Relativity and Cosmology

Of all the various polyhedra, why shaped like a dodecahedron, specifically? Do the researchers say? At any rate, I'm sure Plato would be pleased.

Hi europium,

From what I could gather, they did try out various 3-d shapes, starting with a cylinder and worked upwards in the number of "faces". The dodecahedron apparently gave the closest fit to the WMAP data.

-J

In flat (zero-curvature) 3-D space, only certain regular solids can tessellate space (fill space with copies of themselves, leaving no gaps). The three possibilities are: the cube (a Platonic solid), the truncated octahedron (an Archimedean solid), and the rhombic dodecahedron (an Archimedean dual solid).

http://en.wikipedia.org/wiki/Truncated_octahedron

http://en.wikipedia.org/wiki/Rhombic_dodecahedron

Note that the dodecahedron (a Platonic solid) mentioned in the article can not be used to tessellate flat 3-D space (the dodecahedron has pentagonal faces, while the rhombic dodecahedron has diamond-shaped faces, so these are two different solids). Apparently the authors used distorted dodecahedra, and/or positive space curvature allows dodecahedra to do the job? (difficult to visualize this)

http://en.wikipedia.org/wiki/Dodecahedron

What a fascinating article. In addition to the cosmology, it interests me because of its relevance to cellular automata. I wrote a 3-D version of Conrad's Game of Life in which I used cubes for each cell. The program works fine and produced some very interesting patterns. Now I want to try it with rhombic dodecahedra as cells. This will be more of a challenge to code.

Hi svengali, you wrote: "Apparently the authors used distorted dodecahedra, and/or positive space curvature allows dodecahedra to do the job?"

Interesting observations! My guess is that the slight positive curvature did the trick for the authors. I agree that flat space makes it unlikely.

-J

Hi Jorrie, you wrote:

"My guess is that the slight positive curvature did the trick for the authors."

And a positive curvature of space requires that our universe be finite (since it must curve back onto itself). A very simple and beautiful solution to the problem of accounting for an infinite universe! However, the multiverse could still be infinite -- it could contain an infinite number of finite universes. But I digress.

Hi svengali, you wrote:

"And a positive curvature of space requires that our universe be finite (since it must curve back onto itself). A very simple and beautiful solution to the problem of accounting for an infinite universe!"

Agreed, but how do we define a finite, hyper-spherical (or more generally, closed) universe? One way would be to say it's finite if light can circumnavigate it in finite time. The problem is that accelerating expansion makes that impossible...

So does "finite universe" have any practical meaning?

-J

Jorrie wrote:

So does "finite universe" have any practical meaning?

Finite in mass-energy. Of course we may have no practical way of measuring this. One approach (possible in principle, but not in practice) would have us count the number and types of atoms within our universe. But how would we know when to stop looking for more atoms? I guess that the most practical method to prove we live in a finite universe would be to prove that it has positive curvature -- would this suffice?

Wasn't it Pythagorous who was fascinated by the dodecahedron??

Maybe, but historically the dodecahedron is one of a group of regular polyhedra called the Platonic Solids.

I just want to say I'm a student at OSU and have worked on a similar idea for 6 months over last summer, then an astrophysists said to drop it. I couldn't figure out how to do the math and after random nights of staying awake trying to do the math gave up. I've only done up to 3-D Calculus and am a mere sophmore. I'm happy someone else with a better background has taken up the plan. I believe the reason they pick the Dodecahedron is the reason I did, because the points on the corners and center are equidistant from each other when relativistic length contractions are taken into account with an expansion. I could be wrong though.

I didn't log in before. I forgot my login. However, I did the above post and my basis of the dodecahedron was equidistant neighboring points in 3-D space is impossible unless if there is some sort of length contraction. I figured however this worked, it could account for the expansion of the universe. I was going to try to find out the theoretical velocities needed for this to work but the Geometry with rates bugged me out. A dual pentagonal pyrimid is another possible shape I tried to do since it is only 7.36 degrees off of a closed 3-D shape with equidistant points. This would result in a pancake shaped universe though.

Jorrie,

If all the perturbations of a distant star were to suggest it had a planet with a square orbit would you accept that too?

