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ENTROPY-POTENTIAL ENERGY, How it related to spacetime

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jal

Posted: Sep 28 2006, 06:07 PM

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Good Day!
A short note:
You did notice the following...

QUOTE

...that are spatially separated by distances of the order of the Planck length....

I had used the term "congruence". A better term would be "scaling".
It then leads to the understanding of Nondimensionalization
http://en.wikipedia.org/wiki/Nondimensionalization
jal

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jal	Posted: Sep 29 2006, 02:58 PM Quote
Advanced Member Group: Members Posts: 1459 Joined: 23-October 05 Positive Feedback: 64.29% Feedback Score: -15	Good day! Do you remember these drawings? Can you understand why you cannot have a Planck Sphere less than 2(3(2 pi ))? 1. You would be breaking the speed limit. 2. You would be invalidating relativity. The light cone must bottom out at a 3D spacetime unit. If you reduce the spacetime unit to less than the Planck scale then you will be going faster than the speed of light….that is science fiction. When you use scaling then that spacetime unit could be 10^-18. Don’t forget….. that is only the 2D. You must bring it to a 3d size. Lets see ….. would it also mean that the minimum size would also scale to 2 X of what we had…. 2(3(2 pi ))? (Check my math/logic) Therefore, if they don’t find the “Mexican hat” we must change out understanding/interpretation of the universe. See you at the party. jal -------------------- Moved 10 June 2008 JAL'S BLOG http://www.physicsforums.com/blog.php?b=2
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jal	Posted: Sep 29 2006, 06:44 PM Quote
Advanced Member Group: Members Posts: 1459 Joined: 23-October 05 Positive Feedback: 64.29% Feedback Score: -15	good day! WHAT HAPPENS IF THEY DON’T FIND THE MEXICAN HAT? (Which was arrived by applying uncertainty around zero/a point.) You make a PLAN “B”. There are more amateurs then professionals. You therefore, organize a party and ask everybody to come with anything that they can find that does not break the Planck scale. If there is anything that looks promising then it can be pursued further. The professionals have the abilities and tools to do more than the amateurs. Most professionals will not be interested or inclined to look for a PLAN “B”. Some will be interested. See: http://arxiv.org/PS_cache/quant-ph/pdf/0604/0604008.pdf The mathematical basis for deterministic quantum mechanics By; Gerard ’t Hooft Doing this endeavor might help the professionals and it will certainly help the amateurs. So… tell your friends about the party at http://forum.physorg.com/index.php?showtop...75&#entry127749 For 31 Oct. 2006. They might want to come for a visit. See you at the party. jal ps I sent an invitation to the "professionals" at physicsforums.com -------------------- Moved 10 June 2008 JAL'S BLOG http://www.physicsforums.com/blog.php?b=2
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jal

Posted: Sep 30 2006, 03:21 PM

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Good Day!
SQUARE DANCING
The “old guys” know how to square dance.
Imagine that instead of having 4 couples on a dance floor, that you have 6 couples (that makes 12 people) doing a square dance in space.
They weave in and out, back and forth, round and round, they move in simple patterns.
Everyone has a number of when they are suppose to do their moving. Nobody crashes into each other.
The magic cube would represent their order of movement in 3D.

Once you have done that remember that the “dancing floor” is full of groups of 12 individuals and that nobody can crash into each other and that there must always be groups of 12. Individuals can move from groups to groups as long as each group stays “full”. The snake must eats it’s tail.
Can you find the dance steps/instructions for “the spots”?
What would a dynamic Planck Sphere in 3D look like?
User posted image

New Periodic Orbits for the n-Body Problem
Cristopher Moore

http://arxiv.org/PS_cache/math/pdf/0511/0511219.pdf

QUOTE

Here we exhibit a number of new three-dimensional periodic n-body orbits with equal masses and cubic symmetry.

If you have speakers go to http://www.npr.org/templates/story/story.php?storyId=5670911 and listen to Lee Smolin, Brian Greene.
You can leave it playing in the background while you do your regular computer work.
Window media player has access to visuals…if you have never done it …open it up and do a right click on the screen …see the different lovely visuals.
“THEY” I expect that by the time THEY2 gets to college, teachers will be telling students to go to their media players and it will combine the lesson with music…and an accurate visual of how the universe is made.
Don’t you agree ….. this is fun.

