Algebraic Topology

A first course in Algebraic Topology, with emphasis on visualization, geometric intuition and simplified computations. Given by Assoc Prof N J Wildberger at UNSW. The really important aspect of a course in Algebraic Topology is that it introduces us to a wide range of novel objects: the sphere, torus, projective plane, knots, Klein bottle, the circle, polytopes, curves in a way that disregards many of the unessential features, and only retains the essence of the shapes of spaces. What does this exactly mean? That is a key question... The course has some novel features, including Conway's ZIP proof of the classification of surfaces, a rational form of turn angles and curvature, an emphasis on the importance of the rational line as the model of the continuum, and a healthy desire to keep things simple and physical. We try to use pictures and models to guide our understanding.

See Also:

• Algebraic Topology

Noncommutative standard model

In theoretical particle physics, the non-commutative Standard Model, mainly due to the French mathematician Alain Connes, uses his noncommutative geometry to devise an extension of the Standard Model to include a modified form of general relativity. This unification implies a few constraints on the parameters of the Standard Model. Under an additional assumption, known as the "big desert" hypothesis, one of these constraints determines the mass of the Higgs boson to be around 170 GeV, comfortably within the range of the Large Hadron Collider. Recent Tevatron experiments exclude a Higgs mass of 158 to 175 GeV at the 95% confidence level.^[1] However, the previously computed Higgs mass was found to have an error, and more recent calculations are in line with the measured Higgs mass. ^[2]

Contents   1 Background 2 See also 3 Notes 4 References 5 External links

Background

Current physical theory features four elementary forces: the gravitational force, the electromagnetic force, the weak force, and the strong force. Gravity has an elegant and experimentally precise theory: Einstein's general relativity. It is based on Riemannian geometry and interprets the gravitational force as curvature of space-time. Its Lagrangian formulation requires only two empirical parameters, the gravitational constant and the cosmological constant.

The other three forces also have a Lagrangian theory, called the Standard Model. Its underlying idea is that they are mediated by the exchange of spin-1 particles, the so-called gauge bosons. The one responsible for electromagnetism is the photon. The weak force is mediated by the W and Z bosons; the strong force, by gluons. The gauge Lagrangian is much more complicated than the gravitational one: at present, it involves some 30 real parameters, a number that could increase. What is more, the gauge Lagrangian must also contain a spin 0 particle, the Higgs boson, to give mass to the spin 1/2 and spin 1 particles. This particle has yet to be observed, and if it is not detected at the Large Hadron Collider in Geneva, the consistency of the Standard Model is in doubt.

Alain Connes has generalized Bernhard Riemann's geometry to noncommutative geometry. It describes spaces with curvature and uncertainty. Historically, the first example of such a geometry is quantum mechanics, which introduced Heisenberg's uncertainty relation by turning the classical observables of position and momentum into noncommuting operators. Noncommutative geometry is still sufficiently similar to Riemannian geometry that Connes was able to rederive general relativity. In doing so, he obtained the gauge Lagrangian as a companion of the gravitational one, a truly geometric unification of all four fundamental interactions. Connes has thus devised a fully geometric formulation of the Standard Model, where all the parameters are geometric invariants of a noncommutative space. A result is that parameters like the electron mass are now analogous to purely mathematical constants like pi. In 1929 Weyl wrote Einstein that any unified theory would need to include the metric tensor, a gauge field, and a matter field. Einstein considered the Einstein-Maxwell-Dirac system by 1930. He probably didn't develop it because he was unable to geometricize it. It can now be geometricized as a non-commutative geometry.

See also

 

• Noncommutative geometry
• Noncommutative quantum field theory
• Timeline of atomic and subatomic physics

 

Notes

 

1. ^ The TEVNPH Working Group [1]
2. ^ Resilience of the Spectral Standard Model [2]

 

References

 

• Alain Connes (1994) Noncommutative geometry. Academic Press. ISBN 0-12-185860-X.
• -------- (1995) "Noncommutative geometry and reality," J. Math. Phys. 36: 6194.
• -------- (1996) "Gravity coupled with matter and the foundation of noncommutative geometry," Comm. Math. Phys. 155: 109.
• -------- (2006) "Noncommutative geometry and physics,"
• -------- and M. Marcolli, Noncommutative Geometry: Quantum Fields and Motives. American Mathematical Society (2007).

• Chamseddine, A., A. Connes (1996) "The spectral action principle," Comm. Math. Phys. 182: 155.
• Chamseddine, A., A. Connes, M. Marcolli (2007) "Gravity and the Standard Model with neutrino mixing," Adv. Theor. Math. Phys. 11: 991.
• Jureit, Jan-H., Thomas Krajewski, Thomas Schücker, and Christoph A. Stephan (2007) "On the noncommutative standard model," Acta Phys. Polon. B38: 3181-3202.
• Schücker, Thomas (2005) Forces from Connes's geometry. Lecture Notes in Physics 659, Springer.

 

External links

 

• Alain Connes official website with downloadable papers.
• Alain Connes's Standard Model.

