Greg Egan on Nostr: One way to get the 4-piece dissection is to tile the plane with squares and ...

One way to get the 4-piece dissection is to tile the plane with squares and equilateral triangles, both with area 1.

Find vectors v, w with |v|=the side length of an equilateral triangle with area 1, |w|=2, and det|v,w|=2

v = (2 / 3^{1/4}, 0)
w = (√[4-√3], 3^{1/4})

If we use these vectors to construct a lattice with either the triangles or the squares, suitably offset, they will divide each other into exactly the same four pieces!