christx11
Jedi Master
Are the C's concept of densities anything like Misha Gromov's density model of random groups?
Background:
I came upon an article that seemed to have several coincidences with some of the things the C's sessions have spoken of.
The article is here:
Getting Into Shapes: From Hyperbolic Geometry to Cube Complexes and Back
A proof marks the end of an era in the study of three-dimensional shapes.
By: Erica Klarreich
October 2, 2012
https://www.simonsfoundation.org/quanta/20121002-getting-into-shapes-from-hyperbolic-geometry-to-cube-complexes-and-back/
I was searching for information on the product over all primes being = 4pi^2 and tori and happened on a link to the above article. It is an interesting article in its own right in that it is readable and understandable by someone that does not necessarily have a high level background in math (myself). The article is about the geometrization (a complete taxonomy) of all 3 manifolds as laid out in the vision of mathematician William Thurston. In the end of the journey that the article lays out the task was accomplished and both the virtual Haken conjecture and the virtual fibering conjecture were proved. It was accomplished via a branch of mathematics called Cube Complexes and the fact that the cube complex has a homotopy equivalence to the topological manifold. The equivalence is implied in the name of the article, "From Hyperbolic Geometry to Cube Complexes and Back".
The first thing I found interesting was the title of the article. Once I looked up what a homotopy equivalence was, I thought that the title of the article was one type of example of 'Geometry gets you there; algebra sets you "free."' The article is about the complete geometrization of 3 manifolds and it was the homotopy equivalence between cube complexes and the manifolds that solved the problem at issue. When two things are equal they are exactly the same thing.
The second thing I found interesting was the coincidence of two quotes in the article that were similar to a couple of things that the C's said to Ark.
11-07-1998 - 'A: Our words sing to you. Let them ring.'
-------------
Q: (A) How long can a digestion phase last? We don't have much time!
A: However long it takes. And who says you do not have much "time?" Answer, mi Arkady, answer!!
Q: (A) Well, I agree that I am impatient. But, the point is that I feel that if I would have a little bit more of a clue, I could do much more, and for now...
A: Our words sing to you. Let them ring.
Q: (A) What is the difference between singing and ringing? (L) I don't think that's the point. (A) Ring is to awake? Probably. You mean I am not taking your words seriously enough?
A: No. We meant to let it sink in rejoice. Exult!
In the article, the author quotes two mathematicians:
William Thurston:
[“Many people have an impression, based on years of schooling, that mathematics is an austere and formal subject concerned with complicated and ultimately confusing rules,” he wrote in 2009. “Good mathematics is quite opposite to this. Mathematics is an art of human understanding. … Mathematics sings when we feel it in our whole brain.”]
Walter Neumann:
[Thurston conjectured that many categories of three-manifolds contain one exemplar, a three-manifold whose geometry is so perfect, so uniform, so beautiful that, as Walter Neumann of Columbia University is fond of saying, it “rings like a bell.”]
I found it quite a coincidence that mathematics "sings" and "rings" are both quotes in the article. Perhaps a cigar is just a cigar, but it is none the less a very interesting coincidence. This coincidence kind of spurred me on to look into Cube Complexes. Most of it is way over my head but with a non-technical background I found it interesting that in cube complexes tetrahedrals and pentagons and hexagons and octagons seem quite common. In one respect this is no surprise as this branch of mathematics deals with cubulating shapes, polytopes, etc. and it would be common to encounter faces that are polygons either in the mathematicians hyperplane traces or Dehn diagrams while cubulating. Still it reminded me of the "octagonal complexigram". Complexigram can be viewed as a compound word of "complex" and "diagram".
I wonder if "octagonal complexigram" is a clue about cubulating and Cube Complexes?
