The fine-structure constant might be the most popular target of physics numerologists. α numerology has a long history, such as Eddington's efforts and Feynman's remark. It's a recurring topic in this long thread which might be the high point of Internet-era physics numerology.
Today on vixra there is an article which speculates about how to obtain one of the numerological formulas for α, 4π3+π2+π. It's unusual for two reasons. First, the author (Amir Mulic) speaks the technical language of M-theory; he proposes to "interpret... this expression in terms of the volumes of lp-sized three-cycles on G2 holonomy manifolds". (lp would be the Planck length.)
Second, he mentions that the coupling has to "run", i.e. change its value with energy scale. This aspect of quantum field theory is why particle physics professionals tend to ignore even Koide's relation, to say nothing of the more baroque formulae invented by amateur numerologists. The modern paradigm is that simple relations among particle masses and coupling constants exist at ultra-high energies, but that at low energies these relations will be obscured by complicated corrections, e.g. extra terms containing a logarithm of the energy, described by "beta functions" which can be derived from fundamental theory.
I haven't really gone over Mulic's article (I note that he had a similar one on arxiv years ago), and I am apriori skeptical that this particular idea will work out. But what's noteworthy here is just that someone is making this sort of effort - trying to explain the numerological formulas using the full conceptual apparatus of modern mathematical physics.
Before I comment further, it might help to show how things look without such a bridge. On one side, we have the efforts of someone like Angel Garcés Doz, already mentioned several times on this blog. Garcés Doz works hard, and like Mulic, draws inspiration from 7-dimensional geometry. Still, I find his formulas more interesting than his physics.
On the other side, consider this item of F-theory phenomenology (via Lubos). Here we have a genuine example of how a string-theory background geometry might determine a particular value of α: in this case, it's "the number of fuzzy points" in "a non-commutative four-cycle" wrapped by a 7-brane. But the value of α thereby obtained is the high-energy value, the value at the grand unification scale - perhaps 1/24 or 1/25, says Lubos. It only approaches 1/137 at low energies because of those messy correction terms.
Incidentally, this "fuzzy F-theory phenomenology" played a role at the dawn of my own attempts to make sense of what Marni Sheppeard was doing. One day she exhibited a parametrization of the CKM matrix, in terms of circulant matrices, and I was interested in whether this could fit into an existing framework like F-theory. It was very interesting to see that number 24 appearing as one of her parameters, but at the time none of us knew enough to judge whether Brannen and Sheppeard's circulants, and Heckman and H. Verlinde's fuzzy points, could fit into the same theoretical synthesis.