Calculating The Fine Structure Constant.

[IDMclean]

Fine Structure Constant Varies With Direction in Space, Says New Data

[Don Blazys]

Fine Structure Constant Varies With Direction in Space, Says New Data

 

Yeah, I know.

 

But the really cool thing about this "non-trivial polygonal number counting function"

is that if we use it to estimate the fine structure constant,

then the random fluctuations in the actual count of non-trivial polygonals

guarantees that the fine structure constant will have both an upper and a lower bound.

 

Sloans Online Encyclopedia Of Integer Sequences now references this counting function

and has included some of my comments which you can find here:

 

http://www.research.att.com/njas/sequences/index.html?q=polygonal+order+greater+2&language=english&go=Search

 

I also expanded the table in the article on my website, so that it better reflects the "random fluctuations"

in the actual count of non-trivial polygonals. You can find that table here:

 

http://donblazys.com/on_polygonal_numbers_3.pdf

 

B(x)*(1-a/(u-2*e)) plus or minus 2*x^(1/4)

gives a very realistic upper and lower bound,

and projecting those bounds to x=10^17

shows that the fine structure constant lies between

137.035999092 and 137.035999077.

 

Don.

[IDMclean]

Yeah, I know.

 

But the really cool thing about this "non-trivial polygonal number counting function"

is that if we use it to estimate the fine structure constant,

then the random fluctuations in the actual count of non-trivial polygonals

guarantees that the fine structure constant will have both an upper and a lower bound.

 

Sloans Online Encyclopedia Of Integer Sequences now references this counting function

and has included some of my comments which you can find here:

 

http://www.research.att.com/njas/sequences/index.html?q=polygonal+order+greater+2&language=english&go=Search

 

I also expanded the table in the article on my website, so that it better reflects the "random fluctuations"

in the actual count of non-trivial polygonals. You can find that table here:

 

http://donblazys.com/on_polygonal_numbers_3.pdf

 

B(x)*(1-a/(u-2*e)) plus or minus 2*x^(1/4)

gives a very realistic upper and lower bound,

and projecting those bounds to x=10^17

shows that the fine structure constant lies between

137.035999092 and 137.035999077.

 

Don.

 

I placed this here because I thought it might yield an insight to vectorizing the counting function. If the fine structure varies by direction, the number of nonfiguratives should also vary by direction, no? We could construct a mathematical space representing the space of the nontrivial figuratives and see if we can match results. A Hilbert space seems like a good place to start to me though I don't have the mathematical tools to carry out the work myself.

[Don Blazys]

There is no denying that this non-trivial polygonal number counting function is very accurate.

In fact, it approximates the number of regular figuratives under x far more accurately than

Li(x) approximates the number of primes under x.

 

Thus, as a purely mathematical construct, its integrity is beyond reproach and

considering how extraordinarily difficult it is to actually count how many

regular figuratives there are under x, it is also an extremely useful counting function!

 

For these reasons alone, this counting function deserves further investigation.

 

That said, it should also be remembered that this is the only known mathematical function

that actually requires the physical constants "alpha" and "mu"in order to work,

which makes it both a "rare gem" and a "precious gift", especially since it's discovery was totally unexpected.

 

Anyway, according to several news articles, if we go from "one side of the universe to the other",

then the fine structure constant might go from about 1/137.035989 to about 1/137.036009.

 

Likewise, according to the table in the article on my website, if we go from:

x=300,000,000,000, w(x)=192,108,604,710 to x=400,000,000,000, w(x)=256,144,844,029,

then the fine structure constant goes from 1/137.035946252 to 1/137.03600571447

which is quite reasonable, considering the very limited amount of data that we have on w(x).

 

Don.

[IDMclean]

Variations in fine-structure constant suggest laws of physics not the same everywhere

Evidence for spatial variation of the fine structure constant

 

My only interest in posting these updates is the answer to the question: can we mathematically and deductively reproduce the results?

[Don Blazys]

Quoting IDMclean:

If the fine structure varies by direction,

the number of nonfiguratives should also vary by direction, no?

We could construct a mathematical space representing the space of

the nontrivial figuratives and see if we can match results.

A Hilbert space seems like a good place to start to me though

I don't have the mathematical tools to carry out the work myself.

 

Thanks IDMclean!

 

This is an interesting idea!

I've been thinking about it,

from time to time and

every now and then,

ever since you posted it.

 

I'm gonna start "playing"

with it in my spare time.

 

Also, I got some interesting results applying

absolute values to the function:

[math]B(x)*(1-\frac{\alpha}{\mu-2*e})[/math]

so that it produces negative approximations of:

[math]\varpi(x)[/math]

 

I will also post those results in the near future.

 

Don.

[Don Blazys]

Here is a very interesting article on how prime numbers

might be related to quantum mechanics.

(If they are, then odrer>2 polygonals being related to the FSC

might not be such a far fetched idea after all!)

 

http://www.americanscientist.org/issues/id.3349,y.0,no.,content.true,page.1,css.print/issue.aspx

 

 

Don.