A few trivia about the fine structure constant

Clifford Pickover

In a post I published in this blog over three years ago, I talked about the fine structure constant and some of its physical peculiarities. In this post I am going to talk about some of its mathematical peculiarities. I have taken them from a book I have mentioned on other occasions: A Passion for Mathematics, by Clifford A. Pickover.

As we know, the most exact value of this constant is this:

α = 1/137.035999206

Pickover points out that Eric W. Weisstein, in his World of Physics website, offers two mathematical approximations incredibly close to this value:

α^–1 ≈ 44 π – cos^–1(e^–1) = 137.03600 . . .

α^–1 ≈ 96(e^1/2 + 2^1/3)^1/3  = 137.03599 . . .

Arthur Eddington

Physicist Arthur Eddington, who we have talked about often in this blog, was fascinated by the fine structure constant. Apart from stating that its value should be exactly equal to 1/137 (although the most exact measurements show that it is not), he also maintained that this value is related to the number of electrons in the universe (another statement that many physicists do not accept).

Another curious formula, proposed by a Russian mathematician, relates (approximately) the value of the fine structure constant to the number π and the golden number ϕ, according to the following formula:

As could be expected, this constant has also been associated to the pyramid of Cheops and the megalithic monuments of Stonehenge.

Richard Feynman

Richard Feynman wrote this about the fine structure constant:

[The value of the fine structure constant] has been a mystery ever since it was discovered more than fifty years ago, and all good theoretical physicists put this number up on their wall and worry about it. Immediately you would like to know where this number for a coupling comes from: is it related to pi or perhaps to the base of natural logarithms? Nobody knows. It’s one of the greatest damn mysteries of physics: a magic number that comes to us with no understanding by man. You might say the ‘hand of God’ wrote that number, and ‘we don’t know how He pushed His pencil.’ We know what kind of a dance to do experimentally to measure this number very accurately, but we don’t know what kind of dance to do on the computer to make this number come out, without putting it in secretly!” (Richard Feynman, QED, 1988).

The same post in Spanish

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Manuel Alfonseca

Have a happy vacation. See you by mid-August.