Talk:Fine-structure constant

The de Vries iterative algorithm

i've been studying the de vries algorithm [1]

it has two terms, which may be expressed as:

$\alpha _{\rm {\infty }}=\gamma ^{2}e^{-\pi ^{2}/2}$

$\gamma ={\frac {\alpha ^{0}}{\left(2\pi \right)^{0}}}+{\frac {\alpha ^{1}}{\left(2\pi \right)^{1}}}+{\frac {\alpha ^{2}}{\left(2\pi \right)^{3}}}+{\frac {\alpha ^{3}}{\left(2\pi \right)^{6}}}+{\frac {\alpha ^{4}}{\left(2\pi \right)^{10}}}+{\frac {\alpha ^{5}}{\left(2\pi \right)^{15}}}...$

when executed with sufficient accuracy (at least 150 digits) and enough iterations (over 30 and over 70 triangular terms) [2]

it reaches the following value:

0.0072973525686538583817641878681570233311504125595093

where the inverse is:

137.035999095829623684039688669145107269287109375

here is the 2010 CODATA value [3]

0.0072973525698

that value has a relative uncertainty of 3.2e-10, and the de vries algorithm settles on a value is within 1.6e-10 of that 2010 CODATA value.

now, there are a lot of "algorithms" for $\alpha$. rob munaro maintains a list http://mrob.com/pub/num/n-b137_035.html. all of them with the sole exception of the de vries algorithm are inaccurate, i.e. all of them with the sole exception of the de vries algorithm are outside of the relative uncertainty for the 2010 CODATA value.

given that the de vries algorithm is entirely comprised of primitives (1, 2, e and pi) this is actually very exciting.

2018 CODATA

From the ref NIST, the value is 0.0072973525693

I am unsure what Mathematica gives, but using this value the corrected value for A164040 would be 3.3820235640 * 10^38. Bill McEachen (talk) 22:19, 19 April 2023 (EDT)

• In any case, don't give excessive digits: see the page Decimal expansions on uncertainties. Now it makes sense to wait a few months for CODATA 2022. And anyway, I would give A164040 keyword "less". --Andrey Zabolotskiy (talk) 13:13, 20 April 2023 (EDT)