The Sound Of Billiard Balls

Intuition and Logic in Mathematics by Henri Poincaré

On the other hand, look at Professor Klein: he is studying one of the most abstract questions of the theory of functions to determine whether on a given Riemann surface there always exists a function admitting of given singularities. What does the celebrated German geometer do? He replaces his Riemann surface by a metallic surface whose electric conductivity varies according to certain laws. He connects two of its points with the two poles of a battery. The current, says he, must pass, and the distribution of this current on the surface will define a function whose singularities will be precisely those called for by the enunciation.

It is necessary to see the stance Poincaré had in relation to Klein, and to see how this is being played out today. While I write here I see where such thinking moved from a fifth dimensional perspective, has been taken down "to two" as well as" the thinking about this metal sheet.

So I expound on the virtues of what Poincaré saw, versus what Klein himself was extrapolating according to Poincaré's views. We have results today in our theoretics that can be describe in relation. This did not take ten years of equitation's, but a single picture in relation.

Graduating the Sphere

Imagine indeed a inductive/deductive stance to the "evolution of this space" around us. That some Klein bottle "may be" the turning of the "inside out" of our abstractness, to see nature is endowably attached to the inside, that we may fine it hard to differentiate.I give a example of this In a link below.



While I demonstrate this division between the inner and outer it is with some hope that one will be able to deduce what value is place about the difficulties such a line may be drawn on this circle that we may say how difficult indeed to separate that division from what is inside is outside. What line is before what line.

IN "liminocentric structure" such a topology change is pointed out. JohnG you may get the sense of this? All the time, a psychology is playing out, and what shall we assign these mental things when related to the sound? Related to the gravity of our situations?

You see, if I demonstrate what exists within our mental framework, what value this if it cannot be seen on the very outskirts of our being. It "cannot be measured" in how the intellect is not only part of the "sphere of our influence" but intermingles with our reason and emotive conduct. It becomes part of the very functions of the abstractness of our world, is really set in the "analogies" that sound may of been proposed here. I attach colour "later on" in how Gravity links us to our world.

I call it this for now, while we continued to "push for meaning" about the nature of space and time.

Savas Dimopoulos

Here’s an analogy to understand this: imagine that our universe is a two-dimensional pool table, which you look down on from the third spatial dimension. When the billiard balls collide on the table, they scatter into new trajectories across the surface. But we also hear the click of sound as they impact: that’s collision energy being radiated into a third dimension above and beyond the surface. In this picture, the billiard balls are like protons and neutrons, and the sound wave behaves like the graviton.

While of course I highlighted an example of the geometer in their visual capabilities, it is with nature that such examples are highlighted. Such abstractness takes on "new meaning" and settles to home, the understanding of all that will ensue.

The theorems of projective geometry are automatically valid theorems of Euclidean geometry. We say that topological geometry is more abstract than projective geometry which is turn is more abstract than Euclidean geometry.

It is of course with this understanding that "all the geometries" following from one another, that we can say that such geometries are indeed progressive. This is how I see the move to "non-euclidean" that certain principals had to be endowed in mind to see that "curvatures" not only existed in the nature of space and time, that it also is revealed in the Gaussian abstracts of arcs and such.

These are not just fixations untouched abstractors that we say they have no home in mind, yet are further expounded upon as we set to move your perceptions beyond just the paper and thought of the math alone in ones mind.

Felix Klein on intuition

It is my opinion that in teaching it is not only admissible, but absolutely necessary, to be less abstract at the start, to have constant regard to the applications, and to refer to the refinements only gradually as the student becomes able to understand them. This is, of course, nothing but a universal pedagogical principle to be observed in all mathematical instruction ....

I am led to these remarks by the consciousness of growing danger in Germany of a separation between abstract mathematical science and its scientific and technical applications. Such separation can only be deplored, for it would necessarily be followed by shallowness on the side of the applied sciences, and by isolation on the part of pure mathematics ....

