Are there any (many) algorithmically undecidable problems in computational (combinatorial/discrete) geometry?
Update: the Wang tiles answer the question with "any". (I have somewhat overlooked to count them when I was browsing the answers to general undecidable problems.) But I still suspect that there are not many examples naturally arising in combinatorial/discrete geometry. And I am of course interested in any such example. I do not count artificial reformulations of problems stated (by the authors) in different settings.
For instance, it was open for quite some time whether STRING graphs (intersection graphs of curves in the plane) are recognizable. However, it turned out that they indeed are. If the answer was opposite, it would be an example of problem I seek for.
(Let me also exclude problems very similar to Wang tiles if there are any.)