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Author Topic:   The Big Bang Bamma
cavediver
Member (Idle past 3643 days)
Posts: 4129
From: UK
Joined: 06-16-2005


Message 76 of 80 (265201)
12-03-2005 8:39 AM
Reply to: Message 71 by Tony650
11-29-2005 3:40 PM


Re: Visualising 4D using hypercube
So is a Klein bottle an actual cross-section or a "shadow"?
Depends on context. When I say Klein bottle, I mean the 2d mathematical entity. Others may mean the twisted, self-intersecting 3d representations of the Klein bottle.
With a tesseract, for example, there isn't really a small cube sitting inside a larger one...
I think in this case, tesseract refers to the 3d representation only. The 4d object itself is the 4d-cube, or hypercube.
Why is not worth looking at hexagons and beyond?
I don't really know. I would guess that it's because those more complex are just different combinations of the fundamental polygons?
Ok, what's the minimum number of hexagons with which you can surround a hexagon? Six sides need neighbours so six surrounding hexagons needed. Stick them together and what do you find? They tesselate (tile) flat space perfectly. There's no room to fold up the surrounding haxagons as they are already touching. Heptagons and higher will always overlap when you try to surround them with like polygons. So in flat space, pentagons are going to be the highest polygon that can be used in a regular solid.
The pentagon has five surrounding pentagons, that do not touch when laid flat. But the spaces do not allow for any more pentagons to fit in-between. So five surrounding pentagons is all you can have. Fold them up until they touch and you have the bottom half of a dodecahedron.
The square can have four surrounding squares: one for each face. It can also fit another 4 squares between those squares to form a 3 by 3 grid. But like the hexagons, this is now tiling the flat surface and there is no space to fold them up. So with just the four surrounding squares, fold them up to form the obvious box, and just add the top square: the cube.
The triangle can be surrounded by 3 triangles (one for each face), 6 triangles (1 in each gap), 9 triangles (2 in each gap), and 12 triangles (3 in each gap). The last one tiles the flat surface so has no room to fold up. The first folds up to the tetrahedron. The second folds up to the lower half of an octahedron. And the third folds up to give the lower half of the icosahedron.
And, again, I'm with you all the way up to the final step. That last one just keeps eluding me, it seems
Keep at it

This message is a reply to:
 Message 71 by Tony650, posted 11-29-2005 3:40 PM Tony650 has replied

Replies to this message:
 Message 78 by Tony650, posted 12-05-2005 8:53 PM cavediver has not replied

  
madeofstarstuff
Member (Idle past 5929 days)
Posts: 47
Joined: 08-12-2005


Message 77 of 80 (265536)
12-04-2005 7:26 PM
Reply to: Message 74 by cavediver
12-03-2005 8:24 AM


Re: More questions.
Cavediver:
There's no real processing. It's just a case of adding two functions together
What's the difference between a photon's instantaneous travel and an electron's necessity to travel through time as well, relying on the photon's information for instructions?
Here is what confuses me, from a photon's perspective it's motion is entirely dedicated to motion through space. From our perspective it is also moving, only slightly, through time as well. Is there any meaning or reality to the space in between "here" and "there" to a photon? There obviously is from our perspective, we can influence light "during" its trip. Of course then all we have really done is introduce a new instantaneous ending for the photon that it already knew about. But how could a photon from the other end of the universe instantaneously relocate to something (say a mirror in a telescope on Earth) that didn't even exist when it left?

This message is a reply to:
 Message 74 by cavediver, posted 12-03-2005 8:24 AM cavediver has not replied

  
Tony650
Member (Idle past 4032 days)
Posts: 450
From: Australia
Joined: 01-30-2004


Message 78 of 80 (265875)
12-05-2005 8:53 PM
Reply to: Message 76 by cavediver
12-03-2005 8:39 AM


