You can't get much more elementary than Thurston's idea for letting the arc run from $0$ to $\infty$ and taking the preimage under the (branched) double cover of the Riemann sphere $z\mapsto z^2$. I'm still amazed by this.

I asked a similar question at mathoverflow.net/questions/57766/…

And unknown for $4$-spheres

Okay, so I've figured out how to rationalize "transversal" as both noun and adjective. "Universal" is a word with same property. And, to turn things on their head, "Universe" is a noun.

This reminds me of my first paper, which the referee had very strong stylistic opinions about. I had used the phrase "We induct on n," which the referee really didn't like. He also disliked saying things like "The $\alpha_j$" to mean the collection $\{\alpha_j\}$ varying over an index set. Some people really feel strongly about this kind of thing.

Could it be something along the lines of the following? The universe likes to compose structures along graphs, e.g Feynman diagrams, and as Jeffrey Giansiracusa points out in his answer to Mark's question, cyclic and modular operads are especially suited to that situation.

@Mariano: I have a bad habit of using tortured grammar. I think it is correct as is, but I'll try to rephrase it so it flows off the tongue better.

@Mariano: sorry, what is it missing?

I totally agree with that.

FWIW I just asked Morwen Thistlethwaite about this, and he told me there are examples where the meridian isn't shortest