Update:

•  Resilience of the Spectral Standard Model

Inspirations

Inspired on Escher's works. A free vision on how could be his workplace.

I was made aware of This Youtube video by Clifford of Asymptotia. He also linked, Lines and Colors.

Another Kind of Sideways

 I wanted to expand on where the title,"Another Kind of Sideways." This blog posting  came from an interview with Clifford of Asymptotia by PBS. He had a posting of his own entitled Multiverse Musings about a Nova series on PBS in the Fall related to Brian Greene's book, The Fabric of the Cosmos.

Where would these other universes be in relation to ours? Is there a way to envision it?

Well, we live in three spatial dimensions: We move back and forth, up and down, left to right. And then there's time, so that's our four-dimensional universe. Another universe might be essentially right next to ours by going in another direction that's not one of those four. We might call it "another kind of sideways." See: Riddles of the Multiverse

The whole context of the idea of the Multiverse could have in my layman view be classified as speaking about and argued as the basis of "existing outside of time." I just wanted to say that mathematically this definition of the Multiverse can actually exist in that framework, yet had to be extrapolated to the real universe we live in and how other universes may apply.

SOCRATES: But if he always possessed this knowledge he would always have known; or if he has acquired the knowledge he could not have acquired it in this life, unless he has been taught geometry; for he may be made to do the same with all geometry and every other branch of knowledge. Now, has any one ever taught him all this? You must know about him, if, as you say, he was born and bred in your house.SEE:Meno by Plato

I am always interested in the way a correlation is struck, from a scientist's mind when looking at the world and the comparisons they may find in the real world. I mean, to stand on top of a mountain as I did, you get this sense of the terrain, and how the landscape appears. How from an idealist position, a mathematical position is described and how the universe can be described?

LEE SMOLIN- Physicist, Perimeter Institute; Author, The Trouble With Physics

Thinking In Time Versus Thinking Outside Of Time

One very old and pervasive habit of thought is to imagine that the true answer to whatever question we are wondering about lies out there in some eternal domain of "timeless truths." The aim of re-search is then to "discover" the answer or solution in that already existing timeless domain. For example, physicists often speak as if the final theory of everything already exists in a vast timeless Platonic space of mathematical objects. This is thinking outside of time. See:A "scientific concept" may come from philosophy, logic, economics, jurisprudence, or other analytic enterprises, as long as it is a rigorous conceptual tool that may be summed up succinctly (or "in a phrase") but has broad application to understanding the world.

I find it hard sometimes to try and explain something that is "not outside of time."  That such description of reality while confounding to those like me less able to understand the mathematical world of such truths that contrary to Lee Smolin's opinion such schematics can be found to exist "within each of us." How we build our world from the inside, and how we contain it.

Is the mathematical description of  polytopes any less real as a mathematical basis?

If one is to believe that a mountain top represents some "perfect symmetry" then what said all those places in the valleys can exist and would not represent some genus figure? What are we saying about the possible universes, locations within the universe,  and the creation of?

That a pencil standing on point, could fall one way or another, or a description of a false vacuum to a true could represent something leading away from such symmetry? Why the problem with such mathematical and schematize attributes? Would you as a scientist turn your back on such mathematical interpretations of the world?

Gravitons and Topoi if an illusion, then Where's the Truth?

“Useful as it is under everyday circumstances to say that the world exists “out there” independent of us, that view can no longer be upheld. There is a strange sense in which this is a “participating universe” Wheeler (1983).

Taken from-Valuations in the language of Topos theory

It is always that the representative language current, as it's written, requires some deeper look behind the obvious, a look behind the illusion, to have it contend with the objectification of,  how one can see the truth.

“It’s a bit like a romantic relationship,” says Christopher Isham, describing his collaboration with Andreas Döring. Certainly the two physicists can claim to share their own unique understanding of the world, as many in love do. Together they are proposing a radical new way to view reality—one that takes you into a new "mathematical universe" where notions of "truth" and "falsehood" no longer apply, but where the paradoxes of quantum mechanics suddenly make sense. True Lies: Why Mathematics is an Illusion

Such a graduation which leads one to understand the structure of a 5d world only makes more sense when you combine what you see exists behind the geometric revelations of the real world, to see it apply to some underlying feature of the way the world works in those valleys. How "time variable measures" can be used to describe the landscape of the earth/moonscape's elements in a way not considered before.



Location of the reflector landing sites

Hubble Reveals Potential Titanium Oxide Deposits at Aristarchus and Schroter's Valley Rille

Further to this understanding of pathways through space,  require a firm understanding of how one can perceive the fabric of spacetime and the impression the earth leaves in it(just for demonstrative purposes to understand how indentation can be used in the understanding of the fabric).  To have satellites travel by the planets, to be propelled onto different routes of travel, or, to be held in stationary orbits around. L1 to L5 positions of the three body problem relation help again to orientate how we see the nature of space-time as it structurally allows us to see in these abstract ways . Its as if you look at the space provide and understand that variation in gravity can be understood in a "three dimensional space" given by the universe.



