framework of a cube

the idea for this comes from 'mental gymnastics' by dick hess

a ppt is here

what are the lengths?

the lengths of 1 are the edges of the cube
the lengths of root 2 are on the faces, 2 on each
the lengths of root 3 connect the corners of e.g. the top face to the opposite corner of the lower face

28 lengths altogether (which is 8 choose 2)

only 3, perhaps surprisingly

56 triangles altogether = 8 choose 3
but you could, without loss of generality, have 1 as the first corner to obtain the (correct) probabilities

what is intriguing is that the numbers of different lengths
match the numbers of different triangles
there may be a good reason for this (but I can't see it!)