See you at the party.

jal

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Confused2	Posted: Oct 3 2006, 12:37 PM Quote
Toothpaste salesman Group: Members Posts: 4755 Joined: 8-November 05 Positive Feedback: 63.3% Feedback Score: -31	Hi jal, I hope I don't come across as one of the people pretending to understand your thread .. mostly I haven't a clue what's going on . I'm really not happy with this kissing thing. To keep your space isotropic don't you need any sphere to be able to kiss any point on any adjacent sphere .. is that what you are drawing? .. Once one sphere overlaps into another sphere .. I don't think you (certainly not me) can do any proper sums with it. I've only got a limited number of brain cells.. help needed. The closest I've ever been to this sort of thing is finite element analysis .. the claim seems to be that you can do a perfect analysis (of reality) with a finite element size (Planck length/area/volume) but then geometry/topology(?) leaps out and makes it non-isotropic and wrecks it. Am I seeing things .. or not? -------- Baez again.. ok it's got nothing to do with Planck (or maybe it has), it's a very nice party trick. http://www.math.ucr.edu/home/baez/week235.html Here's a simpler example. Take a ball bearing and drop it into a wine glass. Regardless of its initial position and velocity - within reason - the ball winds up motionless at the bottom of the glass. Lots of different states seem to be converging to one state! But this isn't really true. In fact, information about the ball's position and velocity has been converted into heat: irrelevant information about the motion of atoms. In short: for a fundamentally analogue physical system to keep acting digital, it must dispose of irrelevant information, which amounts to pumping out waste heat. In fact, Rolf Landauer showed back in 1961 that getting rid of one bit of information requires putting out this much energy in the form of heat: kT ln(2) where T is the temperature and k is Boltzmann's constant. That's not much - about 3 x 10-21 joules at room temperature! But, it's theoretically important. ----------- I'm working on a 'Thick as a Planck' outfit. Best wishes, -C2.
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jal	Posted: Oct 3 2006, 06:05 PM Quote
Advanced Member Group: Members Posts: 1459 Joined: 23-October 05 Positive Feedback: 64.29% Feedback Score: -15	Hi Confused2 As usual Baez can be helpfull. I'm working on an answer to help answer the importance of "kissing numbers" and information flow. jal -------------------- Moved 10 June 2008 JAL'S BLOG http://www.physicsforums.com/blog.php?b=2
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jal

Posted: Oct 4 2006, 02:27 PM

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C2!....All
I hope that you are prepared to do some reading.
It’s a long answer.

When you read the following think “SPOT” = ANYONS
The quotes are meant to illustrate the pertinence of the material.
As the authors say…. They are still looking…. For the right model (hint…hint)
Read the papers.
http://www.sciencewatch.com/interviews/frank_wilczek1.htm

QUOTE

Wilczek:
When we think about anyons. We really must think about two-dimensional worlds.
With anyons, as opposed to just fermions and bosons, a fundamentally new possibility opens up.
It is this: there are different kinds of anyons that allow you to range continuously between bosons and fermions.
In fact, it is best to think of the different possibilities as a circle, with bosons on one side of the diameter, fermions on the other side, and anyons all around the perimeter, including fermions and bosons as a special case.
The basic idea is that you start from some known, analyzed case-fermions or bosons- and make small changes in statistics and start to move around in the circle to learn about how the system is going to behave with slight different statistics for its particles.
The most dramatic payoff of that circle of ideas, so far, has been that one gets an additional insight into the fractional quantized Hall state.
And I think that is a very important theoretical tool for understanding two-dimensional states of matter.
I think that in the next few years, in their exploration of the two-dimensional worlds, researchers will find a lot more use-many more realizations and more uses-for these ideas.
April 1991

The relevancy of anyons to spots is obvious as you can see in my drawing.
User posted image

http://online.itp.ucsb.edu/online/glasses_c03/kitaev/
Anyons in a spin model on the honeycomb lattice
Alexei Kitaev

If you do not see the similarity with what I have been saying then go back and read my thread.

http://info.phys.unm.edu/~thedude/topo/sciamTQC.pdf
Computing with Quantum Knots
By Graham P. Collins
COPYRIGHT 2006 SCIENTIFIC AMERICAN, INC.

