Just in case you are not entirely satisfied by the answers given in the P&A board and still interested in the basic question about
+++++++++++++++++++++++++++++
a) what we can say about the "geometrical shape"= topology of our Universe,
b) it's implications for its "ultimate fate"
c) the crucial question of infinity/finiteness
d) what is known experimentally!!
+++++++++++++++++++++++++++++
here is a very neat and well readable summary lecture by Nick Bower from the Astrophysics Dept. of U.Chicago with lots of illustrative graphics:
http://astro.uchicago.edu/home/web/olin ... nbower.htm
Don't miss the Link to Princeton:
http://www.astro.princeton.edu/~dns/nas ... s_neg.html
or to Inflation
http://home.uchicago.edu/~jmdavis1/astro.html
and many other Links to Topology Websites.
+++++++++++++++++++++++++++++
I suggest you read the main lecture first and then I'll try to explain things further if necessary.
++++++++++++++++++++++++++++++
Bye Fridger
The Topology of our Universe
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julesstoop
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Let me just get this straight.
On the one hand Fridger creates a huge fuss about not wanting to be associated with the P&A board which is in practice about 95% science and 5% irrelevant crap.
But then he goes ahead and posts educational, astronomical stuff on a board that specifically for the posting of random noise (which apparently he has no problem frequenting) but doesn't want that moved to P&A where it is actually relevant because he's not posting there anymore?
Anyone else sensing a slight inconsistency here?
. Heck with it, it makes no sense at all to me...
(no doubt Fridger will accuse me of trying to oppress him here. But I'm just saying this because it's a valid comment).
On the one hand Fridger creates a huge fuss about not wanting to be associated with the P&A board which is in practice about 95% science and 5% irrelevant crap.
But then he goes ahead and posts educational, astronomical stuff on a board that specifically for the posting of random noise (which apparently he has no problem frequenting) but doesn't want that moved to P&A where it is actually relevant because he's not posting there anymore?
Anyone else sensing a slight inconsistency here?
. Heck with it, it makes no sense at all to me...
(no doubt Fridger will accuse me of trying to oppress him here. But I'm just saying this because it's a valid comment).
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Topic authort00fri - Developer
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I am rarely acting in an inconsistent manner.
There is a big difference between a board with the serious title "Physics & Astronomy" and another one carrying the joke title "Purgatory".
As I have explained repeatedly there is a simple reason why I refused to post in the "serious P&A board" about "serious P&A matters" once "science entertainment" has been officially endorsed there. This clash does not exist if I post anything in a "fun" board with a glass of beer next to me .
There is a big difference between a board with the serious title "Physics & Astronomy" and another one carrying the joke title "Purgatory".
As I have explained repeatedly there is a simple reason why I refused to post in the "serious P&A board" about "serious P&A matters" once "science entertainment" has been officially endorsed there. This clash does not exist if I post anything in a "fun" board with a glass of beer next to me
.
t00fri wrote:I am rarely acting in an inconsistent manner.
There is a big difference between a board with the serious title "Physics & Astronomy" and another one carrying the joke title "Purgatory".
Funny, I remember when you had a fit at the idea that your Antarctic thread would be moved to Purgatory because you found it offensive...
You justify it however you like. It just seems ridiculous to me though.
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Topic authort00fri
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Malenfant wrote:t00fri wrote:I am rarely acting in an inconsistent manner.
There is a big difference between a board with the serious title "Physics & Astronomy" and another one carrying the joke title "Purgatory".
Funny, I remember when you had a fit at the idea that your Antarctic thread would be moved to Purgatory because you found it offensive...
You justify it however you like. It just seems ridiculous to me though.
Again there was a BIG difference
. It was YOU who proposed to have my thread moved to purgatory.
Also I am in good company now: Chris is also publishing in the Purgatory about Stargazing in Bolivia....After all this nonsense recently, everybody knows meanwhile that the Purgatory has become the best board in town, hi hi...
Last edited by t00fri on 23.06.2006, 23:30, edited 2 times in total.
t00fri wrote:Again there was a BIG difference
Whether it was me or anyone else makes no difference, the fact is that a thread on Antarctica had nothing to do with Celestia and had no business being on the Users board in the first place. So it was quite justifiable to ask for it to be moved to where irrelevant threads go.