Georgi Dvali

"This is the crucial difference between the dark energy and modified gravity hypothesis, since, by the former, no observable deviation is predicted at short distances," Dvali says. "Virtual gravitons exploit every possible route between the objects, and the leakage opens up a huge number of multidimensional detours, which bring about a change in the law of gravity."

Dvali adds that the impact of modified gravity is able to be tested by experiments other than the large distance cosmological observations. One example is the Lunar Laser Ranging experiment that monitors the lunar orbit with an extraordinary precision by shooting the lasers to the moon and detecting the reflected beam. The beam is reflected by retro-reflecting mirrors originally placed on the lunar surface by the astronauts of the Apollo 11 mission.

Coffee and Donut?

A continuous deformation (homeomorphism) of a coffee cup into a doughnut (torus) and back.

Similarly, the hairy ball theorem of algebraic topology says that "one cannot comb the hair flat on a hairy ball without creating a cowlick."

***

This fact is immediately convincing to most people, even though they might not recognize the more formal statement of the theorem, that there is no nonvanishing continuous tangent vectorfield on the sphere. As with the Bridges of Königsberg, the result does not depend on the exact shape of the sphere; it applies to pear shapes and in fact any kind of smooth blob, as long as it has no holes.

In order to deal with these problems that do not rely on the exact shape of the objects, one must be clear about just what properties these problems do rely on. From this need arises the notion of homeomorphism. The impossibility of crossing each bridge just once applies to any arrangement of bridges homeomorphic to those in Königsberg, and the hairy ball theorem applies to any space homeomorphic to a sphere.

Intuitively two spaces are homeomorphic if one can be deformed into the other without cutting or gluing. A traditional joke is that a topologist can't distinguish a coffee mug from a doughnut, since a sufficiently pliable doughnut could be reshaped to the form of a coffee cup by creating a dimple and progressively enlarging it, while shrinking the hole into a handle. A precise definition of homeomorphic, involving a continuous function with a continuous inverse, is necessarily more technical.

Homeomorphism can be considered the most basic topological equivalence. Another is homotopy equivalence. This is harder to describe without getting technical, but the essential notion is that two objects are homotopy equivalent if they both result from "squishing" some larger object.

Equivalence classes of the English alphabet in uppercase sans-serif font (Myriad); left - homeomorphism, right - homotopy equivalence

 

An introductory exercise is to classify the uppercase letters of the English alphabet according to homeomorphism and homotopy equivalence. The result depends partially on the font used. The figures use a sans-serif font named Myriad.

Notice that homotopy equivalence is a rougher relationship than homeomorphism; a homotopy equivalence class can contain several of the homeomorphism classes. The simple case of homotopy equivalence described above can be used here to show two letters are homotopy equivalent, e.g. O fits inside P and the tail of the P can be squished to the "hole" part.

Thus, the homeomorphism classes are: one hole two tails, two holes no tail, no holes, one hole no tail, no holes three tails, a bar with four tails (the "bar" on the K
is almost too short to see), one hole one tail, and no holes four tails.

The homotopy classes are larger, because the tails can be squished down to a point. The homotopy classes are: one hole, two holes, and no holes.

To be sure we have classified the letters correctly, we not only need to show that two letters in the same class are equivalent, but that two letters in different classes are not equivalent. In the case of homeomorphism, this can be done by suitably selecting points and showing their removal disconnects the letters differently. For example, X and Y are not homeomorphic because removing the center point of the X leaves four pieces; whatever point in Y corresponds to this point, its removal can leave at most three pieces. The case of homotopy equivalence is harder and requires a more elaborate argument showing an algebraic invariant, such as the fundamental group, is different on the supposedly differing classes.

Letter topology has some practical relevance in stencil typography. The font Braggadocio, for instance, has stencils that are made of one connected piece of material.

Three-body problem and WMAP

"We all are of the citizens of the Sky" Camille Flammarion

In 1858, by the set of its relations, it will allow Camille Flammarion, the 16 years age, to enter as raises astronomer at the Observatory of Paris under the orders of Urbain the Glassmaker, at the office of calculations.

See:The Gravity Landscape and Lagrange Points



Now there is a reason that I am showing "this connection" so that the jokes that go around at the PI institute in regards to Tegmark( not that I am speaking for him and have absolutely no affiliation of any kind) and the "mathematical constructs are recognized" beyond just the jeering section, that while not being a party too, will and can be shown some light.

Three-body problem

For n ≥ 3 very little is known about the n-body problem. The case n = 3 was most studied, for many results can be generalised to larger n. The first attempts to understand the 3-body problem were quantitative, aiming at finding explicit solutions.

* In 1767 Euler found the collinear periodic orbits, in which three bodies of any masses move such that they oscillate along a rotation line.
* In 1772 Lagrange discovered some periodic solutions which lie at the vertices of a rotating equilateral triangle that shrinks and expands periodically. Those solutions led to the study of central configurations , for which \ddot q=kq for some constant k>0 .

