Pascal Basis For M-Theory?

[tetrahedron]

I've been starting to acquaint myself with the basics of string/superstring/m-theory, and already am noting that there is a spooky resemblance between the claims and the structures and properties of generalized Pascal Triangles.

 

First there are (and have been for a while now) SIX subtheories. The Pascal systems have SIX straight-line relationships that are normally recognized- columns, rows, two symmetrically distributed sets of edge-parallel (classical) diagonals and two symmetrically distributed sets of shallow diagonals.

 

This matches the ways that the m-theory cuts up their subtheories and the dualities between them, IF the one subtheory that is self-dual is duplicated.

 

The patterning of the spatiotemporal dimensionality also parallels that of the Pascal system's generalized Fibonacci sequences, whose terms are formed by summing numerical values found on the shallow diagonals. Thus ...10,16,26...dimensions of the bosonic theory are double Fib. OTOH ...4,7,11... belong to the Lucas sequence.

 

Interestingly the public/hidden dimension dichotomy seems to fall against the odd/even dimensioned shallow diagonals. Is there a real connection?

 

I've also been looking at decimal expansions of straight-line sequences within the triangle. We already know that generalized Fibonacci sequences pattern as whole-number multiples of 1/89, while generalized Fib numbers themselves rationalize as limit Golden Mean. Generalized sequences associated with OTHER Metallic Means (such as Pell for the Silver Mean) are similarly derived- here with multiples of 1/79, 1/69 for the Copper Mean, and so on, all perfectly regularly (and I would guess the same for the system in other bases, which are all triangular numbers minus 1, as in 1/71 for base 9, 1/89 for base 10, and 1/109 for base 11).

 

Within the classical Pascal Triangle interesting things pop out. For example edge-parallel diagonals have decimal expansions that relate to one another as negative powers of 9. Compare this with the powers of 11 for rows.

 

The center column has a limit value, as decimal expansion, of 1/2sqrt15, and ratios between nearest columns are related to this. The ratios for the shallow diagonals, as decimal expansions, have a limit related to sqrt35. An interesting spread, given that sqrt25 is 5, and related to Phi. So as I often say, something very interesting going on here.

 

Whether my hunch that this all relates to m-theory remains to be demonstrated. Yet lots of the bits and pieces DO seem to be present within the Pascal system, which is much more multifunctional than one might expect given the simplicity of its construction. Each generalized Triangle is a SINGLE object, but sampled SIX different ways, like the proverbial elephant by the wise men. There are an infinite number of such generalized Pascal Triangles, so if there IS a connection to m-theory we have our multiverse/omniverse right at hand.

 

Many of these will fall into families that share many properties (rationalization to Phi, sequence overlaps, or multiplications, etc.). Maybe we could figure out how to visit and survive the trip. Others will be radically different from what we know, especially if they are based on higher level analogues of Pascal, using other Metallic Means, or even more hypothetically other Pisot numerical motivations. And here might be the homes of the gods themselves.

[tetrahedron]

I'm still working through more generalized Pascal Triangles to see if my hypothesis bears any relation to fact.

 

In the classical Pascal system and others like it which have identical numerical values on both sides the columns are mirror images except of course for the central one, which is its own mirror. The other straight-line relations are similarly mirrored, that is the edge-parallel diagonals, the shallow diagonals pair, though rows are self-reflective like the central column.

 

The columns are of course perpendicular to the rows, but less visually obvious unless you rotate the figure 120 degrees is that the edge-parallel diagonals are perpendicular to the shallow diagonals. In sum we have three perpendicular pairs.

 

We also have a pair of triangular relations, where the straight-line relations are at 120 degrees from each other. The columns and the shallow diagonals form one triad, and the rows and edge-parallel diagonals form the other.

 

I've found several types of relationships taking sums and ratios that appear to be the same no matter which generalized Pascal Triangle is chosen, though the initial and resulting values are different between Triangles. It appears the Triangles are generalized mathematical objects with their own internal logic. Thus if there is a connection to M-theory the particular numerical values aren't at issue, and are just variations on the same themes.

 

Jess Tauber

[tetrahedron]

New generalization (well, to be honest I bet most of my findings are rather old, but its the first I'VE seen of them- always more fun to work them out on paper on your own rather than get them prechewed).

 

For column values in generalized Pascal Triangles, central column values are always a multiple of the sum of the side/edge values, and the multiples themselves are always the same set: 1,3,10,35... where the latter relate to each other as fractions that increase each time by 4 for the numerator and 1 for the denominator (so 3=1x6/2, 10=3x10/3, 35=10x14/4 etc.).

 

This may be related to the fact that for the decimal expansions of shallow diagonals the resulting numbers are, depending on which direction in the generalized Triangle one is going up to, multiples of the classical Pascal decimal expansion, and the factors are the side values of the Triangle again, but instead of being sums they are over different tens places (so for example if the sides are 4 and 7, then the decimal expansions are multiplied by 47 or 74).