But whatever. Like I said, you just carry on making up your own justifications for what you do as you always do. Meanwhile we'll just have an otherwise interesting educational thread being lost in the noise of Purgatory because you're too stubborn to just suck it up and post in the P&A board where it belongs (and regardless of what you think about the posts in there, most of them still are scientific and educational).
I am leaving this thread now.
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BlindedByTheLight
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Thanks for the post, Fridger! Time permitting me, I'm looking forward to digging in and checking it out. And thanks for being open to questions. Hopefully, I shall be able to find those answers in the "noise of purgatory"...though considering the first page goes back all the way to March 18th, I figure I'll be okay. 
Digging through the noise of why the post should in another forum maybe prove to be a more difficult matter....
Digging through the noise of why the post should in another forum maybe prove to be a more difficult matter....
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Topic authort00fri
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Distinguishing Topology and Geometry of the Universe:
++++++++++++++++++++++++++
It is crucial to understand the basic difference between Geometry and Topology of the Universe!
++++++++++++++++++++++++++
Here is a VERY simple and instructive discussion from U.Oregon of that basic point.
http://abyss.uoregon.edu/~js/cosmo/lectures/lec15.html
The Geometry is a more or less LOCAL attribute of the space while the topology tells us about its global shape AND whether the space is singly or multiply connected.
What does the latter mean?
Flat space (e.g. a plane) is simply connected, meaning there is only one direct path for light to travel from a source to an observer. But the universe might instead be "multiply connected," like a doughnut (torus), in which case there are many different such paths. An observer would see multiple images of each galaxy and could easily misinterpret them as distinct galaxies in an endless space, much as a visitor to a mirrored room has the illusion of seeing a huge crowd!!!
Note how we can make a doughnut from a plane (sheet of paper):
We simply glue together the opposite sides of the sheet: first glue together two to form a cylinder and then you glue the two ends of the cylinder together.
If you place yourself anywhere on the surface of the doughnut you can easily see that light emitted from a source can reach you along different paths along the doughnut's surface! Since the speed of light is finite, you'll see multiple images of the source corresponding to (vastly) different times!
(see the above article for more details and illustrations)
Like a hall of mirrors, the apparently endless universe might be deluding us. The cosmos could, in fact, be finite. The illusion of infinity would come about as light wrapped all the way around space, perhaps more than once--creating multiple images of each galaxy.
This latter point is highly intriguing, though: since the universe is really big, light would take billions of years to go around once and thus the different images of the same galaxy would picture it in entirely different stages of its evolution!! So when looking for mirror images of galaxies this has to be crucially taken into account.
++++++++++++++++++++++++++
I hope these simple examples give you a flavor that Frank's original question in P&A was ALL but TRIVIAL and perfectly well posed!!!
++++++++++++++++++++++++++
Bye Fridger
PS:
==
Topology is a fascinating and most important discipline in mathematics. It deals with characterizing apparently different geometrical shapes in terms of simple common attributes. So you learn that a cube and a sphere are topologically the same thing. They may be deformed into each other. A doughnut is fundamentally different from a sphere, though! It's equivalent to a sphere with a hole! That additional hole is CRUCIAL: on the surface of a sphere, EVERY inscribed loop may be contracted to a point, while this is impossible on a doughnut (why??, example? specify a loop that cannot be contracted to a point! )
Along these lines you may easily find out the basic differences between a teapot and a doghnut, for example
It is crucial to understand the basic difference between Geometry and Topology of the Universe!
++++++++++++++++++++++++++
Here is a VERY simple and instructive discussion from U.Oregon of that basic point.
http://abyss.uoregon.edu/~js/cosmo/lectures/lec15.html
The Geometry is a more or less LOCAL attribute of the space while the topology tells us about its global shape AND whether the space is singly or multiply connected.
What does the latter mean?
Flat space (e.g. a plane) is simply connected, meaning there is only one direct path for light to travel from a source to an observer. But the universe might instead be "multiply connected," like a doughnut (torus), in which case there are many different such paths. An observer would see multiple images of each galaxy and could easily misinterpret them as distinct galaxies in an endless space, much as a visitor to a mirrored room has the illusion of seeing a huge crowd!!!