The three-body problem is much more complicated; its solution can be chaotic. A major study of the Earth-Moon-Sun system was undertaken by Charles-Eugène Delaunay, who published two volumes on the topic, each of 900 pages in length, in 1860 and 1867. Among many other accomplishments, the work already hints at chaos, and clearly demonstrates the problem of so-called "small denominators" in perturbation theory.
The chaotic movement of 3 interacting particles
The chaotic movement of 3 interacting particles

The restricted three-body problem assumes that the mass of one of the bodies is negligible; the circular restricted three-body problem is the special case in which two of the bodies are in circular orbits (approximated by the Sun-Earth-Moon system and many others). For a discussion of the case where the negligible body is a satellite of the body of lesser mass, see Hill sphere; for binary systems, see Roche lobe; for another stable system, see Lagrangian point.

The restricted problem (both circular and elliptical) was worked on extensively by many famous mathematicians and physicists, notably Lagrange in the 18th century and Poincaré at the end of the 19th century. Poincaré's work on the restricted three-body problem was the foundation of deterministic chaos theory. In the circular problem, there exist five equilibrium points. Three are collinear with the masses (in the rotating frame) and are unstable. The remaining two are located on the third vertex of both equilateral triangles of which the two bodies are the first and second vertices. This may be easier to visualize if one considers the more massive body (e.g., Sun) to be "stationary" in space, and the less massive body (e.g., Jupiter) to orbit around it, with the equilibrium points maintaining the 60 degree-spacing ahead of and behind the less massive body in its orbit (although in reality neither of the bodies is truly stationary; they both orbit the center of mass of the whole system). For sufficiently small mass ratio of the primaries, these triangular equilibrium points are stable, such that (nearly) massless particles will orbit about these points as they orbit around the larger primary (Sun). The five equilibrium points of the circular problem are known as the Lagrange points.

So the thing is, that while one may not of found an anomalousness version of what is written into the pattern of WMAP( some Alien signal perhaps in a dimension of space that results in star manipulation), and what comes out, or how string theory plays this idea that some formulation exists in it's over calculated version of mathematical decor.

String Theory

In either case, gravity acting in the hidden dimensions affects other non-gravitational forces such as electromagnetism. In fact, Kaluza and Klein's early work demonstrated that general relativity with five large dimensions and one small dimension actually predicts the existence of electromagnetism. However, because of the nature of Calabi-Yau manifolds, no new forces appear from the small dimensions, but their shape has a profound effect on how the forces between the strings appear in our four dimensional universe. In principle, therefore, it is possible to deduce the nature of those extra dimensions by requiring consistency with the standard model, but this is not yet a practical possibility. It is also possible to extract information regarding the hidden dimensions by precision tests of gravity, but so far these have only put upper limitations on the size of such hidden dimensions.

Bold was added by me for emphasis. See also:Angels and Demons on a Pinhead

This is/was to be part of the hopes of people in research for a long time. I have seen it before, in terms of orbitals(the analogical version of the event in the cosmos) and how such events could gave been portrayed in those same locations in space. Contribute, to the larger and global distinction of what the universe is actually doing. If it's speeding up, what exactly does this mean, and what should we be looking for from what is being contributed to the "global perspective" of WMAP from these locations??

But lets move on here okay.

If you understand the "three body problem" and being on my own, and seeing things other then what people reveal in the reports that they write, how it is possible for a lone researcher like me to come up with the same ideas about the universe having some kind of geometrical inclination?

You would have to know that "such accidents while in privy to data before us all", and what is written into the calculations by hand would reveal? Well, I never did have that information. What I did know is what Sean Carroll presented of the Lopsided Universe for consideration. This coincided nicely with my work to comprehend Poincaré in a historical sense. The relationship with Klein.

As mentioned before, at the time, I was doing my own reading on Poincaré and of course I had followed the work of Tegmark and John Baez's expose' on what the shape of the universe shall look like. This is recorded throughout my bloggery here for the checking.

What I want to say.

Given the mathematics with which one sees the universe and however this mathematical constructs reveals of nature, nature always existed. What was shown is that the discovery of the mathematics made it possible to understand something beautiful about nature. So in a sense the mathematics was always there, we just did not recognize it.:)

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.

Stringy Geometry

fancier way of saying that is that in general, it's okay to model the space around us using the Euclidean metric. But the Euclidean model stops working when gravity becomes strong, as we'll see later. The Euclidean model for space

The magic square of "Albrect Durer" located in my index on the right is fascinating from the point of view that such a symmetry can be derived from the view of moving in an abstract space.

Trying to understand the implication of what is happening in a stronger gravitational field is an abstract journey for me as well, while I hold "thoughts of lensing" in my mind as a accumulative effect of something that is happening naturally out in space.

The move to Lagrangian points out in space is also an accumulative effect of thinking in this abstract way.

I not only think of the "magnetic field as as an associative value for that abstractness," it is a geometry that is the same for me, as I try to unravel the energy valuation of points(KK Tower) of any location in space. While the valuation of a circle on a 2 dimensional screen sees a string vibrating, I am moving this perception to valuations onto mathematical models.

I have nobody to help this way I have to push forward, knowing there will be mistakes, and that hopefully I am grasping the full scope of seeing in a abstract way.


