Marseille workshop on loops and spin foams

[marcus]

Nightcleaner,
the Letralia website has the complete poem in both languages
http://www.letralia.com/58/en02-058.htm
and many more besides this one

Everness

One thing does not exist: Oblivion.
God saves the metal and he saves the dross,
And his prophetic memory guards from loss
The moons to come, and those of evenings gone.
Everything is: the shadows in the glass.
Which, in between the day's two twilights, you
Have scattered by the thousands, or shall strew
Henceforward in the mirrors that you pass.
And everything is part of that diverse
Crystalline memory, the universe:
Whoever though its endless mazes wanders
Hears door on door click shut behind his stride,
And only from the sunset's farther side
Shall view at last the Archetypes and Splendors.


Everness

Sólo una cosa no hay. Es el olvido.
Dios, que salva el metal, salva la escoria
Y cifra en Su profética memoria
Las lunas que serán y las que han sido.
Ya todo está. Los miles de reflejos
Que entre los dos crepúsculos del día
Tu rostro fue dejando en los espejos
Y los que irá dejando todavía.
Y todo es una parte del diverso
Cristal de esa memoria, el universo;
No tienen fin sus arduos corredores
Y las puertas se cierran a tu paso;
Sólo del otro lado del ocaso
Verás los Arquetipos y Esplendores.

...

here is one that Letralia doesn't have:

to see a world in a grain of sand
and a heaven in a wild flower,
hold infinity in the palm of your hand,
and eternity in an hour

I don't know what you are talking about
I know what you are talking about

pick one

 

[marcus]

sorry everybody
I got off topic
it is probably better to start a separate thread for poetry et al. and
let this one stay focused on what John Baez called attention to:
the AJL paper
Dynamical Triangulations

 

[selfAdjoint]

I don't know where you get that about the Planck scale. In the general relativity view, spacetime at any scale is one unified thing.

Go back to Marcus' earlier post about Regge Calculus. Years ago Tullio Regge triangulated GR spacetime and by doing combinatorial things with the edge-lengths of the triangulation he was able to do all the GR curvature math that is usually done with tensors and differential forms and second derivatives. Then Ambjorn and coworkers made all the lengths the same size and revised the combinatorial shuffle to an even simpler form, but they had problems and it was a years-long slog to get to their present causal triangulations which work so splendedly.

Now if you want to use packed spheres instead of triangulations, go to it, but you have to show as Regge did and Ambjorn et al did that it reproduces the world we know, not just at the handwaving level but in the details where god and the devil duke it out.

 

[marcus]

I don't know where you get that about the Planck scale. In the general relativity view, spacetime at any scale is one unified thing.

Go back to Marcus' earlier post about Regge Calculus. Years ago Tullio Regge triangulated GR spacetime and by doing combinatorial things with the edge-lengths of the triangulation he was able to do all the GR curvature math that is usually done with tensors and differential forms and second derivatives. Then Ambjorn and coworkers made all the lengths the same size and revised the combinatorial shuffle to an even simpler form, but they had problems and it was a years-long slog to get to their present causal triangulations which work so splendidly.

Now if you want to use packed spheres instead of triangulations, go to it, but you have to show as Regge did and Ambjorn et al did that it reproduces the world we know, not just at the handwaving level but in the details where god and the devil duke it out.

classic epigrammatical account, wanted to email it to Ambjorn as a kind of
maximally concise statement of their work's place in the q.g. story.
won't though, since they must have plenty to think about without
e-fanmail

 

[marcus]

photo of Tullio Regge

http://www.icra.it/MG/awards/Images/regge.jpg

born 1931 Torino
discovered Regge calculus 1961 while at Princeton
where he worked with John Archibald Wheeler

 

[marcus]

I still think the best detailed introduction to AJL dynamical triangulations
is "Dynamically Triangulating Lorentzian Quantum Gravity"
http://arxiv.org/hep-th/0105267

more than one person at PF has indicated they'd found it useful,
printed it out, etc. Also AJL refer back to it as a basic reference
several times in their recent (2004) papers.

would also be nice to have an online source giving an
introduction to Regge calculus----hopefully would have pictures
since the subject could be presented visually

for basic path integral terminology, here is the Wiki article
on "path integral"
http://en.wikipedia.org/wiki/Path_integral_formulation

if you wish, this will lead you back to contributory Wikis on "action", "Lagrangian" etc.

Here's a brief introduction to Regge calculus (esp. as applied to numerical relativity) by Adrian Gentle
http://arxiv.org/abs/gr-qc/0408006
it really has barely a page actually explaining R's discrete gen. rel.

hope we find more. I will keep looking

 

[nightcleaner]

I don't know where you get that about the Planck scale. In the general relativity view, spacetime at any scale is one unified thing.

Now if you want to use packed spheres instead of triangulations, go to it, but you have to show as Regge did and Ambjorn et al did that it reproduces the world we know, not just at the handwaving level but in the details where god and the devil duke it out.

Yes, in GR spacetime is one unified thing, but my reading has led me to think that GR isn't used to describe the world at the Planck scale, but is considered to be a poor model of events in the very small, very high energy realm. GR is a cosmological paragigm, while the standard model of particles in flat space is more often used in discussions of the very small. Did I miss something?

Now, it is very nice of you to invite me to try to match the work of PH.D's at two major European universities and The Max Planck research institute, backed up by all the departmental machinery and academic freedoms they have available to them, while I am nothing but a nightcleaner in a tourist restaurant. Actually I am gratified by the fact that AJL has done work in the very field I have been unsuccessfully trying to draw attention to here and in previous years on SST.com.

I feel somewhat as a bean farmer must who finds a fertile plot of ground, scratches at it with a stick and makes a little crop for a few years, then finds himself and his tender garden uprooted by the massive machinerey of agribusiness. It seems they want to build a driveway for their factories on top of my little digs, and I may as well get out of the way or get paved over. Huh.

Well, it is no real surprise. And I have the small satisfaction of saying that I, at least, knew where to dig. And I take away something else as well. I may not be able to apply the Regge calculus (at least not yet) but my model is prettier.

nc

 

[marcus]

... I may not be able to apply the Regge calculus (at least not yet) ...

hello NC, i am still groping around for introductory material on Regge calculus and the closely related DT approach.

here are some online page references, if for no other use than my own!
I found parts of these helpful.

Loll 98----pages 8-13 are about standard Regge
pages 14-17 are about DT
http://arxiv.org/gr-qc/9805049
Discrete approaches to quantum gravity in four dimensions
this is a "LivingReviews" survey article that the AEI invited Loll to contribute
it surveys current (1998) research in several related areas and give
a large bibliography. She includes a halfdozen or so introductory sources on Regge calculus but none are online. her own treatment is quite concise.


For a more elemenary discussion: try Loll 02, pages 8-16
see also the summary at the end pages 34 and 35.
http://arxiv.org/hep-th/0212340
A Discrete History of the Lorentzian Path Integral

this is a pedagogical article, to help get graduate students involved.
It is historical, describing difficulties as they were encountered. I find this often helps me understand.
this essay is willing to waste words explaining some simpler points that a normal research article would not explain

However the rest of the article----pages 1-7 and 17-33
is much concerned with the problems that were being encountered in 2002!
since they have gotten past some of that, it is less interesting now IMO.

