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Just wanted to share the link to a mathematical counterexamples site that I develop: Math Counterexamples. Any idea for improvement is more than welcome.
On the way the site looks like, on good ideas for counterexamples, on the way to split counterexamples between level on math... Or anything else!
Nice! It seems you have few geometry (counter)examples. Here is one:
There are 4-dimensional convex polytopes that cannot be realized with rational vertex coordinates: that is, the combinatorial structure can only be realized with one or more irrational vertex coordinates.
In $\mathbb{R}^3$, however, every convex polyhedron's combinatorial type can be realized with rational coordinates. (But it is unknown if the rational coordinates can be chosen to be "small.")
Ziegler, Günter M. "Nonrational configurations, polytopes, and surfaces." The Mathematical Intelligencer. 30.3 (2008): 36-42.