Note how we can make a doughnut from a plane (sheet of paper):
We simply glue together the opposite sides of the sheet: first glue together two to form a cylinder and then you glue the two ends of the cylinder together.
If you place yourself anywhere on the surface of the doughnut you can easily see that light emitted from a source can reach you along different paths along the doughnut's surface! Since the speed of light is finite, you'll see multiple images of the source corresponding to (vastly) different times!
(see the above article for more details and illustrations)
Like a hall of mirrors, the apparently endless universe might be deluding us. The cosmos could, in fact, be finite. The illusion of infinity would come about as light wrapped all the way around space, perhaps more than once--creating multiple images of each galaxy.
This latter point is highly intriguing, though: since the universe is really big, light would take billions of years to go around once and thus the different images of the same galaxy would picture it in entirely different stages of its evolution!! So when looking for mirror images of galaxies this has to be crucially taken into account.
++++++++++++++++++++++++++
I hope these simple examples give you a flavor that Frank's original question in P&A was ALL but TRIVIAL and perfectly well posed!!!
++++++++++++++++++++++++++
Bye Fridger
PS:
==
Topology is a fascinating and most important discipline in mathematics. It deals with characterizing apparently different geometrical shapes in terms of simple common attributes. So you learn that a cube and a sphere are topologically the same thing. They may be deformed into each other. A doughnut is fundamentally different from a sphere, though! It's equivalent to a sphere with a hole! That additional hole is CRUCIAL: on the surface of a sphere, EVERY inscribed loop may be contracted to a point, while this is impossible on a doughnut (why??
, example? specify a loop that cannot be contracted to a point! )
Along these lines you may easily find out the basic differences between a teapot and a doghnut, for example
Last edited by t00fri on 24.06.2006, 10:49, edited 1 time in total.
I think there are two meanings of infinity and t00fri shows a good exemple in his first post about the geometry.
The curve can increased to infinity (e.g. f(x) = x??) or the curve increased "infinitely" to a limit that it never equals to the limit (e.g. the picture above (first t00fri 's post)).
It is difficult for me to explain it in english but I hope you understand what I want to say.
The curve can increased to infinity (e.g. f(x) = x??) or the curve increased "infinitely" to a limit that it never equals to the limit (e.g. the picture above (first t00fri 's post)).
It is difficult for me to explain it in english but I hope you understand what I want to say.
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Topic authort00fri
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Yes, Fightspit,
the next somewhat less simple mathematical issue concerns
Limiting Processes for Products of Functions f1(r) * f2(r).
++++++++++++++++++++
Notably if one factor f1(r) -> 0 tends to zero while the other one tends to infinity f2(r) -> infinity, while the independent variable r becomes large, r -> infinity
++++++++++++++++++++
Let us consider a simple example:
Let d(r) be the number density of UFOs , i.e. their number within a 3d spacial sphere of radius r around the observer. Suppose we knew how that UFO-density depends on the distance r from the observer,
e.g.
Clearly, if r increases towards infinity, d(r) tends to ZERO, since the argument of the logarithm tends to 1 and log(1) = 0.
So if we ask about the density of UFOs in an INFINITE Universe (r -> infinity) the answer is ZERO.
But suppose we want to extract the TOTAL NUMBER of UFOs in the Universe from the function d(r), we must multiply the number density d(r) with the
then tells us the number of UFOs in a spacial volume of radius r. In the limit r-> infinity, N(infinity) then gives the desired result for the whole INFINITE UNIVERSE.
Is N(infinity) ZERO, INFINITE or FINITE?
The tricky point is that
How does the PRODUCT d*V behave?
++++++++++++++++++++++++
What is 0 * infinity in this case?
++++++++++++++++++++++++
A priory the product 0*infinity is undefined. It could have any value. But for our example we can uniquely find the answer as follows:
We must write down explicity how d(r) behaves very close to where it vanishes, i.e. for VERY large r. For that purpose we can expand the logarithm in d(r) in form of a series of terms each one vanishing more rapidly than the previous one, like so
There are explicit algorithms how to find such a series expansion for a given function. Maple knows of course how to do this .
