We introduce a new type of meshes called 5-6-7 meshes. For many mesh processing tasks, lowor high-valence vertices are undesirable. At the same time, it is not always possible to achieve complete vertex valence regularity, i.e., to only have valence-6 vertices. A 56-7 mesh is a closed triangle mesh where each vertex has valence 5, 6, or 7. An intriguing question is whether it is always possible...