Maybe as a sample I will quote some from around page 8.

 

[marcus]

Here is a sample from around page 8 of Loll 20 survey
http://arxiv.org/hep-th/0212340
A Discrete History of the Lorentzian Path Integral

this is just to give the flavor. I will not bother to reproduce the math symbols exactly but will simply drop symbols in some cases---leaving whatever copies easily: the words.

---sample---
“Lorentzian dynamical triangulations”, first proposed in [13] and further elaborated in [14, 15] tries to establish a logical connection between the fact that non-perturbative path integrals were constructed for Euclidean instead of Lorentzian geometries and their apparent failure to lead to an interesting continuum theory. Is it conceivable that we can kill two birds with one stone, ie. cure the problem of degenerate quantum geometry by taking a path integral over geometries with a physical, Lorentzian signature? Remarkably, this is indeed what happens in the quantum gravity theories in d < 4 which have already been studied extensively. The way in which Lorentzian dynamical triangulations overcome the problems mentioned above is the subject of the Sec. 5.

4 Geometry from simplices
The use of simplicial methods in general relativity goes back to the pioneering work of Regge [16]. In classical applications one tries to approximate a classical space-time geometry by a triangulation, that is, a piecewise linear space obtained by gluing together flat simplicial building blocks, which in dimension d are d-dimensional generalizations of triangles. By “flat” I mean that they are isometric to a subspace of d-dimensional Euclidean or Minkowski space. We will only be interested in gluings leading to genuine manifolds, which therefore look locally like an R^d. A nice feature of such simplicial manifolds is that their geometric properties are completely described by the discrete set ... of the squared lengths of their edges. Note that this amounts to a description of geometry without the use of coordinates. There is nothing to prevent us from reintroducing coordinate patches covering the piecewise linear manifold, for example, on each individual simplex, with suitable transition functions between patches. In such a coordinate system the metric tensor will then assume a definite form. However, for the purposes of formulating the path integral we will not be interested in doing this, but rather work with the edge lengths, which constitute a direct, regularized parametrization of the space Geom(M) of geometries. How precisely is the intrinsic geometry of a simplicial space, most importantly, its curvature, encoded in its edge lengths? A useful example to keep in mind is the case of dimension two, which can easily be visualized. A 2d piecewise linear space is a triangulation, and its scalar curvature R(x) coincides with the so-called Gaussian curvature. One way of measuring this curvature is by parallel-transporting a vector around closed curves in the manifold. In our piecewise-flat manifold such a vector will always return to its original orientation unless it has surrounded lattice vertices v at which the surrounding angles did not add up to 2[pi], but [formula omitted]
see Fig.4. The so-called deficit angle [delta] is precisely the rotation angle picked up by the vector and is a direct measure for the scalar curvature at the vertex. The operational description to obtain the scalar curvature in higher dimensions is very similar, one basically has to sum in each point over the Gaussian curvatures of all two-dimensional submanifolds. This explains why in Regge calculus the curvature part of the Einstein action is given by a sum over building blocks of dimension (d-2) which are simply the objects dual to those local 2d submanifolds
---end quote---

Notice how new Causal DT (Lorentzian DT) is! She says it was first proposed only in 1998-----the reference [13] is to a paper by Ambjorn
and her.

I have bolded "...which constitute a direct, regularized parametrization of the space Geom(M) of geometries..."

you have a formless continuum M, and you make a "space" consisting of all the possible geometries you could have on M. this is where the quantum state of the geometry is going to live, or be defined. In another of her writings Loll calls this space of geometries, this Geom(M) the "mother of all spaces" or something like that.

this is a more direct "quantization-ready" approach than some others (e.g. think of the Ashtekar variables). In the straight Regge, there is just this long list of EDGE LENGTHS and that effectively describes a geometry and coordinatizes Geom(M)

now DT insists that all the edges are standard lengths and so instead of a list of edgelengths you have a list of what is next to what, recording the "connectivity"---it should be simple enough: some ways of writing it down would be more efficient than others----some computer data structure that names the vertices and says which ones are vertices of what tetrahedron etc., enough information so you can tell what is a side of what.

 

[nightcleaner]

hello NC, i am still groping around for introductory material on Regge calculus and the closely related DT approach.

here are some online page references, if for no other use than my own!
I found parts of these helpful.

...

Hi Marcus, and thanks for posting your finds here. I do think they are helpful.

However, I have trouble reading the math, and the papers are full of jargon which make them difficult for ordinary English speakers such as myself. I have been reading physics for a couple years and trying to improve my math skills, so I think I have some idea of what AJL are trying to convey. Still, I find myself taking a drubbing on the forehead when trying to read Loll and her collegues.

Are you willing to entertain questions on the math and physics here?

For example, here is a web link from Mathworld. It seems to be relevant, but I am not sure, and will withdraw it from the forum if it is not to the point of this thread.

http://mathworld.wolfram.com/Simplex.html

nc

3790

 

[selfAdjoint]

Cleaner, in 0 dimensions a simplex is a point. In one dimesion, take a 0-simplex and another point not on it, and draw all possible straight lines from the one to the other (there's only one line in this case). The result is the 1-simplex, which is just a line segment, right? Now on to 2 dimensions; take a 1-simplex and a point in the plane not on the 1-simplex, and draw all possible straight lines from one to the other. BTW this construction is called "taking the cone" over the 1-simplex. The result is the 2-simplex, which you should see to be a triangle. For the 3-simplex take the cone over the 2-simplex in 3 dimensions. The result is a 4 sided pyraimid on a triangular base.

And so on, although you can't visualize it. Notice that the 0-simplex had 1 vertex, 0 edges, and 0 faces. The 1-simplex had 2 vertices, 1 edge, and 0 faces. The 2-simplex had 3 vertices, 3 edges, and 1 face, and the 3-simplex has 4 vertices, 6 edges, and 4 faces. Evidently an n-simplex has n+1 vertices. How many edges, faces, and higher dimensional faces does it have? Well each k-dimensional face is a k-simplex itself, and so it has k+1 vertices, as we just said. And the number of k-faces in an n-simplex is the number of combinations of n+1 vertices taken k=1 at a time; the total number in the n-simplex taken the number in a typical k-face at a time.

Mathematicians use simplexes instead of cubes or whatever to triangulate spaces because they have this simple facial property, which leads to simple formulas for combining them.

 

[nightcleaner]

My reading of Regge Calculus: a unique tool for numerical relativity, Adrian P. Gentle, arXiv:gr-qc/0408006 v1 2 Aug 2004 today has brought me to the following notions of what Regge calculus is, and how it is applied to the structure of space time at the Planck scale.

Regge calculus uses objects called simplices , which in general are lower dimensional structures applied to approximate higher dimensional surfaces. For example, a three spatial dimensional object like the event horizon of a black hole seen at an instant of time appears spherical, and has the dimensions given by spherical geometry at the Schwartzchild radius. The positions of points on the Schwartxchild surface of the event horizon can be approximated by applying a large number of triangles to the surface. Each triangle is two dimensional, making it easier to calculate positions of points on the three dimensional surface of the event horizon as approximations to positions of the vertices of the triangles. Given that the triangles have straight edges and that these edges are very close to the curvature at any point, it is possible to calculate the vertices, and these calculations are very close to the values of the more difficult calculations required to obtain the exact three dimensional position on the surface. To get a better fit, the edges of the triangles can be made shorter, so giving a better approximation to the surface.