Joe

Hi Kceum, you wrote: "If all the perturbations of a distant star were to suggest it had a planet with a square orb would you accept that too?"

No.

Who said that I'm accepting the "dodecahedron universe"?

But, what would you say "If all the perturbations of a distant star were to suggest it had a planet with a square orbit"?

I would say that it's intriguing, but rather unlikely!

-J

Very interesting!

As a youth I noticed that soap bubbles were hexagonal. I always thought that this indicated something. If the 2-dimension soap bubbles were expanded to 3-dimensions, would they be dodecahedrons or what?

StandardsGuy wrote:

As a youth I noticed that soap bubbles were hexagonal.

Yes, and note that these shapes occur in other natural systems, such as honeycombs produced by bees:

http://en.wikipedia.org/wiki/Honeycomb

Quoted from Wikipedia:

"There are two possible explanations for the reason that honeycomb is composed of hexagons, rather than any other shape. One, given by Jan Brożek, is that the hexagon tiles the plane with minimal surface area. Thus a hexagonal structure uses the least material to create a lattice of cells with a given volume. Another, given by D'Arcy Wentworth Thompson, is that the shape simply results from the process of individual bees putting cells together: somewhat analogous to the boundary shapes created in a field of soap bubbles."

In the case of soap bubbles it is to minimize surface energy.

I just remembered another interesting natural hexagon -- the hexagonal atmospheric pattern on Saturn which has puzzled astronomers recently:

http://www.jpl.nasa.gov/news/news.cfm?release=2007-034

This might have something to do with soap bubbles. Since a circle accommodates exactly 6 touching equal-radius circles around its circumference (as seen with soap bubbles), maybe the hexagon on Saturn results from the interaction of a central circular storm cell (at the pole) surrounded by 6 more storm cells?

I don't know what connection this might have to the postulated dodecahedral geometry of our universe. But it seems worth some thought.

I'm a little unclear on how this idea would work. It seems to say that space would be flat for a while and then suddenly bend, then be flat for a while again and then bend suddenly again.

Ok, I'm extremely skeptical of this idea, but for fun, here's some conjecture.

One thing that jumps out to me is that in order for space to be flat and then suddenly bend (as is required in this type of universe) it would seem to indicate a inhomogenous distribution of matter on the large scale. Also, I could be wrong here, but I think space wouldn't be isotropic at the bends. So this basically would turn the Cosmological Principle on its head.

http://en.wikipedia.org/wiki/Cosmological_principle

Hi Roger,

I suggest that the WMAP data fit because there are approximately 12 'clumps' of matter (to fit a 12 sided polygon). In the far future when there are only 4 clumps of matter, it will probably fit a cube shaped universe. What does that say about the math they are using, or the assumptions?

S

Hi Roger.

If I have it right, their model says that the closed universe ends at the edges of the dodecahedron. So, there is no change in curvature there. What we observe as being farther than the edge are images of what's on the opposite side of us, with light that wrapped around.

Anyway, I am as sceptical as you are about it, but I will not exclude it as a possibility...

-J

This sounds like a manifold. If there is an edge, then what we see as being farther away should be inverted. There's a book called The Shape of Space by Jeff Weeks which gives some good exercises for envisioning such spaces. In fact, the picture above looks very similar to the book's cover picture.

>>If there is an edge, then what we see as being farther away should be inverted.<<

Interesting point.

Jorrie, wouldn't that give us a 'parity problem'?

Joe

Hi Joe.

You asked: "Jorrie, wouldn't that give us a 'parity problem'?"

I don't understand. What do you mean by "parity problem"?

-J

Jorrie,

What I meant was by inverting every other one, it wouldn't come out right but if as you say in (22) there would be no inversion (which makes more sense) that would not be true.

I guess the fact that I don't totally understand this concept is quite obvious. Would it be fair to compare it to pattern repetition in wallpaper? If this is true then the actual demarcation lines would be arbitrary but still represent, in the case of wallpaper, squares/rectangles. It does however dictate an absolute direction. To leave earth, travel in a straight line and return (within the span of one universe), one would have to travel in one of twelve absolute directions as opposed to any direction in a good old finite curved universe.