Felix Christian Klein (April 25, 1849 – June 22, 1925) was a German mathematician, known for his work in group theory, function theory, non-Euclidean geometry, and on the connections between geometry and group theory. His 1872 Erlangen Program, classifying geometries by their underlying symmetry groups, was a hugely influential synthesis of much of the mathematics of the day.

See:

Inside Out
No Royal Road to Geometry?

The Geologist and the Mathematician

In an ordinary 2-sphere, any loop can be continuously tightened to a point on the surface. Does this condition characterize the 2-sphere? The answer is yes, and it has been known for a long time. The Poincaré conjecture asks the same question for the 3-sphere, which is more difficult to visualize.

On December 22, 2006, the journal Science honored Perelman's proof of the Poincaré conjecture as the scientific "Breakthrough of the Year," the first time this had been bestowed in the area of mathematics

I have been following the Poincaré work under the heading of the Poincaré Conjecture. It would serve to point out any relation that would be mathematically inclined to deserve a philosophically jaunt into the "derivation of a mind in comparative views" that one might come to some conclusion about the nature of the world, that we would see it differences, and know that is arose from such philosophical debate.

Poincaré, almost a hundred years ago, knew that a two dimensional sphere is essentially characterized by this property of simple connectivity, and asked the corresponding question for the three dimensional sphere (the set of points in four dimensional space at unit distance from the origin). This question turned out to be extraordinarily difficult, and mathematicians have been struggling with it ever since.

Previous links in label index on right and relative associative posts point out the basis of the Poincaré Conjecture and it's consequent in developmental attempts to deduction about the nature of the world in an mathematical abstract sense?

Jules Henri Poincare (1854-1912)

The scientist does not study nature because it is useful. He studies it because he delights in it, and he delights in it because it is beautiful.

HENRI POINCARE

Mathematics and Science:Last Essays

8 Last Essays

But it is exactly because all things tend toward death that life is
an exception which it is necessary to explain.

Let rolling pebbles be left subject to chance on the side of a
mountain, and they will all end by falling into the valley. If we
find one of them at the foot, it will be a commonplace effect which
will teach us nothing about the previous history of the pebble;
we will not be able to know its original position on the mountain.
But if, by accident, we find a stone near the summit, we can assert
that it has always been there, since, if it had been on the slope, it
would have rolled to the very bottom. And we will make this
assertion with the greater certainty, the more exceptional the event
is and the greater the chances were that the situation would not
have occurred.

How simple such a view that one would speak about the complexity of the world in it's relations. To know that any resting place on the mountain could have it's descendants resting in some place called such a valley?

Stratification and Mind Maps

Pascal's Triangle

By which path, and left to some "Pascalian idea" about comparing some such mountains in abstraction to such a view, we are left to "numbered pathways" by such a design that we can call it "a resting" by nature selection of all probable pathways?


Diagram 6. Khu Shijiei triangle, depth 8, 1303.

The so called 'Pascal' triangle was known in China as early as 1261. In '1261 the triangle appears to a depth of six in Yang Hui and to a depth of eight in Zhu Shijiei (as in diagram 6) in 1303. Yang Hui attributes the triangle to Jia Xian, who lived in the eleventh century' (Stillwell, 1989, p136). They used it as we do, as a means of generating the binomial coefficients.

It wasn't until the eleventh century that a method for solving quadratic and cubic equations was recorded, although they seemed to have existed since the first millennium. At this time Jia Xian 'generalised the square and cube root procedures to higher roots by using the array of numbers known today as the Pascal triangle and also extended and improved the method into one useable for solving polynomial equations of any degree' (Katz, 1993, p191.)

Even the wisest of us does not realize what Boltzmann in his expressions would leave for us that such expression would leave to chance such pebbles in that valley for such considerations, that we might call this pebble, "some topological form," left to the preponderance for us in our descriptions to what nature shall reveal in those same valleys?

The Topography of Energy Resting in the Valleys

The theory of strings predicts that the universe might occupy one random "valley" out of a virtually infinite selection of valleys in a vast landscape of possibilities

Most certainly it should be understood that the "valley and the pebble" are two separate things, and yet, can we not say that the pebble is an artifact of the energy in expression that eventually lies resting in one of the possible pathways to that energy at rest.