Triangular simplices in two dimensions can be applied as described above to a three dimensional surface. In general, then , the same procedure can be applied to a four dimensional spacetime by applying three dimensional simplices to the four dimensional manifold. The three dimensional analog of the two dimensional triangle is the tetrahedron. So, by building a three dimensional lattice of tetrahedrons, we can model events in four dimensions.

The paper referenced above applies two dimensional triangulations to an embedding diagram of a black hole. The embedding diagram uses artistic perspective on a two dimensional surface to represent the black hole as a gravitational depression in an elastic sheet, the familiar “whirlpool” or vortex shape. The triangles are shown as fitted to the curvature of the surface of the deformed sheet. Far from the edges of the hole, the surface is flat and the triangles make a near perfect fit. At the lip of the hole, the surface is highly curved and the triangles are made smaller to better approximate the curvature. How does this process translate into higher dimensional analyses, so that we can use a three dimensional lattice to model four dimensional space-time events?

If we build an undistorted lattice of tetrahedrons all of which have edges of identical length, we have a three dimensional model of four dimensional spacetime in a region where there is no mass or energy to distort the lengths of the edges. This lattice can be extended to infinity and points in the lattice can easily be calculated using any coordinate system, so the lattice can be said to be background independent.

If we introduce a gravitational field to the lattice, we should see the edges in the vicinity of the gravitational object made shorter as the curvature of spacetime increases near the object. This process breaks the symmetry of the lattice, since the original conditions of the lattice, as seen far from the gravitational object, gradually transition to the reduced conditions near the object. This transition cannot be made in the three dimensional model in a smooth way, but must rely on our making certain choices in the representation in regard to edge lengths and interior angles of the triangles which make up the tetrahedra. These choices are not background independent, since the lattice itself has now become the background for the model.

I will now recommend a solution to this difficulty, which seems to me to be inherent in the Regge calculus, at least as far as I have been able to understand it.

If the lattice is built using a type of simplex which has a naturally occurring and easily calculable expansion symmetry, the choice of which lattice edge and so which internal angle to reduce can be automated, restoring a degree of background independence. For example in the triangular fit to a curved surface we could choose to divide each triangle by a reduced triangle inscribed between the midpoints of the edge lines. Each time we choose to make this division, there is a discontinuity in the accuracy of the fit of the simplex to the surface, but locally at the center of the triangle the fit is improved.
I will now suggest a better way to make the said choice. It is better because it does not rely on the human intervention of someone to decide where to apply the division into smaller triangles.

First, instead of a tetrahedral simplex, use a spherical one. Then, by stacking the spheres in close contact, and using the contact points as vertices, a type of lattice structure is formed. In this case the edges of the lattice are virtual, since they cut through the spherical surfaces in the same way a cord cuts a circle. However the geometry of the stack ensures that the lengths of the edges are the same as the length of the radius of the circle, just as a hexagram can be inscribed in a circle by dividing the circumference with a compass set to the length of the radius. This structure (I imagine groans from some readers) is none other than the isomatrix.

It is to be noted now that there are twelve spheres of equal radius which can be fitted around a central sphere of the same radius. These twelve spheres can be encompassed by a new, larger sphere, with a new radius equal to three times the radius of the original sphere. I suggest then that this three dimensional naturally occurring model be chosen as the most appropriate lattice structure for representing events in four dimensions. The lattice structure can be refined to any number of iterations to fit any curved space, even down to the singularity.

Then, returning to the three dimensional lattice meant to represent a gravitational object in four dimensions, as one approaches the object, the spheres and their contact lattice naturally and smoothly contract without human interference or background dependence. This model has to provide a better fit than any artificially constructed tetrahedral lattice, and I recommend that persons interested in spacetime geometry investigate the relatively simple mathematics of this type of nested spherical stack. Not only is it simpler and more beautiful than the broken glass edges used in AJL, but it is free of the complications introduced by human choice of when and where and by how much to reduce triangular edges in order to conform to increasing curvature.

I am now able to offer a simplified model conforming to the above suggestion which involves four interwoven circles of equal diameter which can be laid on a two dimensional flat surface, then given any degree of deformation up to the three dimensional sphere, without discontinuities. But that will have to wait until I can return to this, and depends upon my not being hit by any trucks.

Be well,

Richard

 

[nightcleaner]

Cleaner, in 0 dimensions a simplex is a point. In one dimesion, take a 0-simplex and another point not on it, and draw all possible straight lines from the one to the other (there's only one line in this case). The result is the 1-simplex, which is just a line segment, right? Now on to 2 dimensions; take a 1-simplex and a point in the plane not on the 1-simplex, and draw all possible straight lines from one to the other. BTW this construction is called "taking the cone" over the 1-simplex. The result is the 2-simplex, which you should see to be a triangle. For the 3-simplex take the cone over the 2-simplex in 3 dimensions. The result is a 4 sided pyraimid on a triangular base.

And so on, although you can't visualize it. Notice that the 0-simplex had 1 vertex, 0 edges, and 0 faces. The 1-simplex had 2 vertices, 1 edge, and 0 faces. The 2-simplex had 3 vertices, 3 edges, and 1 face, and the 3-simplex has 4 vertices, 6 edges, and 4 faces. Evidently an n-simplex has n+1 vertices. How many edges, faces, and higher dimensional faces does it have? Well each k-dimensional face is a k-simplex itself, and so it has k+1 vertices, as we just said. And the number of k-faces in an n-simplex is the number of combinations of n+1 vertices taken k=1 at a time; the total number in the n-simplex taken the number in a typical k-face at a time.

Mathematicians use simplexes instead of cubes or whatever to triangulate spaces because they have this simple facial property, which leads to simple formulas for combining them.

Ok, thanks. Then a 4-simplex has five vertices. This could be shown as five vertices on a circle, a pentagram. It has ten edges. To count the faces, one must look at the structure as a pair of four sided pyramids joined by a common three edged base? One edge is lost, or obscured, by the representation of a higher dimensional object in a lower dimensional space?

then,

n,v,e,f
0,1,0,0
1,2,1,0
2,3,3,1
3,4,6,4
4,5,10,10
5,6,15,20

n an integer
v=n+1
e=nv/2=[(n^2)+n]/2
f=e_(n-1) + f_(n-1)
={ [(n-1)^2]+n-1]/2} + f_n-1
= n^2-n + f_(n-1)?

(working. have to go do evening chores. Be back, iidghbat )

3,815

 

[marcus]

I agree with NC about these numbers of vertices, edges, faces (and they agree with the combinatorics that sA said)

then,

n,v,e,f
0,1,0,0
1,2,1,0
2,3,3,1
3,4,6,4
4,5,10,10
5,6,15,20

...