Joe

I´m in the same ball park, The one on the edge of town where all the players try but cannot quiet connect with the ball!!

To have multiverses connected in such a manner as to form dodecahedron's, would all versi have to be the same size, thus expanding at the same rate and time? would this make all verses parallel having the same cataclysmic events happen simultaneously to keep their equilibrium or have I bashed another ball into the dirt?

Thinking about it, I am on the wrong track but, It´s food for thought!

Mr. Truman,

>>I´m in the same ball park, The one on the edge of town where all the players try but cannot quiet connect with the ball!!<<

One big advantage of not being a scientist (me) is that I can say what I want about the scientists (those other guys). They all seem to think (not you Jorrie) that if it's a theory for them it should be regarded as a fact for us. The plethora of conflicting theories (facts?) make it abundantly clear however that the majority must be wrong and if we can't hit the ball, they are not pitching in the strike zone. (that was fun!)

But anyway:

>>To have multiverses connected in such a manner as to form dodecahedron's, would all versi have to be the same size, thus expanding at the same rate and time?<<

I don't think the implication here is multiple universes but rather the same universe appearing to be in multiple places. As such if you leave one copy you reappear on the opposite side. If the universe were a cube, it would look like many copies stacked high, wide and deep. The actual boundaries between them would be completely arbitrary however as long as the repetition remained.

At least I think that's what they are trying to say???

Joe

Throw me a curve ball eh! so all the copies are the same but stacked up in 3D and you never leave the one you are in because when you leave, you arrive in the same one but on the opposite side! I wonder what the other people in the other copies think about all this! (even if the others are just a figment of my imagination!)

WHOOOOSH...STRIKE 2......can I have my club back please

I once went to a lecture by the Great Patrick Moore, I sat in the middle in the front row! I only understood about 50% of what he said and only about 20% of that made any sense!! ( If you have ever heard him speak you will understand me!)

Also if the universe is expanding, the universe that leaves one side will enter the other side? if so, how do we know it is expanding if we have an expanding universe leaving in all directions and entering in all directions surely the two would kind of counteract each other?

Until this blog I had a clear idea as to what was going on! now I´m not at all sure!

Do I qualify for little league with this or has the umpire called me out?

Hi Mr. Truman, you wrote: "Until this blog I had a clear idea as to what was going on! now I´m not at all sure!"

Take comfort in it that nobody has a very clear idea as to what is going on!

I still like the 'plain old spherical model' best. The one where if you take one space dimension away, you end up with the 'balloon model'. If the darn balloon was not blown up so rapidly, a light beam could have circumnavigated the sphere and appear from the opposite direction. Finite, but without boundary, like the surface of Earth.

Alas, due to the accelerating expansion evidence, there is virtually no difference between this model and the flat, infinitely large model. A photon will never be able to circumnavigate the rapidly expanding sphere...

-J

Hi Jorrie,

on the lines of a dodecahedron shaped universe, would the slight curvature of the faces create a kind of lensing effect making the other side of the universe look closer?

I´m with you on the balloon shaped universe. I also think that if there are other universes close by then by default these would combine without too much trouble, almost like two banks of fog! Maybe this happens on a regular basis and we just haven´t had our telescopes pointed in the right direction!

Hi Mr. Truman ,

I have no idea what slight curvature of the faces would do in the dodecahedron model. Somehow I do not believe there will be a magnifying effect, but this is pure speculation. As I understood it, there is a bit of a rotation effect, but the image of ourselves are looking farther, just like in a mirror.

-J

Hi Jorrie,

You said: "If the darn balloon was not blown up so rapidly, a light beam could have circumnavigated the sphere and appear from the opposite direction."

I'm definitely less than Little League (more like T-ball) in this arena.

If the most distant galaxies in the observable universe are receding at close to the speed of light, seems like the light that they emit would be a lot closer to circumnavigation (from our perspective). In other words since the CMB is receding at slightly LESS than c, then the light emitted by the CMB which, of course, is traveling at c should be circumnavigating the universe slightly faster than the rate of recession (expansion) and should be visible from the opposite direction. In other words, from the baloon theory, the light projected in the direction of recession should be gaining ever so slightly on itself from the opposite direction. Sorta like a cat chasing its tail.