Figure 2. Clebsch's Diagonal Surface: Wonderful.

We are told that "mathematics is that study which knows nothing of observation..." I think no statement could have been more opposite to the undoubted facts of the case; that mathematical analysis is constantly invoking the aid of new principles, new ideas and new methods, not capable of being defined by any form of words, but springing direct from the inherent powers and activity of the human mind, and from continually renewed introspection of that inner world of thought of which the phenomena are as varied and require as close attention to discern as those of the outer physical world, ...that it is unceasingly calling forth the faculties of observation and comparison, that one of its principal weapons is induction, that it has frequent recourse to experimental trial and verification, and that it affords a boundless scope for the exercise of the highest efforts of imagination and invention. ...Were it not unbecoming to dilate on one's personal experience, I could tell a story of almost romantic interest about my own latest researches in a field where Geometry, Algebra, and the Theory of Numbers melt in a surprising manner into one another.

Dr. Kip Thorne, Caltech 01-Relativity-The First 20th Century Revolution

It was the beginning of what might be called (and in fact is called) Stringy Geometry. The point is that strings are not points, and specifically, their extended nature means that in addition to being able to see the usual geometrical properties of a space that the theory like General Relativity can see, the strings can see other, intrinsically stringy, data. There is a quantity in the theory that is called the Kalb-Ramond field (or just the “B-field”) that can be used to measure how much the string can winds on or wraps a piece of the geometry, in essence. The parameter a that measures the size of a piece of the space that collapses when the geometry becomes singular, is essentially joined by another parameter, b, that sort of measures how much the strings have wound or smeared themselves on that piece of the space. The upshot is that a and b naturally combine themselves into a complex parameter that naturally describes the resolution process, solving the puzzle that the Mathematicians faced.

Beyond Einstein: Fixing Singularities in Spacetime

I am always trying to get the "visual models" of such proposals in terms of the B Field. Nigel Hitchin

Can you tell me, if the Dynkin diagrams and the points on a Sylvestor surface/ Cayley model have some value when looking at this subject?

Also, if it would be wrong to see "UV coordinates of a Gaussian arc" can be seen in this light as well?

I am recording this to help me understand how energy windings of the string may be seen as points on the Sylvester Surface?

See: What is Happening at the Singularity?

Inside Out

3.1 As Cytowic notes, Plato and Socrates viewed emotion and reason as in a kind of struggle, one in which it was vitally important for reason to win out. Aristotle took a more moderate view, that both emotion and reason are integral parts of a complex human soul--a theory proposed by Aristotle in explicit opposition to Platonism (De Anima 414a 19ff). Cytowic appears to endorse the Platonic line, with the notable difference that he would apparently rather have emotion win out.

I am trying to "create a image" that will use the one above. It is important that the select quoted comment below is understood. This can't be done without some reference.

So while the exercise may be going on "inside" things are happening on the outside. Scientists have never been completely honest with themselves, while some may concern themselves with whose name said what?

I use Plato as a namesake obviously, because of what I saw of some of our influential minds speaking, all the while making inferences to Plato. When ever you read something that resonates with you, it is of value because it correlates to something that you already know. This is what I tried to get across in the previous post, about what is "self evident." Little do some people recognize that while I may have inferred the point of some philosophical foundations, it is not without recognizing that the "qualitative phrases" have to be reduced as well to a logic. To reason.

How do you do that? Well I'll tell you what I found and then you can think whether I understood reason in it's proper format. Whether I understood the "shadows of Plato" to mean something other then what could have been interpreted as being wrong. What is that analogy of the Cave really mean?

Our attempt to justify our beliefs logically by giving reasons results in the "regress of reasons." Since any reason can be further challenged, the regress of reasons threatens to be an infinite regress. However, since this is impossible, there must be reasons for which there do not need to be further reasons: reasons which do not need to be proven. By definition, these are "first principles." The "Problem of First Principles" arises when we ask Why such reasons would not need to be proven. Aristotle's answer was that first principles do not need to be proven because they are self-evident, i.e. they are known to be true simply by understanding them.

Yes I did not enter the halls of higher learning in the traditional ways. You can converse for many years, does not mean you become devoid of the lessons that spoken amongst the commentors. How is it you can think that while listening to scientists you cannot uncover the the processes they use? If I had given thirty years to study, what exactly had I studied? I am a doctor of nothing.:)

This is a torus (like a doughnut) on which several circles are located. Unlike on a Euclidean plane, on this surface it is impossible to determine which circle is inside of which, since if you go from the black circle to the blue, to the red, and to the grey, you can continuously come back to the initial black, and likewise if you go from the black to the grey, to the red, and to the blue, you can also come back to the black.

My quote at Backreaction on this and that, reveals not only part of the understanding gained through this "infinite regress," but also the understanding we have with the world around us. Some would be better served to see the image of the Klein bottle, but I wanted to show what is going on in a "abstract way" to what is happening inside of us, and at the same time, what is happening outside.



I had used the brain and head as a place of our conscious awareness within context of our environment, our bodies. The topological explanations of the numbers above, and used them in the next paragraph. There will be confusion with the colour lines, please disregard that.

