Manifolds:
The surface of a solid should satisfy some of conditions so that the resulting solid is well-behaved. It is the so-called manifold condition. Whereas recent research has illustrated that the surface of a solid does not contain to be a manifold, we will restrict to manifold surfaces to simplify our discussion.
A surface is a 2-manifold if and only if for each point x on the surface there presents an open ball along with centre x and adequately small radius so that the intersection of this ball and the surface may be continuously deformed to an open disk. An open ball along with centre at the coordinate origin and radius r is described by equation x2 + y2 + z2 < r2. It has all points inside of the sphere x2 + y2 + z2 = r2. An open disk is described similarly x2 + y2 < r2. By continuously deformed, it means one may twist or bend the shape without cutting (that means adjacency relations should be maintained) and gluing (that means a one-to-one relation is needed).
There is a cube & three open balls. Ball 2 contains its centre on the top face. The intersection of this ball & the surface of the cube is an open disk (illustrated in red). Ball 1 contains its centre on an edge. Its intersection along with the surface of the cube is a "bent" open disk, which of course may be "unbent" to make it an open disk. Ball 3 contains its centre at a corner. Its intersection along with the cube's surface is a three-way bent open disk.