
5d is g=m/r^4
but not just for gravity
any effect (like light for eg) that spreads out does so
in the same proportions
in our 3d universe the intensity of light decreases by the inverse of the square
so i looked through every denominator in several thousand pages of textbooks
searching and searching for a denominator that decreases to the inverse of the cube
and i found it
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Jonathan Ainsley Bain forum physicist
 Posts : 188
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Location : Africa
Re: Thinking in 4 dimensions of space
Jon,
I don't see either of your claims. I believe your first claim is correct: i.e. that in 3D space the effects emanating from a point drop off as the square of the distance from the point. This is easily seen and understood if you know the formula for the area of surface of a sphere, which is A = 4 pi r^2 (I'm too lazy and tired right now to learn how to format that formula nicely on this forum.)
Your picture, however, doesn't illustrate that fact. First of all, it isn't clear where that single square on the right is located. Doing a little geometry in your head, and assuming your lines are parallel, square, etc., it looks like it is a third of the way from the central point to the larger square on the left. And, counting the small squares in the larger one gives 9 which happens to be the square of 3. So the big square is three times as far from the source as the small one. Whatever flux goes through the small square gets diluted or reduced by a factor of 9 by the time it gets to the large square.
But the geometry of this picture has a problem because the distance from the source to the center of the large square is less than the distance from the source to any of the outside corners. This error would be negligible if the size of the square was small in comparison to the distance to the source, but why put up with that error anyway? If you simply use a sphere, then the inverse square relationship drops right out of the formula for the surface area.
Your second claim is less convincing. In the place of the sphere that I suggested for your first case you could use a torus with the wire at the center of the smaller toroidial circles (there may be a name for them but I don't know what it is). But the relationship of the surface of the torus to the radius of the small circles might be a little harder to demonstrate. I suspect it is also a power of 2 relationship, but I would have trouble proving it. The formula you cited is unfamiliar to me and your claim of four dimensions escapes my understanding.
I think you are on the right track, though. I think that both higher spatial dimensions and toroidal shapes might be important in expanding our understanding of the universe.
Paul
I don't see either of your claims. I believe your first claim is correct: i.e. that in 3D space the effects emanating from a point drop off as the square of the distance from the point. This is easily seen and understood if you know the formula for the area of surface of a sphere, which is A = 4 pi r^2 (I'm too lazy and tired right now to learn how to format that formula nicely on this forum.)
Your picture, however, doesn't illustrate that fact. First of all, it isn't clear where that single square on the right is located. Doing a little geometry in your head, and assuming your lines are parallel, square, etc., it looks like it is a third of the way from the central point to the larger square on the left. And, counting the small squares in the larger one gives 9 which happens to be the square of 3. So the big square is three times as far from the source as the small one. Whatever flux goes through the small square gets diluted or reduced by a factor of 9 by the time it gets to the large square.
But the geometry of this picture has a problem because the distance from the source to the center of the large square is less than the distance from the source to any of the outside corners. This error would be negligible if the size of the square was small in comparison to the distance to the source, but why put up with that error anyway? If you simply use a sphere, then the inverse square relationship drops right out of the formula for the surface area.
Your second claim is less convincing. In the place of the sphere that I suggested for your first case you could use a torus with the wire at the center of the smaller toroidial circles (there may be a name for them but I don't know what it is). But the relationship of the surface of the torus to the radius of the small circles might be a little harder to demonstrate. I suspect it is also a power of 2 relationship, but I would have trouble proving it. The formula you cited is unfamiliar to me and your claim of four dimensions escapes my understanding.
I think you are on the right track, though. I think that both higher spatial dimensions and toroidal shapes might be important in expanding our understanding of the universe.
Paul
Paul Richard Martin Posts : 20
Join date : 20150725
Location : Seattle, WA USA
a point not fully made
Jonathan,@Jonathan Ainsley Bain wrote:Can anyone else see that if an effect decreases to the power 2
(inverse of square law)
then it is being spread out perfectly in 3d space?
(: like gravity, light intensity, etc)
schematically:
Can anyone see how when the torus is used to hold an electric current,
its field decreases to the power 3?
how can I conclude anything other than the diagram above
is generating a 4d magnetic field?