If this is true then I guess the only problem is deciding what point in the CMB is also visible in the opposite direction.

Also the light from the CMB is also streaming rearward. Seems to me that this would more than make up for the recession rate being almost c. Thus the photons projected "forward" should easily meet up with their littermates projected "rearward", so to speak. Does this make any sense at all?

Regards,

-John

Hi Johnjohn, you wrote:

"If the most distant galaxies in the observable universe are receding at close to the speed of light, seems like the light that they emit would be a lot closer to circumnavigation (from our perspective)."

With the best measurement of Ω so close to 1 (1.01±0.01), meaning that the universe may be just closed, the present circumference of the universe may be more than 50 times that of the observable distance. Even with no expansion, it would take at least 50 times the present age of the universe to be circumnavigated by a photon.

Now add in expansion and worse (accelerated expansion) and there may be no circumnavigation ever.

The dodecahedron group postulates a universe smaller than the Hubble radius, that just looks bigger because of the "reflections", if you like. They are actually looking for "duplicates" on roughly opposite sides in the WMAP data ...

-J

Jorrie,

>>>I still like the 'plain old spherical model' best. The one where if you take one space dimension away, you end up with the 'balloon model'.<<<

The problem I keep having with that darn 'balloon model' is that if someone were a good deal away from you (around the balloon) their inertial reference frame would be tilted with respect to yours. This would seem to have the effect of decreasing their length as well as increasing their mass. Emanating light with a red shift wouldn't surprise me either. Might even cause you to think they were moving away from you. (Nahhhh!)

Joe

Hi Kceum, you wrote:

"The problem I keep having with that darn 'balloon model' is that if someone were a good deal away from you (around the balloon) their inertial reference frame would be tilted with respect to yours."

Taking into account what I wrote to JohnJohn above about the size, it means that someone "a good deal away" may be beyond our particles horizon and never to be known to us!

You must also take into account that the universal space curvature is different to the space curvature caused by a massive object in our vicinity. It is like on a perfectly spherical Earth, where there is no horizontal gravitational effect. However, on a locally sloping piece of Earth, loose things may accelerate in the vertical and horizontal directions (rolling down the slope).

AFAIK, cosmologists reckon that all isolated items that are static relative to the CMB have clocks that run at the same rate - they call it "cosmological time". Our clocks run slower than that due to various reasons, e.g., us being in a gravitational "well" and being "sucked" towards the Great Attractor at great speed.

-J

Hey guys (Johnjohn, Mr. Truman, et al.), what are the chances of Jorrie hitting us up with a Pop Quiz on this stuff later ?

Joe

Hi Kceum, I've tried a "dry-run" on Yani in my post #17 on http://cr4.globalspec.com/blogentry/2192?frmtrk=cr4sd#newcomments, but with little apparent success!

-J

Well since Yani never answered;

>>It gets a bit more interesting if the universe was created with only two objects at some distance from each other. Say one is very massive and is a rest relative to this almost empty universe at large (the CMB is isotropic for it) and the other one is a light-weight that is being dragged through space towards the heavy-weight.

If we ignore gravitational time dilation and concentrate on the velocity time dilation of special relativity, whose clock will record the most elapsed time since the BB? Easy, the one not being accelerated, of course.

But, how would they know which one is being accelerated? I will leave this as an exercise to "de-cramp" some braincells.<<

Aside from the fact that this great big thing was coming at him and getting bigger and bigger, how would the little guy even know anything was happening ?

Why would the big guy care ?

(My mother always told me never to answer a question with a question (course that was nothing compared to me answering an answer with an answer))

Joe

Hi Kceum, you wrote: "I don't think the implication here is multiple universes but rather the same universe appearing to be in multiple places."

Yep, that's what I tried to say, with the operative word in your passage being "appearing ". The multiple copies may only be apparent, while there is only one 'real' universe. At least, this is what the data seem to say. We have no idea of what lies beyond what we can observe...

