Andrzej Zak found that a triangle with sides based on powers of the root $d^6-d^2-1=0$, $(d=1.15096...)$ that can replicate itself with 6 differently sized copies. The numbers are powers of $d$. The full triangle has sides $d^{10}, d^{11}, d^{14}$.
Is there a database somewhere with other self-replicating triangles like this? Ideally, all triangles should be different sizes.
I've posted code at Zak's triangle.