I had something else to mention. Although I think this is a very promising line of research there are still very very few papers in it.
In fact the idea of the LORENTZIAN or causally ordered approach to DT was only first proposed in 1998! so lorentzian or causal DT is newer than several other approaches (e.g. string and loop)

But even tho there are very few papers, why not have a regular way of keeping track? So here is an arxiv search engine table for this line of research:

http://arxiv.org/find/grp_physics/1...gravity+AND+Lorentzian+quantum/0/1/0/1998/0/1

http://arxiv.org/find/grp_physics/1...gravity+AND+Lorentzian+quantum/0/1/0/1999/0/1

http://arxiv.org/find/grp_physics/1...gravity+AND+Lorentzian+quantum/0/1/0/2000/0/1

http://arxiv.org/find/grp_physics/1...gravity+AND+Lorentzian+quantum/0/1/0/2001/0/1

http://arxiv.org/find/grp_physics/1...gravity+AND+Lorentzian+quantum/0/1/0/2002/0/1

http://arxiv.org/find/grp_physics/1...gravity+AND+Lorentzian+quantum/0/1/0/2003/0/1

http://arxiv.org/find/grp_physics/1...gravity+AND+Lorentzian+quantum/0/1/0/2004/0/1

1998   3
1999   3
2000   5 
2001   4
2002   6
2003   4
2004   4

these are not perfect, they get some they shouldn't and probably miss some, but I've found the links are a good way to check for existence of papers I didnt previously know about in this area. As you can see there is little or no growth as yet. Will be interesting to run the same keyword search in 2005 and see if there's any change

 

[nightcleaner]

Ok, I think I see the pattern now.

A zero dimensional simplex is a vertex (or point).
A one dimensional simplex is an edge (or line), composed of two vertices (zero dimensional simplices).
A two dimensional simplex is a triangle (or plane), composed of three edges (one dimensional simplices).
A three dimensional simplex is a tetrahedron, composed of four triangles (two dimensional simplices)
A four dimensional simplex is a hyper-tetrahedron, and should be composed of five tetrahedral ( three dimensional simplices).

The table then would look like this:


N s0 s1 s2 s3 s4
0 1 0 0 0 0
1 2 1 0 0 0
2 3 3 1 0 0
3 4 6 4 1 0
4 5 5 1

Where N is dimension, s0 is number of vertices, s1 is number of edges, s2 is number of triangles, s3 is number of tetrahedrons, and s4 is number of hyper-tetrahedrons. So, to get the right s2 and s1 for n=4, we have to adjoin five 3-simplices, that is five tetrahedrons. We can do this simply by surrounding one tetrahedron with four more, each adjoined to one of the original tetrahedron’s faces. Then we have a three dimensional model of a four dimensional structure, consisting of five co-joined tetrahedrons.

All four of the original tetrahedron’s faces are now interior to the new structure, and are co-joined to one face each of the outer four tetrahedrons. Since five tetrahedrons would have twenty faces, but eight are now co-joined on the interior of the new structure, we are left with twelve exterior s2 triangular faces.

Five tetrahedrons would have thirty edges, but the lines of the central tetrahedron are all co-joined with two other exterior tetrahedrons. We can neglect these interior lines in our count. Then four exterior tetrahedrons would have twenty-four edges, but twelve of these edges are co-joined with one other exterior tetrahedron each, leaving a count of twenty-four minus six, or eighteen exterior edges. Twelve of these edges are acute, and the other six, the co-joined ones, are oblique.

So the completed table would look like this:


N s0 s1 s2 s3 s4
0 1 0 0 0 0
1 2 1 0 0 0
2 3 3 1 0 0
3 4 6 4 1 0
4 5 18 12 5 1


However I now notice that there are stated to be five s0 vertices in the four dimensional simplex. This comes from the rule that the number of vertices is n+1. The described joining of four tetrahedrons to one tetrahedron in the center does not result in a three dimensional structure with five points. Instead, the structure has four external vertices and four internal vertices. Do we throw out the structure or modify the rule?

Is there any way to co-join four tetrahedrons to end up with five points? I haven’t thought of any.

Is there any reason to change the rule? Where did the rule come from? Let’s look at the rule.

We began with a single vertex in otherwise empty space, and noted that it had zero dimension. To obtain one dimension, we had to add another vertex, so that space was no longer empty. This new vertex had to be constrained to be exterior to the original vertex, so that a line was formed.

Then, when we added a third vertex, we had to constrain the placement again, so that the new vertex was placed in the plane, and not on the edge previously constructed. Our new vertex was non-co-linear with the two that formed the definition of the edge.

Then, when we added the fourth vertex, to make a three dimensional simplex, we had to specify that the new vertex was non-co-planar with the existing three. So what is the rule for the fifth vertex? It cannot occupy the same three dimensional space as the existing three vertices. How might this be done?

We might place the new vertex as an offset in time. In other words, we might take our three dimensional simplex, a tetrahedron, and map it onto a tetrahedron extended one instant in time. Each vertex of our t=0 tetrahedron corresponds with a vertex on our t=1 tetrahedron. This would then be eight vertices, four in one instant and four in the offset instant. But we only want five vertices.

Five vertices can be achieved by setting the time interval to infinity. Doing this reduces one of the tetrahedrons to a point. Then, we see, we have five points, four in one instant mapped onto a fifth which, being in another instant, is not in the same space, and so obeys our constraint.

The result is a space-time structure with five vertices. It has four space-like vertices and six space-like edges, and it has four time-like edges which map from the four space-like vertices onto the one time-like vertex. So we see that there are now, as originally predicted, four dimensions, five vertices, and ten edges. But what happened to the five predicted s3 tetrahedral simplices? Are we justified in saying that they are somehow interspersed along the time line, so that we really have a set of five, one in 3space, one at infinite time, and then one each at the half and quarter marks? What is half or a quarter of infinity?

I have, as usual, an alternative proposition, which I think is more elegant. It is this. Any vertex in 4 space has at least four positions in any 3space. It exists at the origin. It exists at infinity. And it exists at least at two points somewhere in between. Those would be the spaces which contain the original tetrahedron we built in 3space at t=0, and the offset tetrahedron we built in another 3space at t=1.

Of course t=1 is not t=infinity, but again, what is half of infinity? From 3space, when we try observe the 5-vertice structure which exists in 4space, all we can see at one glance is ten edges between five points. That is the whole structure, as far as our familiar 3space geometry will allow. But we must conclude that it is not the entire 4space simplex. Parts of it are hidden from our 3space view.


Now, (groan) to return to the isomatrix model. There is one sphere in the center, representing any universe you choose. There are twelve spheres around it, representing the fundamental unit of spacetime in multiple dimensions. This structure is extended to infinity, in, out, and in every conceivable direction, both in time and in space. When an observer notes that there has been a change in the universe, it is not because the spacetime structure of the multiverse, that is the frozen river of 4d, has changed, but it is because the observer has moved through the unmoving spacetime structure. When the observer moves, it is in a direction, just one direction, not every direction possible. This movement results in a loss of information from the direction opposite to the motion. Information from the direction opposite to motion in the multiverse cannot catch up to the observer. Instead, the observer can obtain information only about the universe current to the observers instantaneous position (one sphere, the center sphere of the observer) and the three spheres which are annexed to that central sphere and still in the line of motion of the observer, and then one more bit, from the sphere that is just beyond the three spheres that form the possible next instant. The universe as you know it in this instant, the three universes that are possible in the next instant, and the one universe that has to be beyond those three. That’s five, the very same five vertices, or origins, that make up the structure we view as fourth dimensional from our familiar three dimensional universe.

