The pink strut direction in[ vZome](http://vzome.com) (half way between blue and yellow) and the olive strut direction (half-way between blue and red) both come close to making Rhombic Icosahedra with half of their faces square, but neither one is able to exactly make a square face with just the golden ratio field.
The blue struts around the equator meet at the intersection of pink and olive struts, but no other struts intersect.
The download includes the vZome file.