QUOTE

Quantum computers promise to perform calculations believed to be impossible for ordinary computers. Some of those calculations are of great real-world importance. For example, certain widely used encryption methods
could be cracked given a computer capable of breaking a large number into its
component factors within a reasonable length of time. Virtually all encryption
methods used for highly sensitive data are vulnerable to one quantum algorithm
or another.
The extra power of a quantum computer comes about because it operates on
information represented as qubits, or quantum bits, instead of bits. An ordinary
classical bit can be either a 0 or a 1, and standard microchip architectures enforce that dichotomy rigorously. A qubit, in contrast, can be in a so-called superposition state, which entails proportions of 0 and 1 coexisting together. One can think of the possible qubit states as points on a sphere. The north pole is a classical 1, the south pole a 0, and all the points in between are all the possible superpositions of 0 and 1 [see “Rules for a Complex Quantum World,” by Michael A. Nielsen; Scientific American, November 2002]. The freedom that qubits have to roam across the entire sphere helps to give quantum computers their unique capabilities.
Unfortunately, quantum computers seem to be extremely difficult to build. The
qubits are typically expressed as certain quantum properties of trapped particles,
such as individual atomic ions or electrons.
A machine based on bizarre particles called anyons that represents a calculation as a set of braids in spacetime might be a shortcut to practical quantum computation

Without knowing what the structure looks like and without knowing how the structure moves they have arrived at some surprising results.
When you look at the cover picture keep in mind that the anyons/spots are NOT SPIDERS leaving a thread behind them.
The complete picture should involve 6 anyons/spots in 2D, not three, and in 3D there should be 12 anyons/spots .

For the following paper keep my Planck Scale sphere drawing in mind and you will be able to see what they have missed.
http://arxiv.org/PS_cache/hep-th/pdf/0606/0606062.pdf
Discreteness and the origin of probability in quantum mechanics
Roman V. Buniy,1, ∗ Stephen D. H. Hsu,1, † and A. Zee2, ‡
10 Aug 2006

QUOTE

We argued in Ref. [9] that quantum gravity suggests a discreteness scale of order ǫ ∼ E, where E is the characteristic energy of the system described by ψ, in Planck units. Equivalently, ǫ ∼ L−1, where L is the characteristic size, or Compton wavelength, of the system. We can motivate this result by noting that quantum gravity seems to imply a minimal length [11] of order the
Planck length.
Interestingly, for ǫ ∼ L−1, the condition that discreteness have only a small effect on Ψ, √Nǫ ≪ 1, leads to a condition on the number of degrees of freedom reminiscent of holography [13]:
N ≪ L2 ∼ A, (3) where A is the surface area of the region. This bound implies far fewer degrees of freedom than the usual extensive scaling N ∼ L3. It can be deduced as a constraint
from gravitational collapse [14]. Excluding states from the Hilbert space of the L3 volume which would have already caused gravitational collapse to a black hole, we find the stronger condition N < A3/4 ∼ L3/2.
However, we noted that discreteness of the quantum state space, even if extremely tiny, may restore the validity of the usual arguments. Some may regard discreteness as a radical proposal. We might argue that it is actually less speculative than absolute continuity, something that
can never be experimentally verified.

Here is another paper.

http://arxiv.org/PS_cache/hep-th/pdf/0605/0605052.pdf
Generic predictions of quantum theories of gravity
Lee Smolin ∗
Perimeter Institute for Theoretical Physics,
Waterloo, Ontario N2J 2W9, Canada, and
Department of Physics, University of Waterloo,
Waterloo, Ontario N2L 3G1, Canada
May 20, 2006

QUOTE

3.1 Discreteness of quantum geometry and ultraviolet finiteness

This discreteness of quantum geometry in turn implies that the theory is ultraviolet
finite. The theory has no states in which areas, volumes or lengths smaller than Planck
scale are meaningfully defined. There are consequently no modes with wavelength smaller than the Planck length. It has also been shown that for a class of theories the path integral is ultraviolet finite[4].

Therefore, they should apply the Planck Sphere as I have explained it.
Whenever they talk about a “torus” they still have not realized that the “torus” is the interior of the 6 or the 12 Planck size sphere where there cannot be any “waves” hiding.

QUOTE

3.2 Elimination of spacetime singularities

For example, as in the full field theory, there is no operator corresponding to Ai
a, rather the connection degrees of freedom (which are the variables conjugate to the spatial metric) are represented by the exponential (1).