The mountain, "as a stratification" exists.



Here in mind then, such rooms are created.

The ancients would have us believe in mind, that such "high mountain views do exist." Your "Olympus," or the "Fields of Elysium." Today, are these not to be considered in such a way? Such a view is part and parcel of our aspirate. The decomposable limits will be self evident in what shall rest in the valleys of our views?

Such elevations are a closer to a decomposable limit of the energy in my views. The sun shall shine, and the matter will be describe in such a view. Here we have reverted to such a view that is closer to the understanding, that such particle disseminations are the pebbles, and that such expressions, have been pushed back our views on the nature of the cosmos. Regardless of what the LHC does not represent, or does, in minds with regards to the BIG Bang? The push back to micros perspective views, allow us to introduce examples of this analogy, as artifacts of our considerations, and these hold in my view, a description closer to the source of that energy in expression.

To be bold here means to push on, in face of what the limitations imposed by such statements of Lee Smolin as a statement a book represents, and subsequent desires now taken by Hooft, in PI's Status of research and development.

It means to continue in face of the Witten's tiring of abstraction of the landscape. It means to go past the "intellectual defeatism" expressed by a Woitian design held of that mathematical world.

Stanley Mandelstam

Research Interests

My research concerns string theory. At present I am interested in finding an explicit expression for the n-loop superstring amplitude and proving that it is finite. My field of research is particle theory, more specifically string theory. I am also interested in the recent results of Seiberg and Witten in supersymmetric field theories.

Current Projects

My present research concerns the problem of topology changing in string theory. It is currently believed that one has to sum over all string backgrounds and all topologies in doing the functional integral. I suspect that certain singular string backgrounds may be equivalent to topology changes, and that it is consequently only necessary to sum over string backgrounds. As a start I am investigating topology changes in two-dimensional target spaces. I am also interested in Seiberg-Witten invariants. Although much has been learned, some basic questions remain, and I hope to be able at least to understand the simpler of these questions.

http://www.physics.berkeley.edu/research/faculty/mandelstam.html



Stanley Mandelstam (b. 1928, Johannesburg) is a South African-born theoretical physicist. He introduced the relativistically invariant Mandelstam variables into particle physics in 1958 as a convenient coordinate system for formulating his double dispersion relations. The double dispersion relations were a central tool in the bootstrap program which sought to formulate a consistent theory of infinitely many particle types of increasing spin.

Mandelstam, along with Tullio Regge, was responsible for the Regge theory of strong interaction phenomenology. He reinterpreted the analytic growth rate of the scattering amplitude as a function of the cosine of the scattering angle as the power law for the falloff of scattering amplitudes at high energy. Along with the double dispersion relation, Regge theory allowed theorists to find sufficient analytic constraints on scattering amplitudes of bound states to formulate a theory in which there are infintely many particle types, none of which are fundamental.

After Veneziano constructed the first tree-level scattering amplitude describing infinitely many particle types, what was recognized almost immediately as a string scattering amplitude, Mandelstam continued to make crucial contributions. He interpreted the Virasoro algebra discovered in consistency conditions as a geometrical symmetry of a world-sheet conformal field theory, formulating string theory in terms of two dimensional quantum field theory. He used the conformal invariance to calculate tree level string amplitudes on many worldsheet domains. Mandelstam was the first to explicitly construct the fermion scattering amplitudes in the Ramond and Neveu-Schwarz sectors of superstring theory, and later gave arguments for the finiteness of string perturbation theory.

In quantum field theory, Mandelstam and independently Sidney Coleman extended work of Tony Skyrme to show that the two dimensional quantum Sine-Gordon model is equivalently described by a thirring model whose fermions are the kinks. He also demonstrated that the 4d N=4 supersymmetric gauge theory is power counting finite, proving that this theory is scale invariant to all orders of perturbation theory, the first example of a field theory where all the infinities in feynman diagrams cancel.

Among his students at Berkeley are Joseph Polchinski and Charles Thorn.

Education: Witwatersrand (BSc, 1952); Trinity College, Cambridge (BA, 1954); Birmingham University (PhD, 1956).