Let me reiterate that I do not visualize 3D spaces with full understanding, and have not been able to visualize or even conceptualize higher spaces. I'm open to learning, but so far you've not shown me anything helpful. Here are some questions.
Re: your assertion that an inverse square law spreads out an effect "perfectly" in 3D space. I don't know what perfectly means in this context; i.e. it is a meaningless concept. Then, what's the effect? You use gravity as an example. But due to relativistic effects measurable in the vicinity of a large enough mass, space itself is shaped. How does this effect gravity?
A simpler law such as 1/r, or 1/r[exp 1.5] would also spread out the effect through 3D space. Likewise, inversecube or inversex laws. What's imperfect about those potential geometries, except that they don't match the models?
Your toroid example falls on (in my case) a mind that took its last e/m course in 1963 and never had occasion to apply that knowledge in the field. I don't recall what that particular inverse cube relationship means, what it applies to, or how it was derived. As I proposed before, a careful, detailed exposition would be helpful, perhaps for others as well.
Incidentally, I don't see the toroid. Your example shows half of a loop of wire that presumably continues beneath the "2D" sheet of paper sprinkled with iron filings that is two dimensions shy of a 4D example. A wire loop approximates a 2D geometry. A toroid is a different structure, and requires 3D.
As you've presented these ideas so far, your conclusions follow from your presentation only in your own mind. Perhaps you'd find better educated and more receptive minds who can understand you with less effort from them and from you, on the Physics Forums?
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Greylorn
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Location : Arizona
Re: Thinking in 4 dimensions of space
Beyond my limited knowledge of geometry, but this does seem plausible to me. https://en.wikipedia.org/wiki/Torus
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Re: Thinking in 4 dimensions of space
Can anyone else see that if an effect decreases to the power 2
(inverse of square law)
then it is being spread out perfectly in 3d space?
(: like gravity, light intensity, etc)
schematically:
Can anyone see how when the torus is used to hold an electric current,
its field decreases to the power 3?
how can I conclude anything other than the diagram above
is generating a 4d magnetic field?
(inverse of square law)
then it is being spread out perfectly in 3d space?
(: like gravity, light intensity, etc)
schematically:
Can anyone see how when the torus is used to hold an electric current,
its field decreases to the power 3?
how can I conclude anything other than the diagram above
is generating a 4d magnetic field?
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Jonathan Ainsley Bain forum physicist
 Posts : 188
Join date : 20150524
Age : 49
Location : Africa
Topological Concepts
Greylorn,
Excellent! Thank you. After apparently concluding my conversation with Jon here, I reread my posts and regretted getting into the topological implications. From Jon's responses, it was clear that he had ignored the topological aspects. Once his confusion between spaces and manifolds (although we dropped the labels) was cleared up, I thought that my mention of topology had been superfluous and a distraction. I'm excited that it struck a chord with you.
Yes, I think the topological implications are very important. They play a major role in the question of whether extra dimensions need to be rolled up (as all serious commentators that I am aware of agree) or whether they can be astronomically large, as I believe. What the scientists have concluded is that there is a special class of topological figures on which Einstein's Field Equations have solutions. These are the CalabiYau spaces.
The difficult, if not impossible, task the String Theorists have set for themselves is to find a curled up CY space that meets all their conditions. They rule out looking at large, notcurledup CY spaces for a reason that doesn't make sense in light of the ontology mentioned by Jon. Now the simplest such CY space happens to be a torus (of whatever dimensionality). I think that is what they should investigate, and using methods of visualizing them, along the lines that Jon presented, should be useful for this purpose.
Anyway, thanks for your response.
Paul
Excellent! Thank you. After apparently concluding my conversation with Jon here, I reread my posts and regretted getting into the topological implications. From Jon's responses, it was clear that he had ignored the topological aspects. Once his confusion between spaces and manifolds (although we dropped the labels) was cleared up, I thought that my mention of topology had been superfluous and a distraction. I'm excited that it struck a chord with you.
Yes, I think the topological implications are very important. They play a major role in the question of whether extra dimensions need to be rolled up (as all serious commentators that I am aware of agree) or whether they can be astronomically large, as I believe. What the scientists have concluded is that there is a special class of topological figures on which Einstein's Field Equations have solutions. These are the CalabiYau spaces.
