How to avoid intersections in an interlocking model? I have been trying to model an "interlocking" object similar to the  following references:

  



My last attempt turned out like this:



To model the object, I have done roughly the following: 


Start with a regular solid (dodecahedron or icosahedron in most attempts).
Bevel the entire object slightly,  and delete large faces, and subdivide edges. 
Duplicate the object and rotate
Scale and rotate along normals with proportional edit and
individual origin as pivot point until the object is in place
Add a solidify and subsurf modifier


The problem I am having, is that there are several intersections (some are circled in yellow) throughout the model. 
It seems that the intersections are caused by duplicating and rotating the object as well as the solidify modifier.

Where might I be going wrong in my approach that I could avoid these intersections and achieve an object like the ones shown in the reference images?
I tried to achive that effect as well and I did stick to the modifier workflow. I added some array modifiers with merging enabled to make that shape and then i just modelled a path. I also added a solidify modifier and voilà. Got it.