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Euclidean, hyperbolic, discrete, convex, coarse geometry, metric spaces, comparisons in Riemannian geometry, symmetric spaces.
6
votes
Tiling with similar tiles
It seems you did not restrict to finitely many scaled down copies of the tile. So here is one interesting tile for which you need infinitely many scaled down copies in order to tile the plane. On the …
1
vote
Three squares in a rectangle
Partial solution
Let the rectangle R have the longer side horizontal. It is easy to prove that you can label the squares A,B,C such that A can be shifted (i.e. translated) all the way to the left with …