WFL asked:
So can you go the whole hog and find an integer triply-planar hexad?
Since you suggested it, I tried, and it doesn't look promising. The search is easier -- it just involves iterating over all triangles with integer side-lengths and height -- but now you need three constraints (of the form "n is a perfect square") to be satisfied, which is a tall order. I ran it overnight and made it past height=25000, and found only 28 times where one of the three unspecified distances was rational (the 3-4-5 triangle providing the first two of these); I never saw two or more rationals. So there may be integer triply-planar hexads, but this isn't the way to find them... --Michael -- It is very dark and after 2000. If you continue you are likely to be eaten by a bleen.