Applied geometry: symmetry within a 3D volume

[DaveC426913]

I've got a 3D shape that, um, inherits from a cube (i.e. it retains properties characteristic of cubes). It has high symmetry.

I wish to string all its vertices (of which there are 16) with - I dunno, yarn - forming the edges (of which there are 32). I can do the entire thing with a single piece of yarn (all vertices have an even number of edges, thus no loose ends); AND I can join the loose ends together, making a single loop of yarn. I have physically proven this to be true.

(For comparison, this cannot be done with a regular cube, since all vertices have an odd number of edges, meaning all vertices will need an end of the yarn.)

Now all I want to do is ensure that my pattern of stringing is as symmetrical as possible, i.e. even accounting for the stringing pattern, there should still be at least some symmetry. ISTM, rotational symmetry is the strictest.

Is there a science to this?


Alternately, I could turn this into a teaser and leave you guys to figure it out...

I'll give you a giant hint: its Schläfli symbol is {4,4} (square faces, 4 faces per vertex).

 

[DaveC426913]

Not one taker?

 

[maze]

Could you provide a little more information on what shape this is?

 

[DaveC426913]

Well, I was figuring I'd be looking for general principles and do the heavy-lifting myself, but:
http://en.wikipedia.org/wiki/Tesseract"
and http://davesbrain.livejournal.com/287490.html" .

The pattern I've shown has no organization. I want it to be symmetrical.