Be well. Comments appreciated. And, yes, Love always,


Richard

 

[nightcleaner]

Ok, I think I see the pattern now.

A zero dimensional simplex is a vertex (or point).
A one dimensional simplex is an edge (or line), composed of two vertices (zero dimensional simplices).
A two dimensional simplex is a triangle (or plane), composed of three edges (one dimensional simplices).
A three dimensional simplex is a tetrahedron, composed of four triangles (two dimensional simplices)
A four dimensional simplex is a hyper-tetrahedron, and should be composed of five tetrahedral ( three dimensional simplices).

The table then would look like this:


N s0 s1 s2 s3 s4
0 , 1 , 0 , 0 , 0 , 0
1 , 2 , 1 , 0 , 0 , 0
2 , 3 , 3 , 1 , 0 , 0
3 , 4 , 6 , 4 , 1 , 0
4 , 5 , ? , ? , 5 , 1

Where N is dimension, s0 is number of vertices, s1 is number of edges, s2 is number of triangles, s3 is number of tetrahedrons, and s4 is number of hyper-tetrahedrons. So, to get the right s2 and s1 for n=4, we have to adjoin five 3-simplices, that is five tetrahedrons. We can do this simply by surrounding one tetrahedron with four more, each adjoined to one of the original tetrahedron’s faces. Then we have a three dimensional model of a four dimensional structure, consisting of five co-joined tetrahedrons.

All four of the original tetrahedron’s faces are now interior to the new structure, and are co-joined to one face each of the outer four tetrahedrons. Since five tetrahedrons would have twenty faces, but eight are now co-joined on the interior of the new structure, we are left with twelve exterior s2 triangular faces.

Five tetrahedrons would have thirty edges, but the lines of the central tetrahedron are all co-joined with two other exterior tetrahedrons. We can neglect these interior lines in our count. Then four exterior tetrahedrons would have twenty-four edges, but twelve of these edges are co-joined with one other exterior tetrahedron each, leaving a count of twenty-four minus six, or eighteen exterior edges. Twelve of these edges are acute, and the other six, the co-joined ones, are oblique.

So the completed table would look like this:


N s0 s1 s2 s3 s4
0 , 1 , 0 , 0 , 0 , 0
1 , 2 , 1 , 0 , 0 , 0
2 , 3 , 3 , 1 , 0 , 0
3 , 4 , 6 , 4 , 1 , 0
4 , 5 , 18 , 12 , 5 , 1


However I now notice that there are stated to be five s0 vertices in the four dimensional simplex. This comes from the rule that the number of vertices is n+1. The described joining of four tetrahedrons to one tetrahedron in the center does not result in a three dimensional structure with five points. Instead, the structure has four external vertices and four internal vertices. Do we throw out the structure or modify the rule?

Is there any way to co-join four tetrahedrons to end up with five points? I haven’t thought of any.

Is there any reason to change the rule? Where did the rule come from? Let’s look at the rule.

We began with a single vertex in otherwise empty space, and noted that it had zero dimension. To obtain one dimension, we had to add another vertex, so that space was no longer empty. This new vertex had to be constrained to be exterior to the original vertex, so that a line was formed.

Then, when we added a third vertex, we had to constrain the placement again, so that the new vertex was placed in the plane, and not on the edge previously constructed. Our new vertex was non-co-linear with the two that formed the definition of the edge.

Then, when we added the fourth vertex, to make a three dimensional simplex, we had to specify that the new vertex was non-co-planar with the existing three. So what is the rule for the fifth vertex? It cannot occupy the same three dimensional space as the existing three vertices. How might this be done?

We might place the new vertex as an offset in time. In other words, we might take our three dimensional simplex, a tetrahedron, and map it onto a tetrahedron extended one instant in time. Each vertex of our t=0 tetrahedron corresponds with a vertex on our t=1 tetrahedron. This would then be eight vertices, four in one instant and four in the offset instant. But we only want five vertices.

Five vertices can be achieved by setting the time interval to infinity. Doing this reduces one of the tetrahedrons to a point. Then, we see, we have five points, four in one instant mapped onto a fifth which, being in another instant, is not in the same space, and so obeys our constraint.

The result is a space-time structure with five vertices. It has four space-like vertices and six space-like edges, and it has four time-like edges which map from the four space-like vertices onto the one time-like vertex. So we see that there are now, as originally predicted, four dimensions, five vertices, and ten edges. But what happened to the five predicted s3 tetrahedral simplices? Are we justified in saying that they are somehow interspersed along the time line, so that we really have a set of five, one in 3space, one at infinite time, and then one each at the half and quarter marks? What is half or a quarter of infinity?

I have, as usual, an alternative proposition, which I think is more elegant. It is this. Any vertex in 4 space has at least four positions in any 3space. It exists at the origin. It exists at infinity. And it exists at least at two points somewhere in between. Those would be the spaces which contain the original tetrahedron we built in 3space at t=0, and the offset tetrahedron we built in another 3space at t=1.

Of course t=1 is not t=infinity, but again, what is half of infinity? From 3space, when we try observe the 5-vertice structure which exists in 4space, all we can see at one glance is ten edges between five points. That is the whole structure, as far as our familiar 3space geometry will allow. But we must conclude that it is not the entire 4space simplex. Parts of it are hidden from our 3space view.


Now, (groan) to return to the isomatrix model. There is one sphere in the center, representing any universe you choose. There are twelve spheres around it, representing the fundamental unit of spacetime in multiple dimensions. This structure is extended to infinity, in, out, and in every conceivable direction, both in time and in space. When an observer notes that there has been a change in the universe, it is not because the spacetime structure of the multiverse, that is the frozen river of 4d, has changed, but it is because the observer has moved through the unmoving spacetime structure. When the observer moves, it is in a direction, just one direction, not every direction possible. This movement results in a loss of information from the direction opposite to the motion. Information from the direction opposite to motion in the multiverse cannot catch up to the observer. Instead, the observer can obtain information only about the universe current to the observers instantaneous position (one sphere, the center sphere of the observer) and the three spheres which are annexed to that central sphere and still in the line of motion of the observer, and then one more bit, from the sphere that is just beyond the three spheres that form the possible next instant. The universe as you know it in this instant, the three universes that are possible in the next instant, and the one universe that has to be beyond those three. That’s five, the very same five vertices, or origins, that make up the structure we view as fourth dimensional from our familiar three dimensional universe.

Be well. Comments appreciated. And, yes, Love always,


Richard

3850

 

[nightcleaner]

1998   3
1999   3
2000   5 
2001   4
2002   6
2003   4
2004   4

these are not perfect, they get some they shouldn't and probably miss some, but I've found the links are a good way to check for existence of papers I didnt previously know about in this area. As you can see there is little or no growth as yet. Will be interesting to run the same keyword search in 2005 and see if there's any change

Marcus! This is really great work! If I didn't have an eyestrain, I'd stay up all night reading. Oh well, tomorrow is tomorrow. Thank you very much for these searches.

