Editing Polycube (section)

==Boundary connectivity==
Although the cubes of a polycube are required to be connected square-to-square, the squares of its boundary are not required to be connected edge-to-edge.
For instance, the 26-cube formed by making a 3×3×3 grid of cubes and then removing the center cube is a valid polycube, in which the boundary of the interior void is not connected to the exterior boundary. It is also not required that the boundary of a polycube form a [[manifold]].
For instance, one of the pentacubes has two cubes that meet edge-to-edge, so that the edge between them is the side of four boundary squares.

If a polycube has the additional property that its complement (the set of integer cubes that do not belong to the polycube) is connected by paths of cubes meeting square-to-square, then the boundary squares of the polycube are necessarily also connected by paths of squares meeting edge-to-edge.<ref>{{citation
 | last1 = Bagchi | first1 = Amitabha
 | last2 = Bhargava | first2 = Ankur
 | last3 = Chaudhary | first3 = Amitabh
 | last4 = Eppstein | first4 = David | author4-link = David Eppstein
 | last5 = Scheideler | first5 = Christian
 | doi = 10.1007/s00224-006-1349-0
 | issue = 6
 | journal = Theory of Computing Systems
 | mr = 2279081
 | pages = 903–928
 | title = The effect of faults on network expansion
 | volume = 39
 | year = 2006| arxiv = cs/0404029| s2cid = 9332443
 }}. See in particular Lemma 3.9, p.&nbsp;924, which states a generalization of this boundary connectivity property to higher-dimensional polycubes.</ref> That is, in this case the boundary forms a [[polyominoid]].

{{unsolved|mathematics|Can every polycube with a connected boundary be [[Net (polyhedron)|unfolded]] to a polyomino? If so, can every such polycube be unfolded to a polyomino that tiles the plane?}}
Every {{mvar|k}}-cube with {{math|''k'' < 7}} as well as the Dalí cross (with {{math|1=''k'' = 8}}) can be [[Net (polyhedron)|unfolded]] to a polyomino that tiles the plane.
It is an [[open problem]] whether every polycube with a connected boundary can be unfolded to a polyomino, or whether this can always be done with the additional condition that the polyomino tiles the plane.<ref name="pucc"/>