[math-fun] Complete graph K19 with all distances at powers of d

The root d=1.15096... from d^6−d^2−1=0 seems to have some unique properties. At https://math.stackexchange.com/questions/2390420/ I build K_19 with it so that all distances are integral powers of d. Are there any other values that can build a power-clique this large? Zak's triangle, http://community.wolfram.com/groups/-/m/t/851275 , is a triangle that can be divided into 6 similar but non-congruent copies of itself. The value d can also build odd objects in 2D space, as seen at https://math.stackexchange.com/questions/1828794/ --Ed Pegg Jr