Musings on NATURE Oct 9, 2003

Another world http://www.worldofescher.com/store/P5.html


"Dear Jack and Saul-Paul,
It is possible for a space to have zero curvature and still wrap around.
One instance in three dims is flat metric on cartesian proudct of three
circles. I suspect Dodec space has to have some curvature. Haven't had a
chance to look at the Nature articles directly yet."
Best,
Lou K.


George Ellis writes:

"the Universe is finite"

Does that contradict what Max Tegmark has written? No. See below.

"it has a specific, rather rigid topology"

He qualifies that by "if confirmed".

That the universe may be "Poincare dodecahedral space" first of all is
3D not 2D and is a matter of metric independent "topology" in sense of
the Felix Klein's 1872 "Erlanger Program" like a layered artichoke where
geometry in the generalized sense is defined as the invariants of a
given group of transformations between different frames of reference or
perspectives or points of view as it were. Topology transformations are
continuous transformations without cutting of holes of a continuum that
leave certain global structures invariant and which do not at all
respect the ds^2 = guv(x)dx^udx^v geometrodynamic invariant of
Einstein's two relativity theories. There is also the issue of
topology-changing catastrophes in the above context with cutting of
holes. One can also consider discretums rather than continuums with
finite groups of transformation which also arise in spaces that are
defined using Galois fields of integers mod primes. Hermann Weyl showed
a deep relation between discretums and continuums in his book on group
theory and quantum theory.

"What is its [the universe's] spatial curvature?"

There are only 3 discrete possibilities. Restriction to "large scale"
understood i.e. 10 megaparsecs

k = 0 spatially flat like high school geometry where only one parallel
to a line through a point not on the given line and never intersect

k = -1 negative curvature where parallel lines diverge from each other
and never intersect

k = +1 positive curvature where parallel lines converge and intersect
like the great circle geodesic longitudes on the surface of a sphere
that intersect at the two poles.

"geodesic" is the generalized "straight" for curved spaces.

Inflation cosmology is like Handel's Messiah tenor aria "And the rough
places plane" on the large scale of space. :-)

Scale-dependence needs the "mother wavelet transform" generalization of
the "Fourier transform". Modern quantum field theory and string theory
are too dependent on the Fourier transform for propagators that define
the difference between vacua (exotic or not) and real particles of
matter and radiation.

The value of k "depends on how well the amount of matter in the
Universe, coupled with the driving force of dark energy, balances the
Universe's kinetic energy of expansion. This is usually expressed in
terms of the normalized density parameter Omega zero, which is unity for
flat space sections, for positive spatial curvature Omega zero is
greater than 1"

"The second question is whether the universe is 'open' or 'closed' --
that is, is spatially infinite, containing an infinite amount of matter,
or is spatially finite containing a finite amount of matter? Positively
curved space sections are necessarily closed, but the converse does not
necessarily follow: both flat and negatively curved space sections can
be finite if their connectivity is more complicated than in Euclidean
space, meaning that their topology is quite unusual."

Enter the metrically engineered exotic vacuum Star Gate walled
boundaries of space? See me talk about this in "Time Travel: The Art Of
The Possible"
http://www.dvdworldonline.com/DVD/Sc...yageHomeSE.asp

"For example in a flat toroidal space, as you exit right you enter left,
and space is finite."

This means the 2D boundary wall is an exotic vacuum /\zpf =/= 0 Star
Gate portal so that space is something like an Escher drawing or a giant
Pong Video Game in the sky.

To be continued.

On Saturday, October 11, 2003, at 11:03 AM, Jack Sarfatti wrote:

Thanks Saul-Paul.
I have the 9 Oct 2003 issue of Nature on this in front of me and
hopefully I will soon be able to understand what Ellis is actually
proposing here in some detail. :-)
George Ellis has a paper with references on p. 566 and more detailed
paper on p. 593 by Luminet et-al so that should clear this all up soon.
He is proposing a "finite" universe so how to square that circle with
the "spatially flat" universe is the issue. I suppose that's where the
periodic boundary conditions come in? It's like we are living inside a
big spatial box that is expanding and accelerating but that we come to
an "edge" i.e. a 2D wall and get instantly transported to an opposite
wall many billions of light years on the opposite side of the universe?
Rather like we are simulations in the "Destiny Matrix" Cosmic Computer
Program after all? :-)

On Saturday, October 11, 2003, at 08:42 AM, S-P & M-M Sirag wrote:

Jack,

I'm forwarding this message on the POSSIBLY dodecahedral universe from my
friend Lou Kauffman, who is a well known topologist. This is the best
information I now have on this new wrinkle in cosmololgy, since I have yet
to see the 8 October issue of *Nature*.