-J

Hi DomerDoc.

You wrote: "If there is an edge, then what we see as being farther away should be inverted."

No, I do not think so. Just like a computer game object that flies off the left edge of the screen is not inverted when it appears again on the right side of the screen...

Have you perhaps got a link to The Shape of Space?

-J

Hi Jorrie,

Here's a link to the book at Amazon.

It's been awhile since I looked at the book, so the inversion may depend on the space. However, I seem to recall Weeks discussing the very phenomenon you mentioned regarding the video game.

I believe another example is the mobius strip. If you walk around the surface, you end up flipped (because the structure has no thickness). I believe Weeks also discusses playing tricks on your friends by taking their right shoe on such a trip while they are sleeping so they have two left shoes when they wake up. Another example I remember is playing "tag" with yourself. The hard part is that sometimes if you move towards yourself, you actually end up moving away from yourself.

Anyway, it's very confusing, but that's how I remember some of the descriptions.

Hi DomerDoc.

I had a quick look at the book that you referred to and also the one svengali referred to below.

Without reading the books, I do not think that they are talking about the same things. The illustrations may look similar, but that is pretty standard in topology discussions. The dodecahedron universe fitting the WMAP data seems to be a modern variant in the topology arena.

Nevertheless, I am not too convinced by that "fit". It could be purely coincidental.

-J

Hi Jorrie,

"I had a quick look at the book that you referred to and also the one svengali referred to below.

Without reading the books, I do not think that they are talking about the same things. The illustrations may look similar, but that is pretty standard in topology discussions."

From the table of contents, it does appear the book referenced by svengali is "standard" topology/geometry. Weeks, however, in Part IV of the book gets into the discussion of the shape of the universe and the possibility of it being closed.

As you say, the fit could be coincidental, but it is still a curious match with the "math."

"There's a book called The Shape of Space by Jeff Weeks which gives some good exercises for envisioning such spaces. In fact, the picture above looks very similar to the book's cover picture."

Have a look at the cover of this book -- seems to exactly portray the idea under discussion:

Hi svengali, it's intriguing, but do you think that picture on Thurston's cover actually portrays a dodecahedron?

-J

Jorrie,

So is my reply (#26) correct, incorrect or so totally far off the wall you can't even think of anything to say?

Joe

Hi Joe, you wrote: "So is my reply (#26) correct, incorrect or so totally far off the wall you can't even think of anything to say?"

The truth is that I do not understand this "thing" (the dodecahedron universe) well enough to tell if you are correct or not!

I read the whole http://arxiv.org/abs/astro-ph/0412569 v2, Feb. 2005 paper again and from what I can gather, the surfaces are not flat, but curved so that there are not 12 "preferred" directions. Stated differently, the surfaces are connected "seamlessly", or so it seems.

However, they do talk about 6 matched pairs of "circles in the CMB data" that, if found, may give a clue as to the correctness of the model. But I fail to comprehend much off the details...

-J

Hi svengali, it's intriguing, but do you think that picture on Thurston's cover actually portrays a dodecahedron?

-J

Yes -- looking from the inside out. It has 12 regular pentagonal faces. Seems to fit the description of a dodecahedron as I understand it. All of those "extra" edges visible in the distance are actually the same original 30 edges seen multiple times due to the closed curvature of space. As mentioned before, a person standing inside such a space could (by using a telescope) in principle see multiple images of himherself (although all of the images are of the same single person).

Here's a nice image of a dodecahedron from Wikipedia:

Hi svengali, thanks.

See my reply to Kceum, where it seems that the edges are not really there, although the multiple images may be...

I have not seen much about it in the literature lately, so maybe the model has been canned. Do you know of anything more recent than the 2005 paper?

-J

Let me start by saying that there is much I don't know regarding this subject, but I'm very interested.

Maybe a couple of questions with help my get my bearings.

1. I was under the impression that whether the universe is spherical or flat is dependent upon how the mass is distributed, since it is the mass that curves space. Is this correct or is the curvature of the universe dependent upon different factors?

