Wednesday, June 16, 2010

Some Unimpressive Numerology

The fine-structure constant α is a fundamental constant in physics, and is currently estimated to be approximately .0072973525376.

The physicist Arthur Eddington, who became rather eccentric and believed he could compute the number of protons in the universe accurately, thought it was equal to exactly 1/137, but our current estimate gives something closer to 1/137.03599967899.

The mathematician James Gilson seems to think that α is given by the rather complicated formula (29/π)*cos(π/137)*tan(π/(137*29)). But this is just numerology, and not even particularly impressive numerology. The trick is that tan(x) is very close to x when x is small, and cos(x) is very close to 1 when x is small. So Gilson's formula is just (29/π) times something that is very close to π/(137*29), with an additional fudge factor of something that's very close to 1 thrown in. There is no real surprise, then, that one can find small integers to make this close to α.

Heck, it's obvious that the real value of the fine structure constant is actually 250/34259. Or maybe (cos(2 π/57) - sin(4 π/47))/100? I can't decide which.

8 comments:

JN said...

Nonsense! Although your logic is sound, it just demonstrates that the behavior of the cosine and the tangent for small arguments are as they are simply because Gilson's formula, which is far too complex to be wrong, must result in the value of the fine-structure constant.

Larry Moran said...

Giving a number that's accurate to 13 decimal places and calling it "approximate" is something that this biochemist finds strange.

How many more decimal places are necessary before it becomes "reasonably accurate"? :-)

Jeffrey Shallit said...

How'd you get 13, Larry? I only see 10, intepreted generously.

Larry Moran said...

I told you I'm not a mathematician! I think I see why it has to one less than the total number. I should have said 12 decimal places since the last number is rounded.

But I don't get why it's only ten.

To me, 0.007 is more accurate than rounding it up to 0.01. Is that not correct? It seems to me that the number 0.007 is accurate to two decimal places (seven one-thousandths) and 0.01 is only accurate to one decimal place (one one-hundredth). It seems to me that the number 0.01 could actually be anything between 0.005 and 0.014 - is this correct?

I'd be delighted if you could show me where I'm going wrong about the other two decimal places. I skipped most of first year math in order to play bridge and go skiing.

Are you not supposed to count the zeros? Would it be 12 decimal places if it were 0.1172973525376 but only 10 if it's 0.0072973525376?

Jeffrey Shallit said...

Larry:

It seems that the current known value of the fine-structure constant is .0072973525376.

Gilson's formula gives .00729735253186412.

So they differ at the 10'th significant digit.

Larry Moran said...

Sorry, I misunderstood.

In my first comment I was referring to what you said in the posting when you wrote "approximately .0072973525376"

It was meant to be humorous.

Lew Baxter said...

Some more numerology relating counting of polygonal numbers, fine structure constant and proton to electron mass ratio? See

http://donblazys.com/on_polygonal_numbers.pdf

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