After looking at cube complexes a bit, it wasn't long and several of the papers I encountered spoke of cubulating random groups in Gromov's density model with density less than 1/6, 1/4, etc. This got me to look at what is Gromov's density model of random groups. I think it was Misha (Mikhail) Gromov who introduced the idea of a density model for random groups in a paper in 1993. Now I am getting way over my head, but I found the concept that randomly generated groups can have properties with high probability at specific densities rather interesting. I found it kind of a coincidence also that I guess you would call it Gromov's main unit in his density model is called a "WORD", (Our WORDS sing to you, let them ring). Other interesting things about Gromov's density model remind me of other things from the C's sessions -
Quoting from YANN OLLIVIER AND DANIEL T. WISE paper 'CUBULATING RANDOM GROUPS AT DENSITY LESS THAN 1/6',
["One of the striking facts Gromov proved is that a random finitely presented group is infinite, hyperbolic at density < 1/2, and is trivial or {±1} at density > 1/2, with probability tending to 1 as ℓ → ∞.]
Density 1/2 reminded me of the mentioning of half-life and also phi, with the phi relation being naturally embedded in the 1, 1/2 numbers. The mathematics is over my head, but it was the first time I could envision the idea of a naturally occurring phenomena where groups could form at different densities and perhaps sub groups at one density could be part of a larger super group at a different density and thus natural divisions of perception were bounded.
And this whole thing brought to mind another quote from the sessions.
10-10-1998
-------------
A: All the masters have channeled, whether aware or not. The "who" is not Germaine.
Q: (A) It is not important. Now, he is talking a lot about p-edic numbers which are different from real numbers, and they are related to prime numbers, and it is a whole big area which may be important for development for the right mathematics for the future. What about p-edic numbers? Are they important?
A: Yes.
Q: (A) Should I learn them?
A: With room for alterations the key to quantum jumps is always in discovering "new" mathematics.
The above quote about [A: With room for alterations the key to quantum jumps is always in discovering "new" mathematics.], is from 1998 and Gromov's model was introduced in 1993 (relatively new). I wonder if with the right generators and perhaps a tweak on the density model if it might yield the densities?
So that is how I arrived at my question:
Are the C's concept of densities anything like Misha Gromov's density model of random groups?
Background:
I came upon an article that seemed to have several coincidences with some of the things the C's sessions have spoken of.
The article is here:
Getting Into Shapes: From Hyperbolic Geometry to Cube Complexes and Back
A proof marks the end of an era in the study of three-dimensional shapes.
By: Erica Klarreich
October 2, 2012
https://www.simonsfoundation.org/quanta/20121002-getting-into-shapes-from-hyperbolic-geometry-to-cube-complexes-and-back/
I was searching for information on the product over all primes being = 4pi^2 and tori and happened on a link to the above article. It is an interesting article in its own right in that it is readable and understandable by someone that does not necessarily have a high level background in math (myself). The article is about the geometrization (a complete taxonomy) of all 3 manifolds as laid out in the vision of mathematician William Thurston. In the end of the journey that the article lays out the task was accomplished and both the virtual Haken conjecture and the virtual fibering conjecture were proved. It was accomplished via a branch of mathematics called Cube Complexes and the fact that the cube complex has a homotopy equivalence to the topological manifold. The equivalence is implied in the name of the article, "From Hyperbolic Geometry to Cube Complexes and Back".
The first thing I found interesting was the title of the article. Once I looked up what a homotopy equivalence was, I thought that the title of the article was one type of example of 'Geometry gets you there; algebra sets you "free."' The article is about the complete geometrization of 3 manifolds and it was the homotopy equivalence between cube complexes and the manifolds that solved the problem at issue. When two things are equal they are exactly the same thing.
The second thing I found interesting was the coincidence of two quotes in the article that were similar to a couple of things that the C's said to Ark.
11-07-1998 - 'A: Our words sing to you. Let them ring.'
-------------
Q: (A) How long can a digestion phase last? We don't have much time!
A: However long it takes. And who says you do not have much "time?" Answer, mi Arkady, answer!!
Q: (A) Well, I agree that I am impatient. But, the point is that I feel that if I would have a little bit more of a clue, I could do much more, and for now...
