Let's start with the following hyperbolic tiling.
It has four equilateral triangles and two squares meeting at every vertex. Its edge is about 1.06 absolute units (it's actually identical to the edge of {5,4}, but that's a whole other can of worms).
The issue is this: a numeric search shows that if you put one of these triangles and four of these squares together, there will be a tiny gap (around 2 degrees). This gap just happens to perfectly correspond to an angle of an equilateral triangle whose edge is exactly 7 times the edge of this tiling.
Like so:
Here's a view centered closer to the triangle's vertex so you could see how it fits:
This is one of those "I've seen it but I don't believe it" moments for me. How could I prove that this relation is valid (I only have numeric computations, but they hold for more than 100 decimals)?