Shelling polysurfaces for fabrication and miters

Hi,

I have a geometry question that is challenging my idea of building an object made in rhino from planar surfaces:

I understand intuitively how regular convex polyhedra can have their faces be given thickness such that the sides of those new, thickened side solids are also planar-- i.e. a shell of a cube can be built from six solids that are extruded squares mitered at 45 degrees, or a square truncated pyramid.   These pieces are easily fabricated.


My question is: under what conditions is this "planar miter" goal impossible?   And, following that, is it possible to develop any planar-faceted solid into a thickened shell whose pieces all have planar sides as well as faces? 

I've been using weaverbird's offset mesh to get started, see pictures for clarification...

thanks!

OK!

OK!

NO!

CURVATURE BAD!