If you plot and compare the exact expression and the approximate right hand side you will find that it represents a VERY good approximation of d(r) for large enough values of r.
Now we may just multiply the series term by term with the Volume = 4/3* Pi * r^3 and perform the limit r -> infinity. Then, obviously, all terms but the first one vanish.
The result for the total number of UFOs in the infinite Universe is therefore
++++++++++++++++++++
N_universe = 4/3 * Pi * 10 ~ 42.
++++++++++++++++++++
Not a terribly dangerous number in case of this example!
These were some simple exercises how one calculates with very big and very small things in mathematics...
Bye Fridger
the next somewhat less simple mathematical issue concerns
Limiting Processes for Products of Functions f1(r) * f2(r).
++++++++++++++++++++
Notably if one factor f1(r) -> 0 tends to zero while the other one tends to infinity f2(r) -> infinity, while the independent variable r becomes large, r -> infinity
++++++++++++++++++++
Let us consider a simple example:
Let d(r) be the number density of UFOs
, i.e. their number within a 3d spacial sphere of radius r around the observer. Suppose we knew how that UFO-density depends on the distance r from the observer,
e.g.
Code: Select all
d(r) = log(1+ 10/r^3)
Clearly, if r increases towards infinity, d(r) tends to ZERO, since the argument of the logarithm tends to 1 and log(1) = 0.
So if we ask about the density of UFOs in an INFINITE Universe (r -> infinity) the answer is ZERO.
But suppose we want to extract the TOTAL NUMBER of UFOs in the Universe from the function d(r), we must multiply the number density d(r) with the
Code: Select all
(spherical) spacial volume V(r) = 4/3 Pi r^3.
Code: Select all
N(r) = d(r) * V(r)then tells us the number of UFOs in a spacial volume of radius r. In the limit r-> infinity, N(infinity) then gives the desired result for the whole INFINITE UNIVERSE.
Is N(infinity) ZERO, INFINITE or FINITE?
The tricky point is that
Code: Select all
d(r) -> 0 while V(r) -> infinity for large r!
How does the PRODUCT d*V behave?
++++++++++++++++++++++++
What is 0 * infinity in this case?
++++++++++++++++++++++++
A priory the product 0*infinity is undefined. It could have any value. But for our example we can uniquely find the answer as follows:
We must write down explicity how d(r) behaves very close to where it vanishes, i.e. for VERY large r. For that purpose we can expand the logarithm in d(r) in form of a series of terms each one vanishing more rapidly than the previous one, like so
There are explicit algorithms how to find such a series expansion for a given function. Maple knows of course how to do this
.
If you plot and compare the exact expression and the approximate right hand side you will find that it represents a VERY good approximation of d(r) for large enough values of r.
Now we may just multiply the series term by term with the Volume = 4/3* Pi * r^3 and perform the limit r -> infinity. Then, obviously, all terms but the first one vanish.
The result for the total number of UFOs in the infinite Universe is therefore
++++++++++++++++++++
N_universe = 4/3 * Pi * 10 ~ 42.
++++++++++++++++++++
Not a terribly dangerous number in case of this example
!
These were some simple exercises how one calculates with very big and very small things in mathematics...
Bye Fridger
Last edited by t00fri on 24.06.2006, 10:51, edited 5 times in total.
t00fri wrote:The result for the total number of UFOs in the infinite Universe is therefore
++++++++++++++++++++
n_universe = 4/3*Pi * 10 ~ 42.
++++++++++++++++++++
Not a terribly dangerous number in case of this example
Does anyone else think like me that it's just a little spooky that this is the same value as "the answer to life, the universe, and everything" from Douglas Adam's "Hitchhikers Guide to the Universe":!:
DISCLAIMER: Although this post may contain a question, this does not nescessarily mean that it is a quiz.
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Topic authort00fri
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Telepath wrote:t00fri wrote:The result for the total number of UFOs in the infinite Universe is therefore
++++++++++++++++++++
n_universe = 4/3*Pi * 10 ~ 42.
++++++++++++++++++++
Not a terribly dangerous number in case of this example
Does anyone else think like me that it's just a little spooky that this is the same value as "the answer to life, the universe, and everything" from Douglas Adam's "Hitchhikers Guide to the Universe":!:
Hi hi
even such hidden subtleties are eventually unravelled by the attentive readers of this Purgatory thread ...