Intellectual Defeatism

Just wanted to say it has been quite busy here because of the work having come back from vacation and preparing for my daughter in law and son's twins, which are to arrive any day now.

Intellectual defeatism

This statement reminded me of the idea about what is left for some to ponder, while we rely on our instincts to peer into the unknown, and hopefully land in a place that is correlated somehow in our future.

This again is being bold to me, because there are no rules here about what a schooling may provide for, what allows an individual the freedoms to explore great unknowns for them. For sure education then comes to check what these instincts have provided, and while being free to roam the world, sometimes it does find a "certain resonance" in what is out there.

Is this then a sign of what intellectual defeatism is about?

I want to give an example here about my perceptions about what sits in the valleys in terms of topological formations, that until now I had no way of knowing would become a suitable explanation for me, "about what is possible" even thought this represented a many possibility explanation in terms of outcomes.

Observables of Quantum Gravity

Scientists should be bold. They are expected to think out of the box, and to pursue their ideas until these either trickle down into a new stream, or dry out in the sand. Of course, not everybody can be a genuine “seer”: the progress of science requires few seers and many good soldiers who do the lower-level, dirty work. Even soldiers, however, are expected to put their own creativity in the process now and then -and that is why doing science is appealing even to us mortals.

To Be Bold

One possible way the Higgs boson might be produced at the Large Hadron Collider.


"Observables of Quantum Gravity," is a strange title to me, since we are looking at perspectives that are, how would one say, limited?

Where is such a focus located that we make talk of observables? Can such an abstraction be made then and used here, that we may call it, "mathematics of abstraction" and can arise from a "foundational basis" other then all the standard model distributed in particle attributes?

Observables of Quantum Gravity at the LHC
Sabine Hossenfelder

Perimeter Institute, Ontario, Canada

The search for a satisfying theory that unifies general relativity with quantum field theory is one of the major tasks for physicists in the 21st century. Within the last decade, the phenomenology of quantum gravity and string theory has been examined from various points of view, providing new perspectives and testable predictions. I will give a short introduction into these effective models which allow to extend the standard model and include the expected effects of the underlying fundamental theory. I will talk about models with extra dimensions, models with a minimal length scale and those with a deformation of Lorentz-invariance. The focus is on observable consequences, such as graviton and black hole production, black hole decays, and modifications of standard-model cross-sections.

So while we have created the conditions for an experimental framework, is this what is happening in nature? We are simulating the cosmos in it's interactions, so how is it that we can bring the cosmos down to earth? How is it that we can bring the cosmos down to the level of mind in it's abstractions that we do not just call it a flight of fancy, but of one that arises in mind based on the very foundations on the formation of this universe?

Fathers of Confederation

Robert Harris's painting of the Fathers of Confederation. The scene is an amalgamation of the Charlottetown and Quebec City conference sites and attendees.

Colonial organization

All the colonies which would become involved in Canadian Confederation in 1867 were initially part of New France and were ruled by France. The British Empire’s first acquisition in what would become Canada was Acadia, acquired by the 1713 Treaty of Utrecht (though the Acadian population retained loyalty to New France, and was eventually expelled by the British in the 1755 Great Upheaval). The British renamed Acadia Nova Scotia. The rest of New France was acquired by the British Empire by the Treaty of Paris (1763), which ended the Seven Years' War. Most of New France became the Province of Quebec, while present-day New Brunswick was annexed to Nova Scotia. In 1769, present-day Prince Edward Island, which had been a part of Acadia, was renamed “St John’s Island” and organized as a separate colony (it was renamed PEI in 1798 in honour of Prince Edward, Duke of Kent and Strathearn).

In the wake of the American Revolution, approximately 50,000 United Empire Loyalists fled to British North America. The Loyalists were unwelcome in Nova Scotia, so the British created the separate colony of New Brunswick for them in 1784. Most of the Loyalists settled in the Province of Quebec, which in 1791 was separated into a predominantly-English Upper Canada and a predominantly-French Lower Canada by the Constitutional Act of 1791.
Canadian Territory at Confederation.