Be sure and check out the jpg's.

For me the most intriguing aspect of this development is that the universe
could have the topology of an A-D-E classified gravitational instanton,
especially of type E6, E7, or E8. The "infinity" of these three instantons
are, respectively, SU(2)/TD, SU(2)/OD, and SU(2)/ID. Note that the topology
of SU(2)/ID is exactly that of a dodecahedron with opposite faces
identified.

Yes, interesting indeed. I need to understand in some detail how Ellis
et-al make those inferences from the statistics of the WMAP data. One
thing I am confused on is are they talking about 2D boundaries of 3D
spacelike geometries? By "faces" do you mean literally 2D or a 3D
generalization?


Also it should be noted that these gravitational instantons (C^2/TD, C^2/OD,
and C^2/ID) are also the identity fibers of the catastrophe bundle of types
E6, E7, and E8.

If one wonders where these catastrophe bundles are geometrically located,
the answer is: inside the Lie algebras of types E6, E7, and E8, which are
vector spaces of (complex) dimensions (respectively) 78, 133, and 248.

Since E8 x E8 is also implicated in the heterotic string theory and E7 is
implicated in membrane theory, this development in cosmology MAY have
something to say about string theory.

What I need to understand is how all this powerful math relates to the
relatively simple math of the standard large scale cosmology of the ~
isotropic homogeneous FRW metric with /\ term that the precision
cosmologists use to analyze the WMAP data and come to conclusions like k
= 0 (universe is spatially flat on large scale).

For more information see my paper "Notes on Hyperspace" which I have as a
pdf, and also my 1998 paper "Hyperspace Platonics" (which unfortunately I
don't have in pdf form).

Send me the word.doc for Mac and I will make it pdf.


Note also that the gravitational instanton idea is closely related to the
Penrose twistor space. See especially *Twistor Geometry and Field Theory* by
R.S. Ward and Raymond O. Wells Jr (Cambridge, 1990). See also P.B.
Kronheimer's papers J. Differential Geometry 29, 685-698 (1989); 32, 473-490
(1990). Peter Kronheimer's Ph.D. thesis (advised by Michael Atiyah at
Oxford) proved in detail the A-D-E classification of a class of
gravitational instantons.

I hope you can see why I continue to study what I call ADEX theory, the
application of the A-D-E series of Coxeter graphs to mathematics and
physics. Incidentally the X stands for the vast object (reality, I think)
which underlies the entire set of A-D-E classifications. The X also stands
for the classification of objects beyond the A-D-E series. In fact the three
E graphs (E6, E7, & E8) are doorways into this set of graphs beyond A-D-E.
There is an especially interesting X9 graph which abuts directly to E7 (by
adding two nodes).

This X9 graph is: *--*--*--*--*--*--*
|
*
|
*

More on this later.


Saul-Paul



----------
From: Louis H Kauffman
To: RWGRAY
Subject: the shape of the universe
Date: Sat, Oct 11, 2003, 12:13 AM


Dear Folks,
I have included some images and information from the web as three jpegs.
The Nature article is in fact coauthored by Jeff Weeks.
Best,
Lou

On Fri, 10 Oct 2003, Louis H Kauffman wrote:

Dear Bob,
The issue of structure of 3d universe as a whole is very interesting.
In the abstract it is certainly possible for space to curve back on itself
so that if one goes far enough "out" then one comes back to the starting
point, just like going around the world.

Yes, but it is my understanding from March and April APS and April 2003
Physics Today et-al that this possibility has been ruled out by the
actual data from WMAP, type 1a supernovae etc? Is George Ellis
challenging this? Or, is what he proposes somehow consistent with it. I
thought the data confirms k = 0 FRW metric consistent with chaotic
inflationary models (Max Tegmark May 2003 Scientific American)?


We understand this effect for
surfaces partly because we can stand outside them and see they have this
property (like looking at the modes of circulation on a torus), but also
because we can imagine a situation with no boundary but continual
periodicity like a tv game where the objects move off screen left and
reapear on screen right. That is the sort of description that the
dodecahedral "identify by 1/5 twist the opposite faces" description uses.
In that description a ray of light heading for one face flies right
through and comes in again via the opposite face. But the "faces" are not
there. That is just a way to describe the periodicity, and it is traces of
this periodicity that are supposed to be inherent in the astronomical
data.

Oh, this helps. What I get from this is that

1) they are talking about 3D spacelike geometries.

