Squares can be divided into 45-60-75 triangles: (Ed Pegg on twitter, where he seems to be hanging out these days)

This is an interesting variant on something I looked at a long time ago, triangulating the square to minimize the maximum angle ( But Pegg's subdivision is not edge-to-edge. And the triangles on my page are not all similar. Do squares have edge-to-edge triangulations by similar acute triangles?

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