However, He has not got the minimum size of a Planck Sphere figured out.

QUOTE

3.3 Entropy of black hole and cosmological horizons

Chern-Simons theory is used to describe anyons in 2+ 1 dimensional condensed matter
physics. The states are labeled by punctures on the two dimensional sphere which is the
spatial cross-section of the horizon. The punctures are points where the graphs attach to
the boundary, and serve also as quanta of area on the boundary. As a result of the boundary
conditions that identify the surface as a horizon, the connection is constrained to be flat
everywhere except at the punctures. The physics on a horizon is then identical to that of
a system of anyons, with the area being proportional to the total charge carried by the
anyons.

They have not got the right model and as a result the conclusions are not right.

QUOTE

3.4 Heat and the cosmological constant
For the case of 3 + 1 dimensions, this leads to A being SLq(2) with q = e 2_ı k+2 where the level k is given by[21, 29, 30]
k =6π/GA/[i]
The quantum deformation of the symmetry algebra has a simple physical meaning, at
least for _ > 0. The ground state should be deSitter spacetime, which has an horizon with an area
A =12π/[i] A
A classic result of quantum field theory in curved spacetime is that QFT’s on the
background of deSitter spacetime are thermal, with a temperature
(see formula)

If he was considering the right model the conclusions would be different.

QUOTE

4 The problem of the emergence of classical spacetime

1. Rovelli and collaborators have computed the graviton propagator in spin foam models[35]. They work in the Euclidean theory and fix a boundary, which is a four sphere, large in Planck units.

2. Freidel and Livine have computed the spin foam path integral for 2+1 gravity coupled
to matter[36].

3. Ambjorn, Jurkiewicz and Loll have constructed a simple discrete and background
independent model of spacetime, which implements discreteness and causal structure,
called the causal dynamical triangulations model[8]. They find that it has a continuum
limit which defines a theory which has a large universe limit. They can measure the
dimension of spacetime by several means and it is to within error 3 + 1.

4. Krebs and Markpoulou have proposed new criteria for the emergence of classical
spacetime in terms of quantum information theory[37]. They address the low energy
physics by asking whether there are local excitations that remain coherent in spite of
the fact that they are continually in interaction with the quantum fluctuations in the
geometry.

…..But at the same time, there is as we have described above, a discreteness scale, which is expected to be the minimal length at which a continuous geometry makes sense…..

There are then two questions. Are there consistent interacting quantum theories with
DSR symmetry? And if so, is DSR a generic prediction of background independent quantum gravity theories?
The results mentioned above by Freidel and Livine show that DSR is the correct description for quantum gravity, coupled to matter in 2 + 1 dimensional worlds[36]. This answers the first question positively. What about 3 + 1?
There are heuristic calculations that indicate that LQG in 3 + 1 dimensions has a
semiclassical approximation characterized by DSR[42]. But there is as yet no rigorous proof of this. One reason to expect a DSR theory is to notice that the symmetry group of the ground state of the theory with a non-zero cosmological constant is, by (9), the quantum deformation of the deSitter or Anti-deSitter algebra. The contraction of this is, under plausible assumptions for the scaling of the energy and momentum generators, no longer the Poincare algebra, it is the κ-Poincare algebra that characterizes DSR theories[43].

6 Conclusions
To summarize, the causal spin network theories, including loop quantum gravity and spin foam models, do a number of things that are expected of any sensible quantum theory of spacetime. They are finite, they predict that quantum geometry is discrete, they remove spacelike singularities and explain the entropy of black hole and cosmological horizons as well as the temperature of deSitter spacetime. If one adds to this that there is progress understanding whether and how classical spacetime emerges from the quantum geometry, we see that these continue to show promise as plausible models of quantum gravity. While there is certainly still much to do, the last years have given us a well defined foundation to build on.

There seems to be quite a few papers that are coming out that “inspire” me to think that what I have been presenting is being used by many people and that they are trying to apply it in their “specialties”.
It would be interesting to see more of the anyons/spot on the dance floor for the party.

See you at the party.

jal

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nobody

Posted: Oct 4 2006, 04:28 PM

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If you have speakers go to http://www.npr.org/templates/story/story.php?storyId=5670911 and listen to Lee Smolin, Brian Greene.

Excellent link!