Are these "faces" like 2D boundaries that are like star gate portals?
You pass through the 2D wall in your starship and instantly come out
many light years across to the opposite edge of the 3D universe? Is that
the picture that we are trapped inside a finite box with mirror faces
that are star gates. The "box" is spatially flat on large scale
consistent with k = 0? Or am I misunderstanding here? If so, these 2D
boundary faces or "walls" in space would have to be exotic vacuum for sure.

(I don't think the universe is small enough for any direct evidence
of the periodicity. It must be quite indirect evidence that these guys are
using, and this is going to make the result quite speculative.) So one
wants more global descriptions of the space. One way to get a global
description is to describe the manifold as a locus of solutions to systems
of equations in a higher dimesional Euclidean space. In the case of the
dodec space there is such a description of it in terms of equations in 6
dimensional Euclidean space written as 3 complex variables (X,Y,Z). Note
that each complex variable has two real coords, making the space 6 diml.
The dodec space is then described as the locus of solutions to the
equations:

X^2 + Y^3 + Z^5 = 0
|X|^2 + |Y|^2 + |Z|^2 = 1.

Of course, if the universe were dodec space, this does not mean that the
geometric universe is actually sitting inside flat 6 space, but that could
be a useful way to model it. In general, if we are going to think about a
boundaryless closed universe we might as well represent its curvature and
dynamics in terms of embedding in some higher dimensional space. Theorems
of John Nash and others tell us that abstract manifolds can be so modeled,
and such models were the starting point for studying manifolds to begin
with. Now for the sake of Science Fiction we have to ask, if Universe is
really inside a 6 (or higher) space, then can't we figure out how to get
off the Universe into the higher dims, just like we got off the surface of
the earth into outer space. Here we have the next stage in space
exploration: Voyages into higher space! The the vibrations of our familiar
Universe will be seen as part of the panoply of quantum vibrating string
and higher dimensional membrane interactions in the higher space. And THAT
space, what if that space is itself closed and bounded and suffering
dynamics in an even higher dimension... Fractal seeds of an infinite
hierarchy of membrane universes. And we need not stop at finite
dimensions. Let them all be enfolded in infinite and unfolding dimensions
extending to the uncountable cardinals, the incaccessible cardinals and
beyond, beyond logic, beyond paradox beyond conceptualization into the
absolute.... And think of it man, all spiralling up out of void, into
self-reference, distinction, triality, the trefoil knot, the quintessence
of the trefoil knot, the dodec space, the enfoldment of the dodec in
perfect six and ... But we seem to have reentered the periodicity. Time
does not have a stop, but this letter does stop. Here.
Love,
Lou

By Dr David Whitehouse
BBC News Online science editor



Leonardo da Vinci had the right idea
More precisely, we may inhabit a dodecahedral cosmos. It is, according to
the scientists, the best way to account for the latest satellite
observations.

Dodecahedrons, and similar shapes, have long fascinated mankind. Plato
believed that the Universe was made up of them.

Leonardo da Vinci also studied them, as did the great astronomer Kepler, who
thought the structure of the Solar System was based on geometrical shapes.

Further observations, especially from space probes yet to be launched, may
settle the matter, and may at last reveal the hidden geometry of the
Universe.

Ripples in the sky

The Cosmic Microwave Background (CMB) radiation, the "echo" of the Big Bang,
contains a wealth of data about the early history of the Universe, as well
as its large-scale structure.

If only we had precise enough observations of it to discriminate between
competing ideas of what the Universe is like.


The scientists were writing in Nature magazine
Will it expand forever? Is space infinite? Such profound questions may have
their answers in the CMB.

Specifically, the answers may be found in the ripples in the CMB -
miniscule, regular, fluctuations in its strength over the sky.

Data from the US space agency's Wilkinson Microwave Anisotropy Probe (WMAP),
which maps the CMB, suggests that at the very largest scales its temperature
fluctuations seen across the sky are smaller than would be produced by an
infinite Universe.

It seems the WMAP data shows the Universe is too small for large
fluctuations to be seen in the microwave background radiation.

Positively curved space sections

Astronomers from the US and France suggest that space itself is not big
enough to support such waves.

A small, cosmologically speaking, finite Universe, however, made of curved
pentagons joined together into a sphere, would fit the observations.


The answer could be in the CMB
Writing a commentary in Nature, George Ellis of the University of Cape Town,
says we live in a Universe "with positively curved space sections and a
non-standard topology".

Indeed, a dodecahedral Universe, were you able to traverse it, would have
some interesting properties.

If you went out to the edge of the dodecahedron, you would come back in
through the opposite face.

More precise observations made by WMAP and by its successor, the Planck
satellite, to be launched in 2007, will tell scientists if the cosmos does
have such a shape, or if it is even stranger.