Following the Rebellions of 1837, Lord Durham in his famous Report on the Affairs of British North America, recommended that Upper Canada and Lower Canada should be joined to form the Province of Canada and that the new province should have responsible government. As a result of Durham’s report, the British Parliament passed the Act of Union 1840, and the Province of Canada was formed in 1841. The new province was divided into two parts: Canada West (the former Upper Canada) and Canada East (the former Lower Canada). Ministerial responsibility was finally granted by Governor General Lord Elgin in 1848, first to Nova Scotia and then to Canada. In the following years, the British would extend responsible government to Prince Edward Island (1851), New Brunswick (1854), and Newfoundland (1855).

The remainder of modern-day Canada was made up of Rupert's Land and the North-Western Territory (both of which were controlled by the Hudson's Bay Company and ceded to Canada in 1870) and the Arctic Islands, which were under direct British control and became part of Canada in 1880. The area which constitutes modern-day British Columbia was the separate Colony of British Columbia (formed in 1858, in an area where the Crown had previously granted a monopoly to the Hudson's Bay Company), with the Colony of Vancouver Island (formed 1849) constituting a separate crown colony until its absorption by the Colony of British Columbia in 1866.

John A. Macdonald became the first prime minister of Canada.


The shear number of people in the United States at approx. 200 million,can be an reminder of what "we", in the approx. same land mass of Canada can be compared to the United States. Our paltry 36 million "being overshadowed" might be better understood from that perspective.

Happy Canada Day

Coastal Highway Views

Just before entering Malibu along the coastal highway of California.

Mathematical Structure of the Universe

Although Aristotle in general had a more empirical and experimental attitude than Plato, modern science did not come into its own until Plato's Pythagorean confidence in the mathematical nature of the world returned with Kepler, Galileo, and Newton. For instance, Aristotle, relying on a theory of opposites that is now only of historical interest, rejected Plato's attempt to match the Platonic Solids with the elements -- while Plato's expectations are realized in mineralogy and crystallography, where the Platonic Solids occur naturally.Plato and Aristotle, Up and Down-Kelley L. Ross, Ph.D.

Discover Magazine-06.16.2008-Photography by Erika Larsen-Article-"Is the Universe Actually Made of Math? Unconventional cosmologist Max Tegmark says mathematical formulas create reality."

It makes no difference at this point, which mathematics you choose to delve into the new model perceptions, because if one were to see how a projective geometry was built on previous platforms, then how is it we can see the universe in ways that the WMAP shows unless the mathematics could show that there was more to it then an artists picture displayed? You had to know the depth of the artists skill.

It does not mean that you are devoid of the possibilities of venturing where the philosophies of mathematics or science can venture. It is understanding that by taking yourself to a certain position in mind, an indecomposable one, one that is self evident, then it is understood that the deductive/inductive efforts bring you to a peak realization, contained in the "Aristotelean Arche."

This position is the question that one assumes in life, that having exhausted all efforts, and having seen all the information, one is to move their previous stalled position into a "third revolution" of a kind you might say?:)

See:Backreaction: Discover Interview with Tegmark

Also see:Theoretical Excellence

Wildlife at Home

Yes you may of noticed the date is wrong again on the camera. Every time we take the batteries out for charging, we loose that date. It dawned on me at this moment and wondered if there is another internal battery that retains the time, maybe, dead too? I'll have to look.

Anyway this fellow has been sticking around the last couple of days.

When we first looked at this lot about a year an done month ago today, I noticed a lot of markings that looked like claw marks, were in fact bull moose who rub their racks on the trees and leave these marks.




You'll also notice my work table I set up to do the final work on the outside of the house. I was installing a product called Nailite We did a board and batten around the rest of the house. I am really quite pleased tackling this job for the first time.

I just finished about two days ago, and have been working like heck to finish the jobs around here, while preparing for a trip into the states. My wife, myself with three of our older grandchildren are going for a trip to Disney Land.

We had our youngest granddaughter yesterday(she's sure a sweety) for the night, as my youngest son and his wife are due for twins which will be here not to long after we get back from our trip. She needs the bed rest and she is huge, and my boy is getting run ragged. So we have been extremely busy getting things in order.