While I talked of the emotive and mental realities. I included the spiritual development in the end. The way this interaction takes place, is sometimes just as the mental function(yellow). Other times, it is the emotive realization of the experience. It is coloured by our emotion(red).

While we interact with our environment, there is this turning inside out, continuously. Sometimes we may say that "1" is the emotive realization, while the number 2 is seen as a mental extension of the situation. While the areas overlap each other, an outward progression may mean that the spiritual progress is numbered 4, while the interaction of the emotive, mental and spiritual progression may be number 3. Ultimately the spiritual progression is 4 (Violet). All these colours can mix and are significant in themself. They reveal something about our very constitution.

While some may wonder how could any conceptualization ever integrate the "Synesthesia views" of the world when it sees itself presented with such a comparison? The journey of course leads to the "Colour of Gravity." Discard your body, and one will wonder about the "clear light." What it means, in the "perceptive state of existence." If one is prepared, then one shall not have "to much time on their hands" getting lost in the fog.

Plato and Aristotle, Up and Down by Kelley L. Ross, Ph.D.

Rafael has Plato pointing up and Aristotle gesturing down to indicate the difference in their metaphysics. For Plato, true existence is in the World of Forms, in relation to which this world (of Becoming) is a kind of shadow or image of the higher reality. Aristotle, on the other hand, regards individual objects in this world as "primary substance" and dismisses Plato's Forms -- except for God as a pure actuality, without matter.

However, when it comes to ethics and politics, the gestures should be reversed. Plato, like Socrates, believed that to do the good without error, one must know what the good is. Thus, we get the dramatic moment in the Republic where Plato says that philosophers, who have escaped from the Cave and come to understand the higher reality, must be forced to return to this world and rule, so that their wisdom can benefit the state. Aristotle, on the other hand, says that the "good" is simply the goal of various particular activities, without one meaning in Plato's sense. The particular activities of most human affairs involve phronésis, "practical wisdom." This is not sophía, true wisdom, for Aristotle, which involves the theoretical knowledge of the highest things, i.e. the gods, the heavens, and God.

Thus, for philosophy, Aristotle should point up and would represent a contemplative attitude that was certainly more congenial to religious practices in the Middle Ages. By the same token, Aristotle's contribution to what we now think of as science was hampered by his lack of interest in mathematics. 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.

Therefore, caution is in order when comparing the meaning of the metaphysics of Plato and Aristotle with its significance for their attitudes towards ethics, politics, and science. Indeed, if the opposite of wisdom is, not ignorance, but folly, then Socrates and Plato certainly started off with the better insight.

It is good that you go to the top of the page of the linked quotes of Kelley L. Ross. You must know that I developed this site without really understanding the extent Mr. Ross had taken this issue. There is much that is familiar, and with him, an opposing view too.

See:

Induction and Deduction
Intuitively Balanced: Induction and Deduction

Euler's Konigsberg's Bridges Problem

"Liesez Euler, Liesez Euler, c'est notre maître à tous"
("Read Euler, read Euler, he is our master in everything") - Laplace

I should say here that the post by Guest post: Marni D. Sheppeard, “Is Category Theory Useful ?” over at A Quantum Diaries Survivor, continues to invoke my minds journey into the abstract spaces of mathematics.

The river Pregel divides the town of Konigsberg into four separate land masses, A, B, C, and D. Seven bridges connect the various parts of town, and some of the town's curious citizens wondered if it were possible to take a journey across all seven bridges without having to cross any bridge more than once. All who tried ended up in failure, including the Swiss mathematician, Leonhard Euler (1707-1783)(pronounced "oiler"), a notable genius of the eighteenth-century.

As a lay person being introduced to the strange world of mathematics it is always interesting to me in the way one can see in abstract processes.

The Beginnings of Topology...The Generalization to Graph Theory
Euler generalized this mode of thinking by making the following definitions and proving a theorem:

Definition: A network is a figure made up of points (vertices) connected by non-intersecting curves (arcs).

Definition: A vertex is called odd if it has an odd number of arcs leading to it, other wise it is called even.

Definition: An Euler path is a continuous path that passes through every arc once and only once.

Theorem: If a network has more than two odd vertices, it does not have an Euler path.

Euler also proved the converse:

Theorem: If a network has two or less odd vertices, it has at least one Euler path.

Leonhard Paul Euler (pronounced Oiler; IPA [ˈɔʏlɐ]) (April 15, 1707 – September 18 [O.S. September 7] 1783) was a pioneering Swiss mathematician and physicist, who spent most of his life in Russia and Germany. He published more papers than any other mathematician of his time.[2]

Euler made important discoveries in fields as diverse as calculus and graph theory. He also introduced much of the modern mathematical terminology and notation, particularly for mathematical analysis, such as the notion of a mathematical function.[3] He is also renowned for his work in mechanics, optics, and astronomy.

Portrait by Johann Georg Brucker- Born April 15, 1707(1707-04-15)Basel, Switzerland and Died September 18 [O.S. September 7] 1783
St Petersburg, Russia

You have to understand that as a lay person, my education is obtained through the internet. This is not without years of study(many books) in a lot of areas, that I could be said I am in a profession of anything, other then the student, who likes to learn a lot.