Richard

 

[marcus]

Richard, do you know about the "N-choose-k" number?

it is relevant here.

it is often written
\left(\begin{array}{cc}N\\k\end{array}\right)

and sometimes called the "combinations" of size k taken from a set of size N, the language is awkward but the idea is very simple


if you have a set of N things then how many subsets of size k are there?
N-choose-k

If you have a set of 3 things (N=3) then how many subsets of size 2 (k=2) are there?

3-choose-2 is equal to 3


the N-choose-k numbers are those appearing in the "Pascal triangle"

\left(\begin{array}{cc}4\\0\end{array}\right) = 1

\left(\begin{array}{cc}4\\1\end{array}\right) = 4

\left(\begin{array}{cc}4\\2\end{array}\right) = 6

\left(\begin{array}{cc}4\\3\end{array}\right) = 4

\left(\begin{array}{cc}4\\4\end{array}\right) = 1

 

[nightcleaner]

No, Marcus, my math is pretty limited. I've had some college calc, but didn't do well in it, and that's been long ago. I was pretty good at physics, in the Life Sciences version, which was light on calclulus. I thought yesterday's problem was interesting, and just tried to think my way through it.

I have considered returning to school to improve my maths. University of Minnesota Duluth is closest, but does not offer much of a curriculum.

Meanwhile I have been reviewing, using a GRE text, and trying to get what I can from internet. Any suggestions would be welcome.

Thanks,

Richard

 

[marcus]

\left(\begin{array}{cc}N\\k\end{array}\right) = \frac{N!}{k!(N-k)!}

...

for this, you have to know what N! the factorial of N is,
and you should know the convention that 0!, the zero factorial, equals one.

...

(but these, after all, are not terribly hard facts to learn
or, more precisely, to accept)

the N-choose-k numbers are those appearing in the "Pascal triangle"

\left(\begin{array}{cc}3\\0\end{array}\right) = 1

\left(\begin{array}{cc}3\\1\end{array}\right) = 3

\left(\begin{array}{cc}3\\2\end{array}\right) = 3

\left(\begin{array}{cc}3\\3\end{array}\right) = 1



\left(\begin{array}{cc}4\\0\end{array}\right) = 1

\left(\begin{array}{cc}4\\1\end{array}\right) = 4

\left(\begin{array}{cc}4\\2\end{array}\right) = 6

\left(\begin{array}{cc}4\\3\end{array}\right) = 4

\left(\begin{array}{cc}4\\4\end{array}\right) = 1



\left(\begin{array}{cc}5\\0\end{array}\right) = 1

\left(\begin{array}{cc}5\\1\end{array}\right) = 5

\left(\begin{array}{cc}5\\2\end{array}\right) = 10

\left(\begin{array}{cc}5\\3\end{array}\right) = 10

...


about GRE review, sounds smart, but I can't advise
Duluth likewise.
maybe selfAdjoint, who also lives in midwest, can give wise and kind counsel
I really cant. all what you say sounds sensible and intelligent
(but in our tangled web of hardship and difficulty how can anyone give advice or encouragement to anyone else besides to say take care)

However whatever you do or do not do in your life, you should understand Pascal triangle and N-choose-k. I seriously insist on this.

the number of triangle faces of a tetrahedron is the number
of THREEPOINTED simplices belonging to a FOURPOINTED
four choose three

[edited to moderate language]

 

[nightcleaner]

Hi Marcus.

Thanks for trying to cheer me up. The world has been, uh, dissappointing. I think it was Dr. Seuss whose last words were "we could have done much better."

I have been sleeping and doing chores all day, and havn't gotten back to this except for a few minutes and then I was interupted. Tonight I have to work. But I will work on the n-choose-k thing.

I am enjoying this conversation.

Pascal triangle, I have read of this somewhere. This board's format makes it hard to show here, but you line the numbers up in centered rows, as I recall. The numbers in each row are the sum of the numbers immediately above them, I see, so (1,1,)(1,2,1)(1,3,3,1)(1,4,6,4,1)(1,5,10,10,5,1) (1,6,15,20,15,6,1) and so on. I don't remember what the connection was but I do know this is familiar.

Four choose three? is four? Four ways to choose three things from a set of four things. I am still looking at the triangle, but my first thought is that if I choose three dimensions to be spatial, then there is one left, which we use for time. So, blessed be, there should be four ways to do this, so four different choices of time line in each instant. Of course, we are coming from one line, so we really have a choice of three time lines at any instant, since going back to the previous node is the same as not going anywhere at all. So the three we can choose from become our three spatial dimensions. The critical point here seems to me to be that the multiverse extends beyond what we have to choose from.

The sceptic view of this idea is that if we cannot choose, cannot even know anything about it, why should we believe it exists? Actually you can get along just fine without all those unwieldy extra universes cluttering up the scenery. Does the Donald know or care about the multiverse?

Well, here is the thing. In the isomatrix model, any instant is surrounded by and in contact with twelve other instants. Three of them are in the immediate future, three in the immediate past, and six co-inhabit the present instant. The expansion pushes out in all twelve directions. We only measure the past-future push. That leaves ten other directions to account for the extra dimensions of string theory. That means that when looking at infinities, like the total expansion of the universe, we find a factor of ten masses unaccounted for. Huh. The missing matter is co-instantaneous but found along other time lines, ones that lead, instantaneously, to other time lines, other dimensions, other universes in the multiverse.

I have to tell you that I have found Rees et al and their knob twiddling to be rather trivial. Please don't be mad. There may be other universes where such things as the electron-muon distance are different from ours, but none of those universes is anywhere close to our region of the multiverse. Those choices were made so long ago that we do not have to make them ever again. All the universes that branch out from here, and all the others that have branched out for several billion years now, are nearly identical to our own in choice of these fundamental numbers.

Of course, one may learn a great deal about how the chosen numbers affect our universe, but the anthropological principle is correct, in so far as we only have to be concerned about the universal constants we find ourselves inhabiting.

Anyway, the reason I have been insisting that you and the others look into the isomatrix is because not all the directions are through the triangles. There are also directions that move in a space-like fashion. The isomatrix has as its fundamental simplex the triangle, but then there are also sets of rectangular planes. In fact, there are four sets of triangular planes, but three sets of rectangular planes. The rectangular planes look just like our usual interpretation of 3space, the Cartesian coordinate system.

By the way, I agree that life is torment, or at least, pain. The Budhists point out that the source of pain is desire. Eliminate desire, and the pain is, well, not gone, but not so important. I would not be here today typing at this screen if I had not fallen off a tall roof and shattered my right femur. That was a painful experience, but it gave me pause for reflection. I decided, while laying in a frozen swamp at the bottom of a hill below the building which was so kind as to try to kill me, that I should not keep all these thoughts about the multiverse to myself any more. After all, the Dalai Lamma, who may well be the only real spiritual leader in the world today, has called for all the libraries to be opened. He has played the drum, he has given breath to the flute, and we in the public have heard these things. It is time to open the libraries, which have been kept so close to preserve them, but now must be flung to the winds if they are to be saved.

In our hundred years of solitude, we must rely, at last, on the wings of butterflies to come and carry us away. Gaia is waiting for us.

