A hamiltonian path is a path that visits each vertex exactly once - in a dodecahedron, it can be done with a beautiful cycle.
As an added math bonus, this figure can be composed from 32 equal cuboids with two 45 degree chamfers that create square cross-sections. For those interested: the radius of the circumscribed sphere in relation to each side of the square is about 3.1935.
I think it should be able to print without support, but the support I printed came clean off. Also, there's only a very small contact point with the bed, so I sunk the model a few layers into the bed. The first two tries, the printer head tipped over the model, messing up the whole print.
If you make one, please share the results!