To find connections between the "real world" and what a lot think of as "to abstract to be real."

Any such expansionary mode of thinking, if not understood, as in the Case of Riemann's hypothesis seen in relation to Ulam's Spiral, one might have never understood the use of "Pascal's triangle" as well.

These are "base systems of mathematics" that are describing processes in nature?

See:Euler - 300th anniversay lecture

E8 and the Blackhole

"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."Donald (H. S. M.) Coxeter

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.

See:Pasquale Del Pezzo and E8 Origination?-Monday, March 19, 2007

If I had thought there was a way to describe the "interior" of the blackhole, it would be by recognizing the dimensionality the blackhole had to offer. One had to know where to locate "this place in the natural world." If we had understood the energy values of the particle world colliding(that space and frame of reference, then what were we finding that such a place in dimensionality could exist in the natural world? Yoyu had to accept that there was dynamical moves that werre being defined as a possiility.

Thus RHIC is in a certain sense a string theory testing machine, analyzing the formation and decay of dual black holes, and giving information about the black hole interior.The RHIC fireball as a dual black hole-Horatiu Nastase

So what ways would allow us to do this, and this is part of the idea that came to me as I was thinking about the place where all possibilities could exist. Yet, what existed as "moduli form in the valleys" was being extended. So I am connecting other things here too.

Housebuilding

It all start off as "a dream" or "an idea." Where do these come from? Dialogos of Eide

This is the house similar to what we will be constructing, with some modifications of course.

Most know of my time helping my son last year constructing his home. The journey of pictures that I have here within this bloggery. It has also some "dimensional aspect" in it's development, so I thought this might help those who are working Euclidean coordinates, may help to seal this process in some way, by being introduced to house construction.

This is the home that my wife and I had built in 1998. It was built on ten acres of land with a wide sweeping view of the mountains in the background. Although not seen here, you may have seen some of my rainbow pictures that I had put up over the years to help with the scenery we had.

Well the time has come for my wife and I to be entering into the venture ourselves. You will notice that the model we choose above is one floor. We thought this suitable for the coming years as when we move into retirement.

Here is a picture of my daughter-in-law and son's house in the winter of this last year. He still has some work to do, but as per our agreement, I help him, he is helping me.

I think I am getting the better of the deal, as he has taken the time to write me a 17 page step procedure with which I must follow. I thought this will become part of the journey for my wife and myself, so that everyone may see the process unfolding and maybe learn something about home construction. The plans of course change from country to country, while this plan is unfolding in Canada.

We purchased a 2 acre parcel of land with which to build the new home up top. I went into the bush with the camera and with about 2 feet of snow. It was not to easy to get around, so as time progresses,and as I put in the roadway and cleared site, you will get a better idea of what it looks like.



We had to contend with where we will live. We wanted the freedom and space to be close to where we will be building, so we bought a 19' foot travel trailer and will be putting it on the acreage while we build our new home. We thought of "renting" and our son of course offered for a time to let us live with him. We thought all around with the new baby Maley, we would leave them have their space as well.

Laying the Foundation

Articles on Euclid

See No Royal Road to Geometry?

I would like people to take note of the image supplied on the website of Euclides.Org, as it is one that I have used showing Plato and Aristotle. The larger picture of course is one done by Raphael and is painted on the wall in the "Signatores room in the Vatican."

The Room of the Segnatura contains Raphael's most famous frescoes. Besides being the first work executed by the great artist in the Vatican they mark the beginning of the high Renaissance. The room takes its name from the highest court of the Holy See, the "Segnatura Gratiae et Iustitiae", which was presided over by the pontiff and used to meet in this room around the middle of the 16th century. Originally the room was used by Julius II (pontiff from 1503 to 1513) as a library and private office. The iconographic programme of the frescoes, which were painted between 1508 and 1511, is related to this function. See Raphael Rooms

While one may of talked abut the past, or use a name like Plato of the past does not mean that what is being supplied from that position is not dealing with information for the 21st century. I would like you to think that while speaking about models that what the house is doing in "a psychological sense" is giving you a method by which all that you do in your life will materialize in consciousness and digs deep into the unconscious.

How often had you seen yourself in dream time, doing something or other, in the living room, kitchen, or anything that deals with the current state of mind, that you of course will see in this house? They are the many rooms of the mind.

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)

See also Laying the Foundation with Respect While one indeed had to start somewhere I thought I would start here with, "Foundational Perspectives."

I choose this as an introduction, whilst I will be starting from the ground up. This will include the planning of road way and building site. Since I have this interest about physics and where science is going these days, how could I not incorporate these things into what I am doing currently with my life now? So while I speak about the science end, I am encapsulating "this process" with regard to how I will construct my home.

Is this possible?

Well having spoken of the "Euclidean reference" one would have to know how one departs form such a scheme of Euclid, to know that this graduation to Non-Euclidean geometries was somehow related to the "fifth postulate" written by Euclid.