It just so happens that I will be traveling on Friday to Colloquium 2006: Revolution in Cosmology in Albany NY where Lee Smolin and Brian Greene are two of the speakers at this weekend event, and this NPR audio is a great primer for the subject matter.

I'll be volunteering at the sign in desk for participants of colloquium if anyone from this site happens to attend.

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yquantum	Posted: Oct 4 2006, 07:43 PM Quote
Will we find the Higgs Boson? Group: Members Posts: 1376 Joined: 19-March 05 Positive Feedback: 74.19% Feedback Score: 14	nobody, jal, C2, et al, I wanted to post this before you left for the Colloquium 2006. If given the chance ask, which I am afraid you will not be able to [FAPP] Brain what physical reality is from a QM perspective* I think you just might make him blush a little. ciao_ yquantum *For All Practical Purposes -------------------- disce quasi semper victurus vive quasi cras moriturus + If you can't explain it simply, you don't understand it well enough. This document was prepared as a service to the the physic community. Neither the United States Government nor any of their employees, makes any warranty, expressed or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, product, or process disclosed, or represents that its use would not infringe privately owned rights.
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jal

Posted: Oct 5 2006, 02:26 PM

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Good Day everyone!
I was reading the published papers (HEP) for Sept. There are almost 300.
I concluded that the "big boys" are very nervous....
I got the impression that they are at the big stake roulette table. They are scurring placing bets to cover all possible out come.

It seems a bit late for publishing different versions of the same thing.
I think that there must be 4 or five versions of SUSY.

I did not find any of the high rollers betting on double zero (2D)

---------------------------
Has anyone else had an insight and done any work on “THE SPOT”?
Well…. Read the following attempt…..”THE SPOT” gets renamed to “Giant Magnons”
Is there anybody interested in writing a paper using the 4S model?
You can keep the Nobel prize. Just mention that you got inspired by “a nobody” and don’t forget my beer.

http://arxiv.org/PS_cache/hep-th/pdf/0604/0604135.pdf
Giant Magnons
Diego M. Hofmana and Juan Maldacenab∗
dhofman@Princeton.edu.
malda@ias.edu

2 May 2006

QUOTE

We also find an infinite number of two magnon bound states
at strong coupling, while at weak coupling this number is finite.
Of course, here we are considering a single magnon. Configurations with many magnons can have large excursions into the AdS directions.
Note that the fact that the magnons have a large size (are “giant”) at strong coupling is also present in the Hubbard model description in [44] 12.
Finally, let us point out that our discussion of the classical string solutions focused on an R×S2 subspace of the geometry.
The spacetime picture of these solutions varies considerably depending on parameters p and q. The easiest case to analyze is the solution corresponding to the breather at rest.
This corresponds to choosing maximal p = π. These are strings with one fixed point (ϕ − t = const) which sweep the entire sphere as they evolve in time, see figure 8. At quarter the period they look like two magnons of maximal p. The value of q controls the period of the sweep. Because P = 2p = 2π for this case there is no distance between the endpoints of the string. As we decrease p a gap opens up while the strings still sweep the sphere, see figure 9(a). At p = π
2 the gap is maximal and the solutions change character:they do not sweep the sphere any longer. For small p the solution is bounded to a small region of the sphere, see figure 9(

. q still controls the period. In appendix B,
we discuss the relevant variables and calculate the energies of these solutions. It would be nice to find more explicit expressions for the solutions.

An additional complication is that the S-matrix appears to depend on the two momenta, rather than a single variable (the center of mass momentum).

WHAT…… Say that again…. Not the center of mass?
NEED TO FIX “Giant magnons” so that they do not violate Planck scale
Acknowledgments
We would like to thank N. Beisert, S. Frolov, K. Intrilligator, J. Plefka, N. Seiberg,
M. Staudacher and I. Swanson for useful comments and discussion.
HEY…. You forgot jal
I have been around since Dec 14 2005,…… Ahhhhh you broke my heart.
You also forgot Thursday, April 20, 2006 http://motls.blogspot.com/2006/04/integrab...nt-magnons.html
After all you only published 2 May 2006

User posted image

Don’t forget….. that is only the 2D. You must bring it to a 3d size.
User posted image

See you at the party.

jal

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yquantum

Posted: Oct 5 2006, 05:17 PM

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Don’t forget….. that is only the 2D. You must bring it to a 3d size.

fapp it is necessary to explain how you can do this? This is not criticism but something you will be asked by your professor or anyone in the field.

