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It's been a while since I left math for industry but I once heard that the shape and contents of "the space of all problem sets and solution sets for enumerative geometry on arbitrary n-manifolds" is something that is amenable to investigation through something called Gromov-Witten theory. I did a quick number of searches on GW theory just now and cannot decipher the results so I still don't know if that claim had any merit to it but a sanity check suggests there's no obvious computational reason that it couldn't be true for n=2 or 3.
I enjoyed reading that, but understood absolutely none of it.
Same - I get the gist, but "how many lines lie on a cubic face" doesn't make sense as a question to me, what is a "line" in this context? I struggle to understand the issue as my only understanding is that the face is divided by line width which determines how many lines, but that's nonsensical and non interesting.

As an aside (again, as a layperson) I've had this feeling with most Quanta articles, it's interesting and I feel like I get the gist but that's all. Kinda like it's both too simplified and touching on too deep concepts to tie together the article.

Sorry for the rambling.

Edit0: how many straight lines going through the whole length of the face it is on on a cubic surface. Honestly, I just hadn't really pictured a cubic surface to start with - that was the main part. Had that picture been higher up I think I would have liked the article right away! Thank you peeps

Google is actually still fairly decent for this kind of query. Just type:

27 lines on a cubic surface

Into the search bar and go to the image results. It will 'click' mentally in practically no time at all.

There’s a picture at the bottom. I think the text there is a bit more clear (maybe?): you have a cubic surface and want to see if there’s any “straight line” that lives/lies on the surface. It turns out there’s 27 lol.
For the articles from Quanta where I have a lot of prior knowledge, which is decidedly a small fraction of them, I think they mostly do a good job of accurately conveying the gist. It gives me more confidence in the articles that are outside my wheelhouse.

Quanta is one of the few examples in popular media I can think of where the Gell-Mann Amnesia effect does not seem to be operative. Or at least, if they are shoveling slop then they have a preternatural ability to hide it.

A circle can be in or out, so two states. 2 ^ 3 = 8.

Not a proof but just something visual I noticed.

That would imply that given the number of points (in this case 3) there would always be an answer.

For, example would it hold if we put the restriction to 4, 5 or 10 points?

"Slide whichever circle is smaller entirely inside the bigger one, and now the answer is zero: You can’t draw any lines that touch each circle only once."

This sounds false. If the smaller circle is at the edge of the larger one, it is still entirely inside the larger one while a tangent line could touch both of them at the edge.