Posted 01 February 2012 - 09:06 AM

Hi Jess,

And thanks for all the interesting feedback.

I am of the opinion that the underpinnings of the universe are essentially

both logical and mathematical (and therefore "eternal"). In other words,

I believe that the universe may very well be, at its most fundamental level,

comprised not of "particles" or "strings", but of "self processing information".

I also agree with you that [math]\phi=\frac{\sqrt(5)+1}{2}=1.61803398874...[/math]

is one of the most important numbers in all of nature.

Your recent posts gave me an idea, and I think it's a pretty interesting one.

Here it is, and after you read it, please let me know what you think about it...

There are, at present, many proposed "Grand Unified Theories" and "Theories of Everything",

which involve a very special mathematical object called [math]E_{6}[/math], whose fundamental representation

is [math]27[/math] dimensional. Thus,

[math]27=3^3=5^2+\frac{\sqrt(5)+1)}{2}+\left(\frac{\sqrt(5)+1}{2}\right)^{-2}=5^2+1.61803398...+\frac{1}{2.61803398...}[/math]

may be the most "fundamental" number of all. (At this point, we just don't know.)

However, since [math]\frac{\sqrt(5)+1)}{2}+\left(\frac{\sqrt(5)+1}{2}\right)^{-2}=2[/math],

all positive integers of the forms [math]2N[/math] and [math]2N-1[/math], that is,

*all* positive integers *including* [math]27[/math] can be viewed as involving [math]\phi[/math].

Note that the above representations of the number [math]27[/math] involve the numbers [math]1,2,3,5,\phi,\phi^{-2},[/math]

all of which are either Fibonacci numbers, or closely related to them.

Now, as it turns out, the most *simple* "super accurate" counting function for

**polygonal numbers of order greater than 2** seems to be...

[math] \varpi(x)\approx\left(\left(\sqrt{\left(\left(1-\frac{1}{\left(\alpha*\pi*e+e\right)}\right)*x\right)}-\frac{1}{4}\right)^{2}-\frac{1}{16}\right)*\left(1-\frac{\alpha}{\left(6*\pi^{5}-\pi^{2}\right)}\right) [/math]

where:

[math] \alpha=\left(\left(A^{-1}*\pi*e+e\right)*\left(\pi^{e}+e^{\left(\frac{-\pi}{2}\right)}\right)-\frac{\left(27*\left(\ln\left(x\right)\right)^{-1}+1\right)}{\left(6*\pi^{5}*e^{2}-2*e^{2}\right)}\right)^{-1} [/math]

which results in the table:

[math] x [/math]_______________________[math]\varpi(x)[/math]_________________[math] B(x_{F\alpha}) [/math]____________Difference

10_______________________3______________________5___________________2

100______________________57_____________________60__________________3

1,000____________________622____________________628_________________6

10,000___________________6,357__________________6,364________________7

100,000__________________63,889_________________63,910_______________21

1,000,000________________639,946________________639,963______________17

10,000,000_______________6,402,325______________6,402,362_____________37

100,000,000______________64,032,121_____________64,032,273____________152

1,000,000,000____________640,349,979____________640,350,090____________111

10,000,000,000___________6,403,587,409__________6,403,587,408__________-1

100,000,000,000__________64,036,148,166_________64,036,147,621_________-545

1,000,000,000,000________640,362,343,980________640,362,340,978________-3002

10,000,000,000,000_______6,403,626,146,905______6,403,626,142,382_______-4523

100,000,000,000,000______64,036,270,046,655_____64,036,270,047,407_______752

200,000,000,000,000______128,072,542,422,652____128,072,542,423,321______669

300,000,000,000,000______192,108,815,175,881____192,108,815,179,516______3635

400,000,000,000,000______256,145,088,132,145____256,145,088,131,948_____-197

500,000,000,000,000______320,181,361,209,667____320,181,361,209,475_____-192

600,000,000,000,000______384,217,634,373,721____384,217,634,375,674______1953

700,000,000,000,000______448,253,907,613,837____448,253,907,608,937_____-4900

800,000,000,000,000______512,290,180,895,369____512,290,180,895,206_____-163

900,000,000,000,000______576,326,454,221,727____576,326,454,224,725______2998

1,000,000,000,000,000____640,362,727,589,917____640,362,727,590,399______482

This version of my counting function is also noteworthy in that as [math]x[/math] goes into infinity,

the value of [math]\alpha^{-1}[/math] as measured at the scale of the electron mass is only *approached*

and never goes higher than [math]137.035999083778...[/math], thereby strongly implying that

the value of the fine structure constant at zero energy can be neither observed,

*nor* deduced, and will therefore **always** remain a mystery.

Don.