For 2D, I asked the question Points with power distances. For 3D, I asked about Points at Integer Distances in 3-space. Combining these, I was able to construct $K_{19}$ so that all distance between points are powers of $d=1.15096...$ from $d^6 -d^2-1=0$, the same as in Zak's triangle. For smaller cliques, see Powered Clique Polyhedra.
Here's the grid of power distances between points. For example, the 16-17 distance is $d^0$. Values are 0 to 17 sans 1 and 16.
Points 1-3 can be placed at the following, with the root value about 4.54932.
{{d^6 /2, Root[-19+72 #1^2 -1328 #1^4 +64 #1^6 &,2], 0}, {0,0,0}, {d^6,0,0}}
That leads to the first question -- is there a natural way of representing this pyramid, so that all coordinates are expressed in terms of $d$, or so that it is placed symmetrically on the $(x,y,z)$ axis? Is it possible to get larger cliques?
Are there any other $d$ values that can even get close to power-distance clique this large?
Consider the following set of 19 triples:
Use each triple in $sign(k) \sqrt{|k|}$ where $k=a p^0 +b p^1 + c p^2$ and $p$ is the plastic constant, $p^3-p-1=0$ and $p=1.32472...$. All the distances between points are powers of $\sqrt{p}$.
Here's a 2D figure for Points at power distances. Divide the triples $(a,b,c)$ in each point by 4 first before plugging into $sign(k) \sqrt{|k|}$ where $k=a p^0 +b p^1 + c p^2$. The larger numbers give the distance $\sqrt{p^n}$ between the points.