So of course, we had those who were involved in this development historically, which serve to remind us about where someone like Dali may of been as a visionary, in terms of Time. Or "geometrically inclined" to higher dimensional figures.

It definitely had it's connotations to "points of view." I mentioned religion, but for the nature of Salvador Dali, and his lifestyle, one would have to wonder where he was going with the Tesserack and his painting of Jesus on the Cross?

While I do not subscribe to any religion per say, I do subscribe to the finger of Plato pointing up. Have you for one moment you thought to roll your eyes up in your head, and think of what is up their in your mind? Assign our highest values to goodness. Surely you would enlist the "Colour of gravity" in all situations as you choose to live your life? It's there for the choosing.

Surely, that if you wore a hat on your head, or thought, to think of the roof of your house, you may indeed think of the highest ideals with which you choose to live your life. It's not my job to tell you what that is, that is yours alone.

You will be involved with aspects of the "universal language" that knows no boundaries, no matter your race, gender, or nationality. Yet, it will be specific to you. It will have "probabilistic outcomes" according to the life you are living regardless.

The Secret of the Golden Flower

When ever you walk the pathways in your mind of what ever model, you are laying the road work for that which you will travel through. Why, I may have referred to the title of the "Golden Flower in the Bee story," is a result, that the probabilistic outcome of life calls upon this "chance meeting" to come to what is held in mind. So what's new having the honey of the Bee community?

Do the Bee dance, and you learnt from others what this model is doing. So you travel. You get the benefits of the honey sometimes in new thoughts? There had to be a point "like the blank slate, glass room, a pen and paper ready" in order for the mind to be receptive to what already exists out there in the "form of ideas." How will these manifest? So indeed, it came from deep inside/outside you?

I never thought this inductive/deductive method while thinking it topological smooth in it's orientation, was not the exchange going on with our environment. That if you live your life according to your principles, then the principles would become part of your life. That on a level not understood to clearly, the "colour of gravity" was what we could evolve too? What is our own dynamical makeup, to become part of the ideals we had set for ourselves. We set our own ship in life. The boat or vehicle, becomes part of the way we will travel in our dream time. The airplane we ride.

Nurturing Creativity

See here and here


It looks as if moderation, or maybe technical problems, has set in for me at Cosmic Variance. So I have to go from the last statement made there by Lee that I was allowed to contribute. To continue with the points I am making.

I was glad to see Jacques was continuing where David B seems to have decided the futility of dealing with these issues of the String theory backlash.

Lee Smolin:When there was little selection we naturally got a wide diversity of types of scientists, which was good for science. My view is that we need that diversity, we need both the hill climbers and the valley crossers, the technical masters and the seers full of questions and ideas.

Raphael Bousso and Joseph Polchinski in "The String Theory Landscape" September 2004 Scientific issue speak exactly to what Lee is saying and descriptively allow us to see the pattern underlying Lee's comment. Maybe George Musser will release it for the group to inspect here

Take full note of the diagrams.

See OFF THE HOOK. Line-by-line crocheting instructions that tell where to increase or decrease numbers of stitches create the global shape of the Lorenz manifold.Univ. of Bristol

Clifford:Hooking Up Manifolds

The artlcle goes a great deal into the story of how mathematician Hinke Osinga and her partner mathematician Bernd Krauskopf got into this, and why they find it useful. You’ll also hear from mathematicians Carolyn Yackel, Daina Taimina, and Sarah-Marie Belcastro. This has been going on for a while, and there are even published scientific papers with crocheting instructions for various manifolds! How did I miss out on this?! This is great!

If you did not continue with understanding the "topography of the energy involved" in terms of what the string theory landscape was doing, then you would have never understood the "hills and valleys" in the context of string theory landscape being described?

HYPERBOLIC FABRIC. Many of the lines that could be inscribed on this crocheted hyperbolic plane curve away from each other, defying Euclid's parallel postulate.
Taimina

IN retrospect decisions we make will always resound with what we should have done, but that misses the boat when coming to the "creative abilities?" What we see may "institute a productive research group?" You exchange one for another?

Lee Smolin:Is string theory in fact perturbatively finite? Many experts think so. I worry that if there were a clear way to a proof it would have been found and published, so I find it difficult to have a strong expectation, either way, on this issue.

The fact that a way had been describe in terms of developing the "Triple Torus" speaks to the continued development of the string theory landscape? How could you conclusively finish off this statement and then from it describe the state of the union, when this had already been explained technically?

We say that E8 has rank 8 (the maximum number of mutually commutative degrees of freedom), and dimension 248 (as a manifold). This means that a maximal torus of the compact Lie group E8 has dimension 8. The vectors of the root system are in eight dimensions, and are specified later in this article. The Weyl group of E8, which acts as a symmetry group of the maximal torus by means of the conjugation operation from the whole group, is of order 696729600.

You had to see the context of the triple torus in relation too where the string landscape places were placing these modular forms. If I had said E8 and the continued development of modular form, what would this represent?

The complexity of the forms themself are limited and finite so how could one claim that such work on the landscape is futile in regards to infinities?