2. When we say the universe is shaped like a "dodecahedron", do we mean it in the same way we would say "the universe is shaped like a "sphere"", that is I mean, are we actually saying Hypersphere and Hyperdodecahedrons?

3. The cosmological constant is cited as a way of quantifying the shape of the universe. Why does the cosmological constant tell us this?

I realize these questions may be phrased poorly, answer as best as any of you can and I'll work from there.

Really interesting discussion.

Roger

Hi Roger, attempted answers to your questions:

1. The curvature actually depends on energy density, including all elements, like mass and the cosmological constant (vacuum energy density). It is curved if the density at any time differs from the critical density for that time.

2. As I understand it, they say that the universe could be a 3-d dodecahedron, while it is just closed (over the critical energy density), meaning it is also a hypersphere. Weird, but I think that's what they say.

3. The cosmological constant does its part (~70%) in the shaping of space by adding in the extra energy to make the universe flat (Ω~1) and at the same time causing accelerated expansion, because vacuum energy density remains constant while other forms of energy decreases in density.

I know you like equations to enable understanding, so maybe you should take another look at Cosmology Equations in my CR4 Blog.

However, I would not put my "reputation" on the line by propagating a dodecahedron shaped universe - too risky, I think!

-J

Roger brings up a very good point - the mass in the universe causes space to bend. This is the essence of Einstein's theory of gravity. So since there is mass, doesn't the universe have to be closed?

Hi SG.

Further to my comments on Roger's questions: the issue you mentioned is a very common misunderstanding in cosmology.

Locally, space is bent by mass-energy. On the cosmological scale, those small "indentations" are still there, but on average the universe can be spatially flat. Just like one can treat Earth globally as a perfect sphere (or rather ellipsoid), but it has many imperfections on its surface.

-J

I´m abit out of my depth here but, if the 3D universe is a dodecahedron which is connected on all of its sides by other dodecahedron's, each of which has a slight curvature on the faces, wouldn´t that mean at every union between universes you would have a zone in the two universes? Or if the two joining surfaces were to press flat, would this not form some kind of positive pressure (even if only small) within the universes?

I may well be missing the point on this one!

Hi Mr. Truman , you asked: "... wouldn´t that mean at every union between universes you would have a zone in the two universes? Or if the two joining surfaces were to press flat, would this not form some kind of positive pressure (even if only small) within the universes?"

The dodecahedron hypothesis says nothing about multiple universes, I think! When you go out at any "edge" you just join our universe from an opposite direction. It is one universe that is 'multiply connected', whatever that may mean...

-J

Little league here, Basic answers to basic questions! (remember, little league)

Lets turn time back to the beginning if I may be so bold.

In the beginning there was nothing, then came the BB, and after that, expansion. Are we any closer to finding out what all started this? There must have been something to make that first spark, or was it the first?

In our universe, there are some v.large masses, but still only minute on the universal scale. Are there limits to the maximum size of a mass? for example, if a mass was a million times the size of the sun, how would that generally effect the universe?

This is probably basic stuff to you lot, but to me, It´s rocket science!

Hi Mr. Truman,

"There must have been something to make that first spark, or was it the first?"

See the thread: What's the shape of the Universe? 05/17/2007 6:26 PM:

The following is from post #32 of that thread by me (just rambling):

"Who knows! If we can develop time travel we may just be able to go back in time to some earlier point. I realize there's a host of well- known problems associated with backward time travel as opposed to traveling ahead in time. But, just but, if we let our imaginations soar for a moment and somehow overcome these annoyances why couldn't we go back far enough to go beyond the "big bang"? Think about it! If we went beyond the "big bang" which is where time began (for us) we would not then be doing time travel because we went beyond time itself. Wow! What a concept.

Then, theoretically, we would be in that "bigger" universe to which I earlier referred. It seems only logical that "that" universe would somehow have a time reference of some sort although it may not be anything like the time that we're accustomed to. Anyway, if we landed in that larger universe we just might be able to catch a glimpse of many "galactic universes" that had already been spawned. And even that larger universe might just be a drop in the bucket, just another everyday universe in a larger one still."

Just food for rainy day thought.

Cheers,

-John