A: Our words sing to you. Let them ring.
Q: (A) What is the difference between singing and ringing? (L) I don't think that's the point. (A) Ring is to awake? Probably. You mean I am not taking your words seriously enough?
A: No. We meant to let it sink in rejoice. Exult!
In the article, the author quotes two mathematicians:
William Thurston:
[“Many people have an impression, based on years of schooling, that mathematics is an austere and formal subject concerned with complicated and ultimately confusing rules,” he wrote in 2009. “Good mathematics is quite opposite to this. Mathematics is an art of human understanding. … Mathematics sings when we feel it in our whole brain.”]
Walter Neumann:
[Thurston conjectured that many categories of three-manifolds contain one exemplar, a three-manifold whose geometry is so perfect, so uniform, so beautiful that, as Walter Neumann of Columbia University is fond of saying, it “rings like a bell.”]
I found it quite a coincidence that mathematics "sings" and "rings" are both quotes in the article. Perhaps a cigar is just a cigar, but it is none the less a very interesting coincidence. This coincidence kind of spurred me on to look into Cube Complexes. Most of it is way over my head but with a non-technical background I found it interesting that in cube complexes tetrahedrals and pentagons and hexagons and octagons seem quite common. In one respect this is no surprise as this branch of mathematics deals with cubulating shapes, polytopes, etc. and it would be common to encounter faces that are polygons either in the mathematicians hyperplane traces or Dehn diagrams while cubulating. Still it reminded me of the "octagonal complexigram". Complexigram can be viewed as a compound word of "complex" and "diagram".
I wonder if "octagonal complexigram" is a clue about cubulating and Cube Complexes?
After looking at cube complexes a bit, it wasn't long and several of the papers I encountered spoke of cubulating random groups in Gromov's density model with density less than 1/6, 1/4, etc. This got me to look at what is Gromov's density model of random groups. I think it was Misha (Mikhail) Gromov who introduced the idea of a density model for random groups in a paper in 1993. Now I am getting way over my head, but I found the concept that randomly generated groups can have properties with high probability at specific densities rather interesting. I found it kind of a coincidence also that I guess you would call it Gromov's main unit in his density model is called a "WORD", (Our WORDS sing to you, let them ring). Other interesting things about Gromov's density model remind me of other things from the C's sessions -
Quoting from YANN OLLIVIER AND DANIEL T. WISE paper 'CUBULATING RANDOM GROUPS AT DENSITY LESS THAN 1/6',
["One of the striking facts Gromov proved is that a random finitely presented group is infinite, hyperbolic at density < 1/2, and is trivial or {±1} at density > 1/2, with probability tending to 1 as ℓ → ∞.]
Density 1/2 reminded me of the mentioning of half-life and also phi, with the phi relation being naturally embedded in the 1, 1/2 numbers. The mathematics is over my head, but it was the first time I could envision the idea of a naturally occurring phenomena where groups could form at different densities and perhaps sub groups at one density could be part of a larger super group at a different density and thus natural divisions of perception were bounded.
And this whole thing brought to mind another quote from the sessions.
10-10-1998
-------------
A: All the masters have channeled, whether aware or not. The "who" is not Germaine.
Q: (A) It is not important. Now, he is talking a lot about p-edic numbers which are different from real numbers, and they are related to prime numbers, and it is a whole big area which may be important for development for the right mathematics for the future. What about p-edic numbers? Are they important?
A: Yes.
Q: (A) Should I learn them?
A: With room for alterations the key to quantum jumps is always in discovering "new" mathematics.
The above quote about [A: With room for alterations the key to quantum jumps is always in discovering "new" mathematics.], is from 1998 and Gromov's model was introduced in 1993 (relatively new). I wonder if with the right generators and perhaps a tweak on the density model if it might yield the densities?
So that is how I arrived at my question:
Are the C's concept of densities anything like Misha Gromov's density model of random groups?