Bye Fridger
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Topic authort00fri
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Contemplating A Finite, Multiply Connected Universe
Now that you have become topology experts I think it's about time to illuminate a bit the breathtaking possibilities arising from a possible finite, yet multiply connected spacial topology of our Universe
+++++++++++++++++++++++++++
We proceed VERY slowly and intuitively!
+++++++++++++++++++++++++++
Let me start from a 2d Universe first since there we can still draw easily what we mean.
The simplest finite and geometrically flat, singly connected 2d Universe is the blue square you can see below on the top left (Fig 1). In this flat universe the Pythagoras (triangle) theorem would clearly hold.
Look at the 2 galaxies depicted in Fig. 1), which I have adapted from the above recommended lecture
http://astro.uchicago.edu/home/web/olin ... nbower.htm
An observer in the red galaxy can perceive the light emitted from the yellow galaxy only along ONE path, the yellow one in Fig. 1)! This corresponds to the fact that our assumed space topology so far is singly connected.
Now, let us make a multiply connected 2d space out of our finite-sized Universe "sheet"! Abstractly speaking this happens if we just identify the opposite sides of our flat rectangle. Look e.g. at the white path of light emitted from the yellow galaxy! It can now reach our observer in the red galaxy: when the light hits the right-hand vertical border of our universe in Fig. 1) it reappears immediately at the opposite left-hand border as depicted (on account of the border identification). Eventually the white path hits the observer's eyes! Analogously, the red path and the top-bottom borders...
This identification of the opposite borders of our finite square Universe may sound somewhat artificial to you at first, but it is NOT at all.
Look at Figs. 2) and 3) what it actually implies. By identifying the opposite borders we have just made a nice doghnut shaped surface (space) out of our original flat rectangle! Figs. 2) and 3) illustrate the steps intuitively. You immediately see how the white yellow and red paths of light from the yellow galaxy can be perceived by our observer in the red one. That 2d doghnut Universe is now multiply connected and correspondingly more than 1 image of the yellow galaxy can be seen by the observer. Since the distance s also differ, the different galaxy images correspond to largely different evolution stages of our yellow galaxy, given the long light travelling times involved.
Of course, the doghnut surface represents a curved 2d space unlike the rectangle we have started with!
Correspondingly the Pythagoras theorem does not hold anymore for triangles inscribed on the doghnut surface.
++++++++++++++++++++++
Next we increase the spacial dimension of our finite universe by 1 and realistically consider 3d. Now it's really becoming fun!
++++++++++++++++++++++
Instead of our 2d sheet in Fig. 1) we now start from a finite cubic 3d Universe as shown here
Analogously to the procedure above , let us now make a 3d-doughnut (torus) out of the cube by identifying the opposite walls of the cube. This is indicated by the matching colors in the figure.
Unfortunately a 3d-doghnut cannot be drawn anymore but conceptionally everything is as in the 2d example above.
Still we may intuitively proceed: The identification may be simulated by replacing our identified walls by mirrors. You, the observer now stands in the center of a finite-sized mirror hall, which quite a few of you must have experienced in reality already.
To you the finite multiply connected 3d doghnut Universe appears as if it was infinite, since through the mirrors you perceive in all directions infinitely many copies of yourself and of the finite cubic "base" Universe. By looking towards the right-hand mirror wall you may e.g. see the back of your own head!
That seems to be a good point to let you contemplate a bit about the amazing implications. Please don't hesitate to ask if there is something unclear at this point.
Cheers,
Fridger
Now that you have become topology experts
I think it's about time to illuminate a bit the breathtaking possibilities arising from a possible finite, yet multiply connected spacial topology of our Universe
+++++++++++++++++++++++++++
We proceed VERY slowly and intuitively!
+++++++++++++++++++++++++++
Let me start from a 2d Universe first since there we can still draw easily what we mean.
The simplest finite and geometrically flat, singly connected 2d Universe is the blue square you can see below on the top left (Fig 1). In this flat universe the Pythagoras (triangle) theorem would clearly hold.