The difficult, if not impossible, task the String Theorists have set for themselves is to find a curled up CY space that meets all their conditions. They rule out looking at large, notcurledup CY spaces for a reason that doesn't make sense in light of the ontology mentioned by Jon. Now the simplest such CY space happens to be a torus (of whatever dimensionality). I think that is what they should investigate, and using methods of visualizing them, along the lines that Jon presented, should be useful for this purpose.
Anyway, thanks for your response.
Paul
Paul Richard Martin Posts : 20
Join date : 20150725
Location : Seattle, WA USA
ah ha!
Paul,@Paul Richard Martin wrote:That is precisely the point I was trying to make.@Jonathan Ainsley Bain wrote:Of course ontologically it must have a 3rd dimension in order to bend.
Paul
You and I have had several discussions about these concepts that I never fully understood. In the course of following your conversation with Jonathan, more lights have flickered on. Bringing basic topological concepts into this was surprisingly helpful, and set me wondering if perhaps topology might be relevant to the structures found within our physical space. Something to think on.
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Greylorn
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Location : Arizona
Re: Thinking in 4 dimensions of space
That is precisely the point I was trying to make.@Jonathan Ainsley Bain wrote:Of course ontologically it must have a 3rd dimension in order to bend.
Paul
Paul Richard Martin Posts : 20
Join date : 20150725
Location : Seattle, WA USA
Re: Thinking in 4 dimensions of space
Yes, once we bend it we bring in the next dimension.
Initially we start with a 2d object,
and bend it thereby making it into a 3d object.
The bend circulates completely to complete the representation visually.
This is a normal process that should be easily followed.
Then the bend is rerepresented implying an extra dimension.
Just note that these images are all representations on your 2d monitor or
piece of paper. The final object is thus not actually a 4d object,
but a representation of such.
Just as the ordinary representation of the 3d tube, is not actually a 3d object,
but a representation of such.
When I describe a 4d object, it is implied that it therefore must have x,y,z & w axes.
The object in its 2d form, has no third dimension to it,
until we change it into a 3d object.
Of course ontologically it must have a 3rd dimension in order to bend.
Initially we start with a 2d object,
and bend it thereby making it into a 3d object.
The bend circulates completely to complete the representation visually.
This is a normal process that should be easily followed.
Then the bend is rerepresented implying an extra dimension.
Just note that these images are all representations on your 2d monitor or
piece of paper. The final object is thus not actually a 4d object,
but a representation of such.
Just as the ordinary representation of the 3d tube, is not actually a 3d object,
but a representation of such.
When I describe a 4d object, it is implied that it therefore must have x,y,z & w axes.
The object in its 2d form, has no third dimension to it,
until we change it into a 3d object.
Of course ontologically it must have a 3rd dimension in order to bend.
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Jonathan Ainsley Bain forum physicist
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Re: Thinking in 4 dimensions of space
@Jonathan Ainsley Bain wrote:There is no math difference between what you call space or manifold.
In fact there are math differences, but for our discussion we needn't care.
OK. Let's not use labels.@Jonathan Ainsley Bain wrote:Whatever label you put on it.
@Jonathan Ainsley Bain wrote:Either we require an X,Y,Z & W axis to describe the entity.
Or we do not.
True, but which is it in this case? You are not consistent.
In your post from Jul 1 2:29PM you established "the entity" we are talking about with your premise, "If we take a flat plane of two dimensions,...". That entity requires only an X and a Y axis which is inconsistent with your claim that
@Jonathan Ainsley Bain wrote:The entity has an X,Y,Z & W axis.
Which is it? If the entity has only X and Y axes, then you cannot bend it. But if your two dimensional sheet of paper is embedded in a space of higher dimensions, say a room with X, Y, and Z coordinates, then you can bend it. You could also bend it if it existed in even higher dimensional space containing, say, X,Y,Z and W axes. That is the difference I was trying to explain.
Paul
Paul Richard Martin Posts : 20
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Location : Seattle, WA USA
Re: Thinking in 4 dimensions of space
There is no math difference between what you call space or manifold.
Either we require an X,Y,Z & W axis to describe the entity.
Or we do not.
Whatever label you put on it.
The entity has an X,Y,Z & W axis.