Re: Visualising 4D using hypercube
cavediver writes:
Depends on context. When I say Klein bottle, I mean the 2d mathematical entity. Others may mean the twisted, self-intersecting 3d representations of the Klein bottle.
Yes, I was referring to the latter. I'm still having a hard time understanding intrinsic curvature of a flat 2D surface, so in cases like this I will generally be referring to the three-dimensional figures. In the case of the Klein bottle, I usually imagine the common 3D depictions of it. Something along these lines.
I am thinking that, if this kind of depiction is not an actual cross-section but a perspective shadow (in the same way that you can have a perspective shadow of a hypercube, showing cubical "surfaces" with distorted sizes and shapes), then the segment that appears to pass inside the bottle is actually just passing "behind" (along the fourth axis) what we see as the outside of the bottle.
That is, the narrow segment inside the 3D figure is just farther away along the fourth axis. It passes "behind" the other so that our edge-on view of its 3D shadow appears to be self-intersecting. But if we could just look "up" at the whole thing, we would see that there is no intersection... it simply loops around and connects back to itself like the Mobius strip, just with an extra dimension.
Am I on the right track?
cavediver writes:
I think in this case, tesseract refers to the 3d representation only. The 4d object itself is the 4d-cube, or hypercube.
Yes, that's what I meant. I think I'm confusing my terms. I always thought that "tesseract" was the name given to the actual four-dimensional figure, just as "cube" is the name given to the three-dimensional one. I thought that the term "hypercube" referred merely to its 3D shadow.
You're saying it's the other way around? "Hypercube" refers to the hypothetical 4D construct, and "tesseract" refers to its 3D shadow. Is that correct?
In other words, this is a tesseract, and the thing I've spent all these years trying to picture is a hypercube. Correct?
Just making sure my terminology is accurate.
Thanks for your explanation of the Platonic solids. It would appear that I actually started out correctly (trying to work out how many ways you can surround each regular polygon) but I went off on the wrong track. Anyway, your explanation makes things quite a bit clearer. Thank you!
So, four-space has the analogues to all of these plus the 24-cell. From this page...
24-Cell -- from Wolfram MathWorld
mathworld writes:
The 24-cell is self-dual, and is the unique regular convex polychoron which has no direct three-dimensional analog.
Now, help me out here. "Self-dual"? Does this refer to the same concept as dual solids? If so, is this a quirk of 4D or the 24-cell itself... or does it simply mean that it is its own dual when appropriately rotated?
Unless I'm greatly mistaken, this is the case with the tetrahedron, is it not? You take a tetrahedron, flip it upside-down, and that gives you the dual of the solid you just had, yes? Is that the same concept that the mathworld page is referring to in the case of the 24-cell?
cavediver writes:
Keep at it
Always, my friend. Thanks again for all of your help. I'm really enjoying this. I've learned a lot in this relatively short time, and I greatly appreciate the patience you display in your efforts to teach me.

This message is a reply to:
 Message 76 by cavediver, posted 12-03-2005 8:39 AM cavediver has not replied

  
Alfred Maddenstein
Member (Idle past 3967 days)
Posts: 565
Joined: 04-01-2011


Message 79 of 80 (612565)
04-17-2011 3:27 AM
Reply to: Message 49 by Tony650
11-18-2005 5:31 AM