Wave-function/Decoherence and Consistent Histories just to name one obstacles.
This is considering one as a human Geiger counters=

.

ciao_
yquantum

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+ If you can't explain it simply, you don't understand it well enough.

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Confused2	Posted: Oct 5 2006, 05:26 PM Quote
Toothpaste salesman Group: Members Posts: 4755 Joined: 8-November 05 Positive Feedback: 63.3% Feedback Score: -31	jal, y!, Thanks yquantum, that's been worrying me too. Some of the 2D solutions we've had reallly were for 2D .. graphite and copper oxide.. and may not press out nicely. Personally I'm still worried about isotropy, I don't know if the professor would ask about that. Best wishes, -C2 .
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jal

Posted: Oct 5 2006, 11:33 PM

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yquantum.... you're fishing.... I cannot spoil the party
We are going where no man has gone before.
http://arxiv.org/PS_cache/quant-ph/pdf/0604/0604008.pdf
The mathematical basis for deterministic quantum mechanics
Gerard ’t Hooft

QUOTE

In realistic theories, one therefore must impose a “superselection rule”, projecting out a subspace of Hilbert space where all energies are non-negative. How exactly to do this will be described here.

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

QUOTE

Decoherence does not provide a mechanism for the actual wave function collapse; the quantum nature of the system is simply "leaked" into the environment so that a total superposition of the wavefunction still exists, but exists beyond the realm of measurement; rather decoherence provides a mechanism for the appearance of wavefunction collapse.

Who knows what we will find...??? a mechanism ???

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

QUOTE

The interpretation based on consistent histories is used in combination with the insights about quantum decoherence. Quantum decoherence implies that only special choices of histories are consistent, and it allows a quantitative calculation of the boundary between the classical domain and the quantum domain.

In some views the interpretation based on consistent histories does not change anything about the paradigm of the Copenhagen interpretation that only the probabilities calculated from quantum mechanics and the wave function have a physical meaning. In order to obtain a complete theory, the formal rules above must be supplemented with a particular Hilbert space and rules that govern dynamics, for example a Hamiltonian.

The key words....rules that govern dynamics...lets look maybe we will find something

Confused2

QUOTE

when the properties of a material are the same in all directions, the material is said to be isotropic. For many polycrystalline materials the grain orientations are random before any working (deformation) of the material is done. Therefore, even if the individual grains are anisotropic, the property differences tend to average out and, overall, the material is isotropic.
When the properties of a material vary with different crystallographic orientations, the material is said to be anisotropic.

That should be a good toy to bring to the party.
You could explain how it can apply rather than taking the negative of why it does not apply.
yquantum since you like fishing....and I know that you're a good fisherman

.... I'm preparing a post just for you.

jal

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yquantum	Posted: Oct 5 2006, 11:53 PM Quote
Will we find the Higgs Boson? Group: Members Posts: 1376 Joined: 19-March 05 Positive Feedback: 74.19% Feedback Score: 14	jal, C2, nobody, et al, http://www.fortunecity.com/emachines/e11/86/qphil.html I am not worth the time to prepare a crack in the universe to deal with me, I am just putting some thoughts out. My intentions are not to spoil any party remember I might be funding this one. The above brings in a subject that I read about while I was away and it has given me much concern dealing with reality Good Elf if you are still around I will respond to you and I do thank you for your kind words but David Bohm and Basil Hiley was mentioned. FYI jal, I like your way of thinking -- you must have a plan B. ciao_ yquantum This post has been edited by yquantum on Oct 5 2006, 11:54 PM -------------------- disce quasi semper victurus vive quasi cras moriturus + If you can't explain it simply, you don't understand it well enough. This document was prepared as a service to the the physic community. Neither the United States Government nor any of their employees, makes any warranty, expressed or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, product, or process disclosed, or represents that its use would not infringe privately owned rights.
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Confused2

Posted: Oct 6 2006, 01:18 PM

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You could explain [of isotropy ] how it can apply rather than taking the negative of why it does not apply.

Ouch!

I was rather hoping for divine inspiration.

I don't see a nice 'kissing' solution.

Kiss all over with infinite dimensions? .. probably an irreversible step.
Kiss at random? kind of spoils the elegance .. 'boiling spots' .. sounds like a disease.

More thoughts will follow when available.

Best wishes,

-C2.

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