The pillars in front of the house will be done when I get back and the backdrop to the front door will have a colonial look which we are trying to keep to a cottage look.



The aggregate walkways around the house were done at the end of May. We are please with how this has turned out. Unfortunately, the snow came before we had a chance to pour them last November, after having readied the forms, and compacted the crush.

Who said it?

At this point in the development, although geometry provided a common framework for all the forces, there was still no way to complete the unification by combining quantum theory and general relativity. Since quantum theory deals with the very small and general relativity with the very large, many physicists feel that, for all practical purposes, there is no need to attempt such an ultimate unification. Others however disagree, arguing that physicists should never give up on this ultimate search, and for these the hunt for this final unification is the ‘holy grail’. Michael Atiyah

"No Royal Road to Geometry?"

Click on the Picture

Are you an observant person? Look at the above picture. Why ask such a question as to, "No Royal Road to Geometry?" This presupposes that a logic is formulated that leads not only one by the "phenomenological values" but by the very principal of logic itself.

All those who have written histories bring to this point their account of the development of this science. Not long after these men came Euclid, who brought together the Elements, systematizing many of the theorems of Eudoxus, perfecting many of those of Theatetus, and putting in irrefutable demonstrable form propositions that had been rather loosely established by his predecessors. He lived in the time of Ptolemy the First, for Archimedes, who lived after the time of the first Ptolemy, mentions Euclid. It is also reported that Ptolemy once asked Euclid if there was not a shorter road to geometry that through the Elements, and Euclid replied that there was no royal road to geometry. He was therefore later than Plato's group but earlier than Eratosthenes and Archimedes, for these two men were contemporaries, as Eratosthenes somewhere says. Euclid belonged to the persuasion of Plato and was at home in this philosophy; and this is why he thought the goal of the Elements as a whole to be the construction of the so-called Platonic figures. (Proclus, ed. Friedlein, p. 68, tr. Morrow)

I don't think I could of made it any easier for one, but to reveal the answer in the quote. Now you must remember how the logic is introduced here, and what came before Euclid. The postulates are self evident in his analysis but, little did he know that there would be a "Royal Road indeed" to geometry that was much more complex and beautiful then the dry implication logic would reveal of itself.

It's done for a reason and all the geometries had to be leading in this progressive view to demonstrate that a "projective geometry" is the final destination, although, still evolving?

Eventually it was discovered that the parallel postulate is logically independent of the other postulates, and you get a perfectly consistent system even if you assume that parallel postulate is false. This means that it is possible to assign meanings to the terms "point" and "line" in such a way that they satisfy the first four postulates but not the parallel postulate. These are called non-Euclidean geometries. Projective geometry is not really a typical non-Euclidean geometry, but it can still be treated as such.

In this axiomatic approach, projective geometry means any collection of things called "points" and things called "lines" that obey the same first four basic properties that points and lines in a familiar flat plane do, but which, instead of the parallel postulate, satisfy the following opposite property instead:

The projective axiom: Any two lines intersect (in exactly one point).

If you are "ever the artist" it is good to know in which direction you will use the sun, in order to demonstrate the shadowing that will go on into your picture. While you might of thought there was everything to know about Plato's cave and it's implication I am telling you indeed that the logic is a formative apparatus concealed in the geometries that are used to explain such questions about, "the shape of space."

The Material World

There are two reasons that having mapped E8 is so important. The practical one is that E8 has major applications: mathematical analysis of the most recent versions of string theory and supergravity theories all keep revealing structure based on E8. E8 seems to be part of the structure of our universe.

The other reason is just that the complete mapping of E8 is the largest mathematical structure ever mapped out in full detail by human beings. It takes 60 gigabytes to store the map of E8. If you were to write it out on paper in 6-point print (that's really small print), you'd need a piece of paper bigger than the island of Manhattan. This thing is huge.