Re: More questions.
Tony650 writes:
Hi cavediver. Let me quickly echo CS's appreciation for the time you take explaining this stuff to us poor uneducated plebs.
Ok, I think I understand your point regarding intrinsic vs. extrinsic curvature. My problem is that I simply don't understand how intrinsic curvature can exist without that other dimension.
In fact, your example is a perfect illustration of the mind jolt that I get when trying to follow this to its logical conclusion. No matter what you do, you will never get any piece of the football to lie flat. So, if we then say that the third dimension simply doesn't exist, then I can't see how the football can possibly exist. By its very nature, it can't exist in two dimensions, can it?
cavediver writes:
The football is by its very nature curved... the extra dimension just enables you to observe that curvature. There are several 2d surfaces that are curved, but 3d is not sufficient for viewing them: the klein bottle and RP2 are the two prototypes of this behaviour. Both are doughnut like surfaces, but with "mobius strip"-like twists. You cannot visualise them in 3d.
This is kind of what I'm getting at. I was under the impression that, without a fourth spatial dimension in which to exist, a true Klein bottle simply can't exist. The best we can do is to create three-dimensional "shadows" of it. Are you saying that a Klein bottle could in fact exist, even if there is no fourth dimension to contain it? It is purely a matter of the object having the necessary intrinsic curvature, even if we could never observe its extrinsic curvature?
Incidentally, I'm familiar with the Klein bottle, but what is the RP2? Can't say I've heard of that one.
cavediver writes:
Put an arrow on the north pole pointing south (obviosuly ) towards London. Slide it down to the equator along the Greenwich meridian. Now slide it sideways until it is at New York's longitude, but still on the equator. Slide it back up to the pole. It is now pointing towards New York. Conclusion - the Earth's surface is curved. No need to mention a third dimension.
I believe I understand the principle you're referring to. What I can't figure out is what it means without another dimension. I know there's no need to mention another dimension, but, in the example of the Earth's surface, it does exist.
I'm having trouble finding the words for this. I think I know what I'm trying to ask, but ugh...
How about this? I understand that the curvature of a body can be shown without referencing any dimension outside of the body, but how can the body possess that characteristic without the other dimension?
This is one of those things that, ultimately, you can really only understand mathematically, isn't it?
P.S. On re-reading my post, I think I have a better way of expressing it. Simply put, how can a body have an intrinsic curvature but no extrinsic curvature? How can one property exist without the other?
Such things as two-dimensional spheres may exist on paper only, so "scientific" papers is where all the possible expansion of the universe is confined to. In real terms though all the spatial dimensions come as a package. That is, all the three either exist at once or none of them will. Any model and 2-D representation of a sphere is a good case in point, is a static snapshot of a process that is dynamic by definition.
Dynamic means everything to do with motion while that two dimensional surface of a sphere is nothing but space and time in an ideal static representation. That ideal space is reckoning without the relative bodies in motion, the motion the space can exist only as an attribute or aspect of. Space may not exist as such and as a thing apart from the motion it is filled with and measured by. Space is only a relative quality of motion and distances are but quantitative expressions of that intrinsic aspect of motion called space. You may say that space is momentum in action. In the human, observational terms distance is the visible momentum. The question here is can anything with no depth possess any width or length? Yes, of course, it can, but as a mathematical object and on paper only. As a static representation. In real terms a square on a piece of paper is but a bottom plane of a cube that could be potentially drawn in the air above the square. The cube in the air is the foundation the plane on the paper is resting upon implicitly and it may not exist otherwise. Zero depth in reality may allow only zero angle to the other two dimensions. A thing lacking any one of the three dimensions may lack all the three necessarily. A single dimension lacking it instantly shrinks into a dimensionless point. Even such a point is again possible only as an imaginary static entity. A purely mathematical object.
Any motion started kills the point on the spot instantly filling it up with all dimensions at once. Straight line is point in motion and the reality of motion, that is all gravity after all, is such that straight lines are not sustainable for any too long tending to curve thus straight line in motion is a circle and circle in motion is a sphere.
In other words straight line is a circle at rest while a point is a motionless straight line. With rest and motion being strictly relative to each other and mutually dependent, neither may exist on its own. Any curve may need the backdrop of a plane real or imaginary to remain a curve.
Therefore all motion implies all the three dimensions, and that point outside the piece of paper as soon as it is set in motion may progressively describe all the curves intrinsic in its motion ending up as a sphere necessarily.
From that reasoning a few natural conclusions could be logically arrived at necessarily.
Time assumed as another necessary attribute of motion in space and given space and time equivalence the universe thought of as either motion in space-time or space-time in motion which are interchangeable descriptions of one and the same process, must be a moving sphere bounded by itself.
That would make it infinite and finite at once; for example bounded in space yet lacking a boundary in time or the other way round.
That, of course, precludes the very possibility of any beginning, end or centre either to space or time making all such assumptions perfectly meaningless and naive.
Neither any expansion is possible following that necessary line of thinking, for, with both space and time being the intrinsic attributes of motion inseparable from the motion itself, an expansion or accumulation of space as a function of time would mean an expansion of motion itself which is absurd as it may imply that the sphere containing all the relative motion or rather being itself that motion can possibly shrink to a point of absolute rest and that such a motionless point may be the source and a universal beginning of all motion to be expanded later.
I'd like to add a quote from Alexander Franklin Mayer's lecture as it expresses so well the unity of space and motion I was talking about.
"Mass-energy creates and shapes space like words that make up a sentence. Take away all the words (galaxies) and you have nothing, not even space that is empty. On a cosmic scale there is no such thing as 'empty space', just as there is no such thing as an 'empty sentence' that has no words."
Edited by Alfred Maddenstein, : Addition of a quote.

This message is a reply to:
 Message 49 by Tony650, posted 11-18-2005 5:31 AM Tony650 has not replied

  
Alfred Maddenstein
Member (Idle past 3967 days)
Posts: 565
Joined: 04-01-2011


Message 80 of 80 (612566)
04-17-2011 3:27 AM
Reply to: Message 49 by Tony650
11-18-2005 5:31 AM


