This thing is an experiment in printing. I realized that a several polyhedra should be printable without support, if cut properly, because their dihedral angles are less than 45 degrees from vertical. As an experiment I created this open-frame dodecahedron, cut into two parts, printed without support, and glued together.
Conclusion: yes, it's possible. I need to work on the minimum frame thickness, because some edges were too thin and sagged a bit. I also need to come up with a better way to fasten the two halves, because 1) superglue requires more-or-less flat surfaces and 2) the raw print has some pretty significant gaps in it, because of the sagging and angles of the edges.
Print two identical copies of the half dodecahedron, both with one face flat on the print bed. Again, this was an experiment in printing polyhedra without support, so don't print Support material.
Glue the two halves together. As I mention in the Summary section, I found using superglue didn't work very well. I'm still searching for a good PLA glue.
I was surprised how easy it was: I created an open-frame pentagon, then created a 116 degree cutter (116 = the dodecahedron dihedral angle) to bevel each edge of the pentagon. The easy part (in FreeCAD) was that once I had the cutter and pentagon in place for cutting the first edge, I could cut the next edge by rotating the pentagon 72 degrees and cutting again. I repeated that process for all edges.
Similarly, once I placed a second pentagonal face at a 116 degree angle to the first, I could fuse those two pieces, rotate the result by 72 degrees, then fuse with the original angled face again. Again, I repeated that process for all 5 edges of the original pentagon.
As I mentioned above, I'm not happy with the way this design fastens the two halves of the dodecahedron together: the edges to glue are too rough for my taste.