Look at the 2 galaxies depicted in Fig. 1), which I have adapted from the above recommended lecture
http://astro.uchicago.edu/home/web/olin ... nbower.htm
An observer in the red galaxy can perceive the light emitted from the yellow galaxy only along ONE path, the yellow one in Fig. 1)! This corresponds to the fact that our assumed space topology so far is singly connected.
Now, let us make a multiply connected 2d space out of our finite-sized Universe "sheet"! Abstractly speaking this happens if we just identify the opposite sides of our flat rectangle. Look e.g. at the white path of light emitted from the yellow galaxy! It can now reach our observer in the red galaxy: when the light hits the right-hand vertical border of our universe in Fig. 1) it reappears immediately at the opposite left-hand border as depicted (on account of the border identification). Eventually the white path hits the observer's eyes! Analogously, the red path and the top-bottom borders...
This identification of the opposite borders of our finite square Universe may sound somewhat artificial to you at first, but it is NOT at all.
Look at Figs. 2) and 3) what it actually implies. By identifying the opposite borders we have just made a nice doghnut shaped surface (space) out of our original flat rectangle! Figs. 2) and 3) illustrate the steps intuitively. You immediately see how the white yellow and red paths of light from the yellow galaxy can be perceived by our observer in the red one. That 2d doghnut Universe is now multiply connected and correspondingly more than 1 image of the yellow galaxy can be seen by the observer. Since the distance s also differ, the different galaxy images correspond to largely different evolution stages of our yellow galaxy, given the long light travelling times involved.
Of course, the doghnut surface represents a curved 2d space unlike the rectangle we have started with!
Correspondingly the Pythagoras theorem does not hold anymore for triangles inscribed on the doghnut surface.
++++++++++++++++++++++
Next we increase the spacial dimension of our finite universe by 1 and realistically consider 3d. Now it's really becoming fun!
++++++++++++++++++++++
Instead of our 2d sheet in Fig. 1) we now start from a finite cubic 3d Universe as shown here
Analogously to the procedure above , let us now make a 3d-doughnut (torus) out of the cube by identifying the opposite walls of the cube. This is indicated by the matching colors in the figure.
Unfortunately a 3d-doghnut cannot be drawn anymore but conceptionally everything is as in the 2d example above.
Still we may intuitively proceed: The identification may be simulated by replacing our identified walls by mirrors. You, the observer now stands in the center of a finite-sized mirror hall, which quite a few of you must have experienced in reality already.
To you the finite multiply connected 3d doghnut Universe appears as if it was infinite, since through the mirrors you perceive in all directions infinitely many copies of yourself and of the finite cubic "base" Universe. By looking towards the right-hand mirror wall you may e.g. see the back of your own head!
That seems to be a good point to let you contemplate a bit about the amazing implications. Please don't hesitate to ask if there is something unclear at this point.
Cheers,
Fridger
t00fri wrote:Telepath wrote:t00fri wrote:The result for the total number of UFOs in the infinite Universe is therefore
++++++++++++++++++++
n_universe = 4/3*Pi * 10 ~ 42.
++++++++++++++++++++
Not a terribly dangerous number in case of this example
Does anyone else think like me that it's just a little spooky that this is the same value as "the answer to life, the universe, and everything" from Douglas Adam's "Hitchhikers Guide to the Universe":!:
Hi hi
even such hidden subtleties are eventually unravelled by the attentive readers of this Purgatory thread ...
Bye Fridger
I tend to think thats where Mr. Adams got the number in the first place... And turned it into a sort of joke... That numbers are the basis of all things weither infinite or finite...
I'm trying to teach the cavemen how to play scrabble, its uphill work. The only word they know is Uhh and they dont know how to spell it!
t00fri wrote:Contemplating A Finite, Multiply Connected Universe
Analogously to the procedure above , let us now make a 3d-doughnut (torus) out of the cube by identifying the opposite walls of the cube. This is indicated by the matching colors in the figure.
Unfortunately a 3d-doghnut cannot be drawn anymore but conceptionally everything is as in the 2d example above.
Thank you for a fun and interesting thread, Fridger!
- rthorvald
P.S.:
Allow me, somewhat on the sideline, to point to Heinleins classic comedy - And he built a crooked house:
http://www.scifi.com/scifiction/classic ... lein1.html
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