Your Friendly Neighborhood Voderberg Tile

We present a generalization of the Voderberg tile, which, in addition to admitting periodic and nonperiodic spiral tilings of the plane, has the property that just two copies can surround 1 or 2 copies of the tile. We construct a generalization of this tile that admits periodic and nonperiodic spiral tilings of the plane while enjoying the property that any number of copies of the tile can be surrounded by just 2 copies. In doing so, we solve two open problems posed in the classic book Tilings and Patterns by Grünbaum and Shephard.