 

Anyway it also turns out that for the flanking columns sums of pairs of them related by reflection symmetry are also multiples of the sums of the side values of the Triangle they are. Very highly organized system, and still working out the dynamics.

 

Jess Tauber

[CraigD]

I can’t imagine a connection between M theory and Pascal triangle, but my grasp and understanding of this kind of physics is slight, so my lack of imagination in the subject shouldn’t surprise.

 

First there are (and have been for a while now) SIX subtheories. The Pascal systems have SIX straight-line relationships that are normally recognized- columns, rows, two symmetrically distributed sets of edge-parallel (classical) diagonals and two symmetrically distributed sets of shallow diagonals.

I’m pretty sure this is coincidental, and that the number of subtheories of any physics theory isn’t something that can be derived for elementary number theory or geometry, but an artifact of the process of trying new theories and categorizing old.

 

I'm curious, Pascal: what are the 6 subtheories of M-theories you mention? Do you have a reference explaining them, and that number?

 

I'm still working through more generalized Pascal Triangles to see if my hypothesis bears any relation to fact.

I’m curious, Pascal, if you’ve tried generalizing beyond the 2 dimensional Pascal triangle, considering the 3 dimensional Pascal 's pyramid, of the n-dimensional cases? These are all easy to calculate the terms of, though like most 3+ dimensional arrangements, not easy to draw on paper or visualize. In the same way the triangle is convenient for quickly finding coefficients of the integer powers of a binomial (ie: [imath]( a + b )^n[/imath]), the pyramid is convenient for finding coefficients of the integer powers of trinomials, ([imath](a+b+c)^n[/imath]), the higher-dimensional cases the coefficients of the interger powers of a polynomial of any number of terms ([imath](a_1+a+2 \dots +a_m)^n[/imath])

[tetrahedron]

I'm still getting my sea legs in this arena, Craig. I've been in touch with several folks who deal with number sequences (as over at the OEIS site). I've GOT to start learning to think in the ways they (and you) do, but it is alien to me. I never had any formal training in it, so am still unfamiliar with the basic premises and conventions. But I keep plugging away, probably re-inventing the wheel at each turn. At this stage it is probably better that I DO do this, before trying to talk the talk.

 

I'll venture a guess that at some point I'll have to delve into multidimensional systems. As you know I've looked at sequences relating to the Metallic Means other than the Golden, and found base 10 systematicity with regard to decimal expansions and fractions as we have discussed regarding 1/89 and Fib. I've wondered what these would look like in bases other than 10.

 

Anyway, I'm still just learning the basics of M-theory myself, and make no claim that it IS related to Pascal math. I'm just asking the question and trying to figure out if it is possible, and seeing if transformations of the Pascal system parallel those of the theory. But first I have to have the internal properties. Here is the Wiki page: http://en.wikipedia.org/wiki/M-theory

 

If things are as simple as I hope then it might (in an ideal omniverse :) ) be possible to relate each one of the different straight-line relations cited in the Triangle to a different subtheory. There ARE more complex straight-line relations, and I've started to examine them as well. It may well be that the generalized Triangle is a quantized field of such relationships, more showing up the bigger the Triangle grows (downward), and then one might expect (again if there IS a link) that we should see MORE subtheories pop up (heck this could go on forever). However in the simplest cases it would still boil down to just the handful.

 

What is of some interest is that the M/string/superstring folks make claims about the maximum possible dimensions for branes etc. And it is of equal interest that the same range (depending on what you're looking at) of 9/10/11 is one of the two sets of forms that keep showing up in the base 10 Triangles (the other relating to square roots of 15,25,35).

[tetrahedron]

Quick update: If you square (5+sqrt35)/10 you add precisely .1 to the original 1.091607978... to get 1.191607978.... Interestingly if you square (5+sqrt15)/10 you SUBTRACT precisely .1 from the original 0.8872983346... to get 0.787293346... Thus these numbers relate to the two values of Phi, where if you square the larger, 1.61803398..you add exactly 1, for 2.61803398.., but if you square the smaller, 0.61803398..., you subtract exactly 1, giving .38196601...

 

(5+sqrt35)/10 (or the inverse) is the limit for an infinitely large Pascal system for the ratio of decimal expansions of two consecutive shallow diagonals, while (5+sqrt15)/10 (or ditto) is the same for for two consecutive columns. Columns and shallow diagonals have a triangular relationship in the system, their angles separated by 120 degrees.

 

For the edge-parallel diagonals the ratio has limit 1/9 or inverse, while for rows (again the triangular relationship) we have 1/11 or the inverse, for decimal expansions.

 

These two systems seem to be cut from very different cloths, and I'm having a bit of a time trying to find a coherent link.