Anyway, I am still working on the meaning of the factorial formula. I have to go get ready for the menial labor which keeps me paying my bills, now. But I will copy the factorial formula onto paper and carry it with me to think about while scrubbing. I believe in the virtue of physical work. Especially physical work in service at the root of things. That is where the real differences are made. But I am getting tired. I don't know how much longer I can keep this up. For a while, at least, I guess. Only I feel sorry for my hands, which are gradually becoming lumps covered with a net of scars. They were beautiful, once, I guess.

Be well, Marcus, whoever you are. Why don't you email me and tell me about yourself? I am mostly harmless.

Richard.

 

[Lacy33]

Hi Marcus.

After all, the Dalai Lamma, who may well be the only real spiritual leader in the world today, has called for all the libraries to be opened. It is time to open the libraries, which have been kept so close to preserve them, but now must be flung to the winds if they are to be saved.
Richard.

3899

Richard I hope that one of the books you intend to "fling into the wind" , I'm guessing is teach to the world is not the Sefer Yetzirah which you took from this library.
Anyone suffering from mental illness such as the "deep, serious, permanent depression" you have now shared with the world is a most dangerous combination to the Kabbalah and any works and attempts to comprehend these concepts as I see much of your offerings are directly connected can cause irreversible damage.
Before you go any further with your calculations in the areas you are exploring now and have been. For fear of danger, do what ever you can to make yourself permanently positive and reasonably happy. You are exploring things of a potentially explosive nature.
It is true, "they are out to get you" ...But I am not one of them.
I believe in your work.
Your work however far from being organized is something that may one day make a substantial breakthough in physical science simply because it IS beginning to reveal the mysteries and this I state as your arguments are revealing the secrets of the ancient books of Kabbalah thousands of years old, that I know you had not seen previously.
Suzanne

 

[marcus]

...
... mostly harmless.

a pleasure to hear Douglas Adams quoted. the HH's Guide verdict on our planet

...
I have to tell you that I have found Rees et al and their knob twiddling to be rather trivial. Please don't be mad.

certainly not angry
it is a bit trivial from my perspective too, but I would like some better
articles found or written about everyday life interpretation of basic constants-----havent seen Rees: he may be watered down which would make it blah

...
This board's format makes it hard to show here, but you line the numbers up in centered rows, as I recall. The numbers in each row are the sum of the numbers immediately above them, I see, so (1,1,)(1,2,1)(1,3,3,1)(1,4,6,4,1)(1,5,10,10,5,1) (1,6,15,20,15,6,1) and so on.
...

You should learn the "code" format for writing TABLES and MATRICES here at PF. it is elementary and easy
just look at this post, where I will write a table, and press "quote" and it will show you how it is typed

in essence you just say [kode] at the beginning and [/kode] at the end, but spell code right, with a c.

1     1
1     2     1
1     3     3    1

all that happens is that it is forced to take it seriously when you type spaces between things. It hears the spacebar.

you can use "code" format to make pascal triangle, for example

             1
         1     1
     1      2     1
1       3      3      1

 

[marcus]

from the Babylonian Talmud:

"It is never advisable for anyone to speculate on these four questions:

What is above?
What is below?
What was there before the world?
What will there be after it?

It would have been better for him had he never been born."


this is from the part of the Babylonian Talmud called
Hagigah 2:1, I have just seen this as the reference. I am not an expert about this and have not read it.

this is a bitter wisdom

if everybody would pay attention to this there would be no cosmologists because cosmologists are always asking these unwise questions

One of the young Quantum Gravity researchers has chosen this quote from the babylonian talmud to begin his thesis.
He is French and he has used the French version of this Hagigah 2:1 passage.
His name is Etera Livine and his thesis is here:
http://arxiv.org/gr-qc/0309028

the french version is this:
Quiconque s’est jamais avisé de spéculer sur ces 4 questions :
– Qu’y a-t-il au-dessus ?
– Qu’y a-t-il en-dessous ?
– Qu’y avait-il avant le monde ?
– Qu’y aura-t-il après ?
Il aurait mieux valu pour lui qu’il ne fut jamais né.

Talmud de Babylone

the title of Livine's thesis, in case anyone is curious about that, is
"Boucles et Mousses de Spin en Gravité Quantique"

 

[nightcleaner]

Hi Marcus. Hi S.

Well, I tried to work out the factorial formula, but muffed it. I'll have a look at it again in the daylight. Thanks, Marcus, for the code. That will no doubt turn out to be useful. And thanks, Shoshanna, for the book. That is still protected, unopened, under covers three. It is a dirty world, after all. Lots of work for a night cleaner.

So I should be afraid that asking these questions will make me more depressed? Not likely. Entertainments don't lift my spirit, and focused thought does not drag it down. It is what it is. I am not trying to change it, but to accept it as a gift. And it is not a matter of being negative or even of being unhappy. I am happy enough. And I always try to take what I find ugly and make it better somehow. Not to hide it. Just to make it better.

I am not ashamed of my depression. It is pain. Should I be ashamed of my broken leg? Thanks to medicine and surgery, I can walk, and still want to. Without help, I would not be here. Should I be ashamed of that? Then who would be here if not for help? Only a blind fool thinks he or she stands alone in this world. None of us got here on our own. I am neither proud nor ashamed of myself. On the whole, however, I am ashamed of the Human species. We could have done much better. For human individuals, I feel a great love, and a great pride, but it is not for their humanity, it for their transcendence. We find ourselves in the muck. We try to rise. It is no sin to fail to rise, but it is terrible not to try.

So, onward and upward. To paraphrase Oscar Wilde, 'I may be lying in the gutter, madam, but while you are looking down on me, I am looking upward, at the stars.' And of course, the next day, he was sober, although I am sure he wanted to get drunk again, while she, never having been drunk, was still stupid and ugly.

Thank you for the warnings, but I think I know the danger of star gazing. I already think it would have been better, for me at least, not to have been born, so what have I to lose? I am here. I will do the best I can with what I have been given. And among the things given to me was a curiosity about what is above the above, and what is beneath that which is below. That curse, if it is one, has already been cast.

As for the explosions, there was a time when I thought I should keep what I know to myself, for fear of the big W. Not Dubya, no, but Weaponization. I thought it would be terrible if someone found out how to use my ideas to make a more terrible bomb with which to destroy the world. But what use was that thought to me? And, what use was it to the world? I am a lowly servant of idiots, so it has done me no good to hide. And as for the world, there is already enough explosive power to destroy us all. If we are going to blow ourselves up, then it will happen without my little bit of wisdom. What we need now is not to hide the light under a basket, but to bring it out and use it to make tools for our common survival.

My mind is still open, so is my heart. I will not let fear shut either down. If you want me to shut up, give me good reason, and put away your bug-a-bears, I am not frightened. Time will shut me up soon enough. While we are here, let's converse a while. Let's learn what we can from each other. Tell me if I am wrong, I will not be angry.

Let us be well,

in the name of Love,

Richard

 

[Lacy33]

Hi Marcus. Hi S.
My mind is still open, so is my heart. I will not let fear shut either down. If you want me to shut up, give me good reason, and put away your bug-a-bears, I am not frightened. Time will shut me up soon enough. While we are here, let's converse a while. Let's learn what we can from each other. Tell me if I am wrong, I will not be angry.