Either we require an X,Y,Z & W axis to describe the entity.
Or we do not.
Whatever label you put on it.
The entity has an X,Y,Z & W axis.
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Jonathan Ainsley Bain forum physicist
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Re: Thinking in 4 dimensions of space
You propose a good thought exercise, Jon, but I think you go off the rails, so to speak, with your very first bend. You have muddled the distinction between a space and a manifold. Here's what I mean:
"If we take a flat plane of two dimensions" we need to be clear whether we are talking about a space or a manifold. If it is a space of two dimensions, then we can't make even the slightest bend in it so we couldn't proceed to your second step.
If, on the other hand, our two dimensional space happens to be a manifold embedded in a space of higher dimension, say in our three dimensional room, then we can bend it exactly as your recipe specifies.
When you form the cylinder, you have not really constructed a three dimensional body, as you correctly noted later. What you have is still a two dimensional manifold embedded in three dimensional space. It just happens to have a different topology than the folded but unjoined sheet of paper does.
When you fold the cylinder into a torus, you have not constructed a 4D object, or even a 3D object. It is still a 2D manifold embedded in 3D space. But you have changed the topology once again. Think of topology as how many different holes does the object have that you could tie a loop of string through.
The plain or even bent sheet of paper has no holes, so you can't tie even one loop of string around a sheet of paper (so that it stays put).
The cylinder has one hole so you can tie one loop of string through it. (The string goes into the cylinder, out the other side, and then comes back and ties together with the other end on the outside of the cylinder.) That string stays put so the count is one. If you tied a string around the outside of the cylinder, it would just slip off and wouldn't count.
The torus has two holes in it that you can tie string around. One you could do only if you are inside the torus. In that case, you run your string around the central hole and tie it to itself when you got back to your starting point. The second one is accessible only on the outside of the torus. In this case, you stick your string through the central hole, bring it up and over (or down and under, or around either side) and back again to your starting point where you tie the ends together.
Topologies are important for some purposes, but I suspect they are not for the purposes of investigating higher dimensions and visualizing structures that might exist in them. That is the subject that interests me a great deal and I have much to say about them if anyone shows the slightest interest.
Paul
"If we take a flat plane of two dimensions" we need to be clear whether we are talking about a space or a manifold. If it is a space of two dimensions, then we can't make even the slightest bend in it so we couldn't proceed to your second step.
If, on the other hand, our two dimensional space happens to be a manifold embedded in a space of higher dimension, say in our three dimensional room, then we can bend it exactly as your recipe specifies.
When you form the cylinder, you have not really constructed a three dimensional body, as you correctly noted later. What you have is still a two dimensional manifold embedded in three dimensional space. It just happens to have a different topology than the folded but unjoined sheet of paper does.
When you fold the cylinder into a torus, you have not constructed a 4D object, or even a 3D object. It is still a 2D manifold embedded in 3D space. But you have changed the topology once again. Think of topology as how many different holes does the object have that you could tie a loop of string through.
The plain or even bent sheet of paper has no holes, so you can't tie even one loop of string around a sheet of paper (so that it stays put).
The cylinder has one hole so you can tie one loop of string through it. (The string goes into the cylinder, out the other side, and then comes back and ties together with the other end on the outside of the cylinder.) That string stays put so the count is one. If you tied a string around the outside of the cylinder, it would just slip off and wouldn't count.
The torus has two holes in it that you can tie string around. One you could do only if you are inside the torus. In that case, you run your string around the central hole and tie it to itself when you got back to your starting point. The second one is accessible only on the outside of the torus. In this case, you stick your string through the central hole, bring it up and over (or down and under, or around either side) and back again to your starting point where you tie the ends together.
Topologies are important for some purposes, but I suspect they are not for the purposes of investigating higher dimensions and visualizing structures that might exist in them. That is the subject that interests me a great deal and I have much to say about them if anyone shows the slightest interest.
Paul
Paul Richard Martin Posts : 20
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Re: Thinking in 4 dimensions of space
Can you see how significant this image is:
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Jonathan Ainsley Bain forum physicist
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Thinking in 4 dimensions of space
Of course I am talking of 4 dimensions of space.
Time is another phenomenon altogether.
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Jonathan Ainsley Bain forum physicist
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