Is Bevel the Holy Grail of undip basket making?

The map operation bevel, Be(M), converts any map into a trivalent, highly hamiltonian map. That makes it describable in undip, in fact it guarantees that it has many shape synonyms, and thus offers many options in choosing a working order for making the basket.

What does a map transformed by bevel look like? The surfaces below were all triangle meshes before bevel got its hands on them.