Re: More questions.
Tony650 writes:
Hi cavediver. Let me quickly echo CS's appreciation for the time you take explaining this stuff to us poor uneducated plebs.
Ok, I think I understand your point regarding intrinsic vs. extrinsic curvature. My problem is that I simply don't understand how intrinsic curvature can exist without that other dimension.
In fact, your example is a perfect illustration of the mind jolt that I get when trying to follow this to its logical conclusion. No matter what you do, you will never get any piece of the football to lie flat. So, if we then say that the third dimension simply doesn't exist, then I can't see how the football can possibly exist. By its very nature, it can't exist in two dimensions, can it?
cavediver writes:
The football is by its very nature curved... the extra dimension just enables you to observe that curvature. There are several 2d surfaces that are curved, but 3d is not sufficient for viewing them: the klein bottle and RP2 are the two prototypes of this behaviour. Both are doughnut like surfaces, but with "mobius strip"-like twists. You cannot visualise them in 3d.
This is kind of what I'm getting at. I was under the impression that, without a fourth spatial dimension in which to exist, a true Klein bottle simply can't exist. The best we can do is to create three-dimensional "shadows" of it. Are you saying that a Klein bottle could in fact exist, even if there is no fourth dimension to contain it? It is purely a matter of the object having the necessary intrinsic curvature, even if we could never observe its extrinsic curvature?
Incidentally, I'm familiar with the Klein bottle, but what is the RP2? Can't say I've heard of that one.
cavediver writes:
Put an arrow on the north pole pointing south (obviosuly ) towards London. Slide it down to the equator along the Greenwich meridian. Now slide it sideways until it is at New York's longitude, but still on the equator. Slide it back up to the pole. It is now pointing towards New York. Conclusion - the Earth's surface is curved. No need to mention a third dimension.
I believe I understand the principle you're referring to. What I can't figure out is what it means without another dimension. I know there's no need to mention another dimension, but, in the example of the Earth's surface, it does exist.
I'm having trouble finding the words for this. I think I know what I'm trying to ask, but ugh...
How about this? I understand that the curvature of a body can be shown without referencing any dimension outside of the body, but how can the body possess that characteristic without the other dimension?
This is one of those things that, ultimately, you can really only understand mathematically, isn't it?
P.S. On re-reading my post, I think I have a better way of expressing it. Simply put, how can a body have an intrinsic curvature but no extrinsic curvature? How can one property exist without the other?
Such things as two-dimensional spheres may exist on paper only, so "scientific" papers is where all the possible expansion of the universe is confined to. In real terms though all the spatial dimensions come as a package. That is, all the three either exist at once or none of them will. Any model and 2-D representation of a sphere is a good case in point, is a static snapshot of a process that is dynamic by definition.
Dynamic means everything to do with motion while that two dimensional surface of a sphere is nothing but space and time in an ideal static representation. That ideal space is reckoning without the relative bodies in motion, the motion the space can exist only as an attribute or aspect of. Space may not exist as such and as a thing apart from the motion it is filled with and measured by. Space is only a relative quality of motion and distances are but quantitative expressions of that intrinsic aspect of motion called space. You may say that space is momentum in action. In the human, observational terms distance is the visible momentum. The question here is can anything with no depth possess any width or length? Yes, of course, it can, but as a mathematical object and on paper only. As a static representation. In real terms a square on a piece of paper is but a bottom plane of a cube that could be potentially drawn in the air above the square. The cube in the air is the foundation the plane on the paper is resting upon implicitly and it may not exist otherwise. Zero depth in reality may allow only zero angle to the other two dimensions. A thing lacking any one of the three dimensions may lack all the three necessarily. A single dimension lacking it instantly shrinks into a dimensionless point. Even such a point is again possible only as an imaginary static entity. A purely mathematical object.
Any motion started kills the point on the spot instantly filling it up with all dimensions at once. Straight line is point in motion and the reality of motion, that is all gravity after all, is such that straight lines are not sustainable for any too long tending to curve thus straight line in motion is a circle and circle in motion is a sphere.
In other words straight line is a circle at rest while a point is a motionless straight line. With rest and motion being strictly relative to each other and mutually dependent, neither may exist on its own. Any curve may need the backdrop of a plane real or imaginary to remain a curve.
Therefore all motion implies all the three dimensions, and that point outside the piece of paper as soon as it is set in motion may progressively describe all the curves intrinsic in its motion ending up as a sphere necessarily.
From that reasoning a few natural conclusions could be logically arrived at necessarily.
Time assumed as another necessary attribute of motion in space and given space and time equivalence the universe thought of as either motion in space-time or space-time in motion which are interchangeable descriptions of one and the same process, must be a moving sphere bounded by itself.
That would make it infinite and finite at once; for example bounded in space yet lacking a boundary in time or the other way round.
That, of course, precludes the very possibility of any beginning, end or centre either to space or time making all such assumptions perfectly meaningless and naive.
Neither any expansion is possible following that necessary line of thinking, for, with both space and time being the intrinsic attributes of motion inseparable from the motion itself, an expansion or accumulation of space as a function of time would mean an expansion of motion itself which is absurd as it may imply that the sphere containing all the relative motion or rather being itself that motion can possibly shrink to a point of absolute rest and that such a motionless point may be the source and a universal beginning of all motion to be expanded later.

This message is a reply to:
 Message 49 by Tony650, posted 11-18-2005 5:31 AM Tony650 has not replied

  
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