Now that I have external evidence that someone is still hoping for new posts, I thought I better write one.

Here’s a result that not only would have I not have guessed, but I would have assumed the opposite is obviously true. There are convex polytopes that cannot be presented in Rⁿ as a polytope all of with rational coordinates. I would have assumed that this is wrong because you can always take the vertices, and perturb them slightly so that they become rational. This argument doesn’t work in general, but you can prove using other techniques that in 3 dimensions that every convex polytope can be written with rational coordinates. There already exist counterexamples in dimension 4.

The survey paper Non-rational configurations, polytopes, and surfaces, by Günter M. Ziegler, gives an explicit construction in 13 dimensions. The paper provides a decent overview of many other, related, topics.