[pascal]

Dunno anything about the stuff you've written as yet. Gotta empty my old head a little first. But, here's an interesting problem.

You may remember my discovery (probably rediscovery) that when one stacks the equations describing the powers of the Metal Means one atop the other, the individual terms have powers and numerical coefficients that come directly out of the (2,1)-sided Pascal Triangle sister.

I was discussing today with the fellow who just got his doctorate in Linguistics working on Phi-related behaviors in syntax, who is looking deeper into the issues involved. He now believes that a slew of Pisot-Vijayaraghavan numbers might describe some syntactic branching phenomena.

My Metal Means are just those PV numbers where we have X^2-NX-1, where N is the N as in 'the Metal Mean of N' when one has the equation (N+sqrt(N^2+4))/2.

What I want to know is this- if the (2,1)Pascal describes Metal Mean powers, might the OTHER Pascal sisters similarly describe other PV powers???

Jess Tauber

[Turtle]

pascal, on 10 May 2012 - 06:42 PM, said:

Dunno anything about the stuff you've written as yet. Gotta empty my old head a little first. But, here's an interesting problem.

acknowledged.

pascal said:

You may remember my discovery (probably rediscovery) that when one stacks the equations describing the powers of the Metal Means one atop the other, the individual terms have powers and numerical coefficients that come directly out of the (2,1)-sided Pascal Triangle sister.

i haven't been able to "see", grok, get-a-grip-on or otherwise internalize exactly what a "Pascal Triangle sister" is or Metal Means are. here's where some drawings would find great utility.

but otheriwse, are you just seeing/describing binomial coefficients?

pascal said:

I was discussing today with the fellow who just got his doctorate in Linguistics working on Phi-related behaviors in syntax, who is looking deeper into the issues involved. He now believes that a slew of Pisot-Vijayaraghavan numbers might describe some syntactic branching phenomena.

will have to find some info on those numbers as i'm unacquainted. do you have a list handy?

but fear not, i just thought of a 3-dimension stacking phi-related behavior that i explored and craig added some computer calculations and observations on area & volume you will find interesting. it's short, but schweet. Fibonacci bricks

pascal said:

My Metal Means are just those PV numbers where we have X^2-NX-1, where N is the N as in 'the Metal Mean of N' when one has the equation (N+sqrt(N^2+4))/2.

What I want to know is this- if the (2,1)Pascal describes Metal Mean powers, might the OTHER Pascal sisters similarly describe other PV powers???

Jess Tauber

PV? i've apparently missed the definition of that acronym.

[pascal]

Methinks that if you look these things up on Wiki (except for the sister Pascal stuff, which isn't there) most will be made clearer to you PV=Pisot-Vijayaraghavan numbers. Pretty famous. Don't fret- until a year or two ago I didn't know any math beyond advanced placement calculus from high school.

Jess Tauber

[Turtle]

pascal, on 10 May 2012 - 10:01 PM, said:

Methinks that if you look these things up on Wiki (except for the sister Pascal stuff, which isn't there) most will be made clearer to you PV=Pisot-Vijayaraghavan numbers. Pretty famous. Don't fret- until a year or two ago I didn't know any math beyond advanced placement calculus from high school.

Jess Tauber

so too apparently is pisano "pretty famous", and previously unknown to either of us. anyway, , seeing as all integers are "PV" it doesn't change anything regarding the modular arithmetic or the sets of polygonal and non-polygonal numbers as i have presented them. just as phi shows up in nature a great many ways, we peoples have devised a great many ways to represent it. some propositions, even if true, simply cannot be proved.

don't fret on your short time in the field; there's plenty of time to learn to draw.

[pascal]

Well, in the past days I've been exploring other connections between these numbers and Pascal Triangle sisters. So far I've found repeated motifs in the deep diagonals. For example, in analogues of the tetrahedral diagonal in higher Pascal systems, the values in the diagonals are themselves composed of original tetrahedrals in combination. One has summed contiguous pairs of classic tetrahedral numbers as its tetrahedral analogue values, making it act like Fibonacci numbers. The next higher analogue has a classical tetrahedral number added to TWICE the next contiguous tet number, making it analogous to Pell numbers. And so on. Thus while the actual Fiblike series generated by these Pascal sister triangles are all related only to the Golden Ratio, their deep diagonals relate more to the entire series of Metal Means. Even the classical Pascal Triangle, since its tetrahedral numbers sum with zero times the next contiguous one, this relates to the first Metal Mean, which is 1.00000....

All the Pascal sisters I've mentioned above have sides defined by numbers that are separated by one: (1,0),(2,1),(3,2) and so on. But if one were to use other differences between side values, would that mean that the internal diagonal values be related more to the PV numbers?

The Golden Ratio is a Metal Mean, and the Metal Means are PV numbers, but in each case we see a broadening of classification, with reduction of shared prototypical characteristics with each larger class.

I've found many relations in the periodic system to the Golden Ratio, a couple it seems to Metal Means. If there are higher level PV-related periodic behaviors, that would be very interesting indeed. Why should Nature limit herself to one-size-fits-all motivations?

Jess Tauber
goldenratio@earthlink.net