Let us be well,

in the name of Love,

Richard

Good day Marcus and Richard,
Marcus, The Talmud says the things you say but it is also written that if a person seeks out the mysteries and can not leave it alone than "They belong to him". Seeking the mysteries are discouraged to test a persons strength and need to engage. It is not the Rabbis, who wrote the Talmud, who discourage the seeking of the mysteries, but the nature of the mysteries themselves. Certianaly we could fill pages of Talmud here arguments for both.
The Mystic will find their way through while the Scholars of the Talmud go back and forth and the Scholars of Blessed Memory know this!
Richard's work, by his own intuitition has led him to the same findings of the Mystics of long ago, (in that it hints at what is written in the Kabbalah). Again it is not complete in the way it resembles the Kabbalah nor do I think it is a complete work in physics. (That part I can not comment on as I am not a physicsist). BUT it does have Merit.

Richard, we note the way you choose to represent youself personally on this open physics forum.
I would not add one way or the other if personal truth and intellectual honesty are requirements here.
Again I want to say that I support your efforts to communicate your vision. This seems to be a safe place to do that as you do not claim any professionalism and you have only to gain from the educated and tolerant people willing to help. As we all know, finding assistance for creative thinkers is at best maddening.
I will conclude with more Talmud as you seem to be communicating with someone who has studied such and is willing to post it on a physics forum... "Three went into the Garden only one came out unharmed". This means to say that three were smart enough to get into the garden or receive information, but only one was fortified enough to endure the light.
Suzanne

 

[marcus]

\left(\begin{array}{cc}N\\k\end{array}\right) = \frac{N!}{k!(N-k)!}

for this, you have to know what N! the factorial of N is,
and you should know the convention that 0!, the zero factorial, equals one.


the N-choose-k numbers are those appearing in the "Pascal triangle"

\left(\begin{array}{cc}3\\0\end{array}\right) = 1

\left(\begin{array}{cc}3\\1\end{array}\right) = 3

\left(\begin{array}{cc}3\\2\end{array}\right) = 3

\left(\begin{array}{cc}3\\3\end{array}\right) = 1



\left(\begin{array}{cc}4\\0\end{array}\right) = 1

\left(\begin{array}{cc}4\\1\end{array}\right) = 4

\left(\begin{array}{cc}4\\2\end{array}\right) = 6

\left(\begin{array}{cc}4\\3\end{array}\right) = 4

\left(\begin{array}{cc}4\\4\end{array}\right) = 1



\left(\begin{array}{cc}5\\0\end{array}\right) = 1

\left(\begin{array}{cc}5\\1\end{array}\right) = 5

\left(\begin{array}{cc}5\\2\end{array}\right) = 10

\left(\begin{array}{cc}5\\3\end{array}\right) = 10

\left(\begin{array}{cc}5\\4\end{array}\right) = 5

\left(\begin{array}{cc}5\\5\end{array}\right) = 1



the number of triangle faces of a tetrahedron is the number
of THREEPOINTED simplices belonging to a FOURPOINTED
is four choose three, namely 4.

and to take another example the number of edges of a tetrahedron, the number of
TWOPOINTED simplices belonging to a FOURPOINTED
is four choose two, namely 6.
==============

now we go to 4 dimensions. the basic simplex in 4D, the socalled "4-simplex" is a FIVEPOINTED simplex

how many of ITSELF does the fivepoint simplex have?
five choose five, namely 1.

how many tetrahedrons does it have, as its threedimensional "sides"?
well a tetrahedron is a FOURPOINTED so to specify one you have to choose 4 points from the 5
so it is five choose four, namely 5

and how many triangles?
five choose three, namely 10

and how many edges?
five choose two, namely 10

and how many vertices?
we already said,
five choose one, namely 5

just as a game, to complete the sequence, let us ask
a further question
and how many NOTHINGS does the basic fivepoint simplex have?
five choose zero, namely 1

 

[nightcleaner]

Hi Marcus and Shoshana

Thanks for the wisdom, S., always appreciated, even when not well taken. And Marcus, always appreciated even when not understood. I am still trying. I have to go make coffee and do chores, but will return to Pascal and the N! formulation this afternoon offline.

Last night I tried to fit the N!'s into the formula to come out with the triangle, and it worked some of the time, but not for every trial. Probably I don't remember some of the details of factorials and will have to go to Wickipedia for a refresher. I thought about the zero factorial, and also about how to make a factorial of a negative.

3!=1+2=3
4!=3!+3=6
5!=4!+4=10

so,
2!=1!+1=1
1!=0!+0=0

0!=-1!-1 so -1! should equal 1?

Hmmmm.

No that can't be right. Will have to have coffee.

By the way, just finished reading "The man who loved only numbers" about Paul Erdos, a famous mathematician whom I had never heard of. I guess from the reading that he is only famous among mathematicians. An inspiring story for impoverished seekers of truth. However the title is misleading. The way i read it, Erdoes loved many people, and loved to talk, but numbers were the only thing that made any sense to him. Apparently he often began conversations, even with old friends whom he had not seen for long periods of time, with something like "Hello. Let n be an even integer..." He was a brilliant, kind, and funny man who was a beloved pain in the ass to most everyone he knew, especially to those who understood him.

btw, Marcus i think Rees is the real stuff and the others are the imitators, but I could be wrong. Sir Martin Rees, Lord High Astronomer to the British Crown or some such fooferall. He is at Cambridge, I think, and was one of Brian Greene's professors.

Be well. Coffee calls.

nc

 

[marcus]

0! = 1
1! = 1
2! = 2
3! = 6
4! = 24
5! = 120

to get the next you always multiply the number you are at.

so, start at 0!= 1 and to get the next, namely 1!, multiply by one
then to get the next, namely 2!, multiply what you have already by two
then to get the next, namely 3!, multiply what you have already by three

Act III of "Lady Windermere's Fan"
We are all in the gutter, but some of us are looking at the stars.
brevity and generality

 

[nightcleaner]

Oh yeah, do I feel dumb. I just went to Wikipedia and still have not gotten coffee. Ugh. So I'll try it again with multiples instead of adds. Cheese. BRB

Ahhh, the bitter and the sweet.

0! = 1    
1! = 1     =0!x1
2! = 2     =1!x2    
3! = 6     =2!x3
4! = 24   =3!x4
5! = 120  =4!x5

So we could imagine 0!= 1 = -1!x0. So -1! is the number which, when multiplied by zero, equals one. Now there is an insistant explosion for you.

5! = 120  =4!x5
4! = 24   =3!x4
3! = 6     =2!x3
2! = 2     =1!x2 
1! = 1     =0!x1
0! = 1     =-1!x0
-1!= n     =-2!x-1

comments?

We could multiply both sides of the last equation by -1, then 1!=-n=2!. Not so good. Could be a problem with multiplication. Maybe -1! x -1 does not equal 1!

-1! = n = -2! x -1
-1 x (-1!) = -n = -1 x (-2! x -1)
-1 x (-1!) = -n = -1 x (-2!) x -1 = -2!

I suppose we could say that n is infinity, so 0 x infinity is unity. Unity is no infinities? Sort of seems to make sense. One is to infinity as zero is to one?

I am getting cross-eyed. Maybe there just are no factorials of negative numbers, as Wikipedia said. Anyway I promised to work on Pascal and the dimensional stuff for Marcus. I'll go offline and do that now instead of trying to invert infinities.

nc