Polytopes and allotrope are examples to me of "shapes in their formative compulsions" that while very very small in their continuing expression, "below planck length" in our analysis of the world, has an "formative structure" in the case of the allotrope in the material world. The polytopes, as an abstract structure of math thinking about the world. As if in nature's other ways.



This illustration depicts eight of the allotropes (different molecular configurations) that pure carbon can take:

a) Diamond
b) Graphite
c) Lonsdaleite
d) Buckminsterfullerene (C60)
e) C540
f) C70
g) Amorphous carbon
h) single-walled carbon nanotube

Review of experiments

Graphite exhibits elastic behaviour and even improves its mechanical strength up to the temperature of about 2500 K. Measured changes in ultrasonic velocity in graphite after high temperature creep shows marked plasticity at temperatures above 2200 K [16]. From the standpoint of thermodynamics, melting is a phase transition of the first kind, with an abrupt enthalpy change constituting the heat of melting. Therefore, any experimental proof of melting is associated with direct recording of the temperature dependence of enthalpy in the neighbourhood of a melting point. Pulsed heating of carbon materials was studied experimentally by transient electrical resistance and arc discharge techniques, in millisecond and microsecond time regime (see, e.g., [17, 18]), and by pulsed laser heating, in microsecond, nanosecond and picosecond time regime (see, e.g., [11, 19, 20]). Both kind of experiments recorded significant changes in the material properties (density, electrical and thermal conductivity, reflectivity, etc. ) within the range 4000-5000 K, interpreted as a phase change to a liquid state. The results of graphite irradiation by lasers suggest [11] that there is at least a small range of temperatures for which liquid carbon can exist at pressure as low as 0.01 GPa. The phase boundaries between graphite and liquid were investigated experimentally and defined fairly well.

Sean Carroll:But if you peer closely, you will see that the bottom one is the lopsided one — the overall contrast (representing temperature fluctuations) is a bit higher on the left than on the right, while in the untilted image at the top they are (statistically) equal. (The lower image exaggerates the claimed effect in the real universe by a factor of two, just to make it easier to see by eye.)

See The Lopsided Universe-.

#36.Plato on Jun 12th, 2008 at 10:17 am

Lawrence,

Thanks again.

“I’m a Platonist — a follower of Plato — who believes that one didn’t invent these sorts of things, that one discovers them. In a sense, all these mathematical facts are right there waiting to be discovered.”Harold Scott Macdonald (H. S. M.) Coxeter

Moving to polytopes or allotrope seem to have values in science? Buckminister Fuller and Richard Smalley in terms of allotrope.

I was looking at Sylvestor surfaces and the Clebsch diagram. Cayley too. These configurations to me were about “surfaces,” and if we were to allot a progression to the “projective geometries” here in relation to higher dimensional thinking, “as the polytope[E8]“(where Coxeter[I meant to apologize for misspelling earlier] drew us to abstraction to the see “higher dimensional relations” toward Plato’s light.)

As the furthest extent of the Conjecture , how shall we place the dynamics of Sylvestor surfaces and B Fields in relation to the timeline of these geometries? Historically this would seem in order, but under the advancement of thinking in theoretics does it serve a purpose? Going beyond “planck length” what is a person to do?

Thanks for the clarifications on Lagrange points. This is how I see the WMAP.

Diagram of the Lagrange Point gravitational forces associated with the Sun-Earth system. WMAP orbits around L2, which is about 1.5 million km from the Earth. Lagrange Points are positions in space where the gravitational forces of a two body system like the Sun and the Earth produce enhanced regions of attraction and repulsion. The forces at L2 tend to keep WMAP aligned on the Sun-Earth axis, but requires course correction to keep the spacecraft from moving toward or away from the Earth.

Such concentration in the view of Sean’s group of the total WMAP while finding such a concentration would be revealing would it not of this geometrical instance in relation to gravitational gathering or views of the bulk tendency? Another example to show this fascinating elevation to non-euclidean, gravitational lensing, could be seen in this same light.

Such mapping would be important to the context of “seeing in the whole universe.”

See:No Royal Road to Geometry
Allotropes and the Ray of Creation
Pasquale Del Pezzo and E8 Origination?
Projective Geometries