In this section, we consider closed curves in R³. It is a somewhat surprising fact that not every curve in R³ with the GTPP is itself a planar curve. However, a result very similar to the two-dimensional problem discussed in the previous section gives some very strong restrictions. We first define the GTPP in higher dimentions.

Definition. An n-dimensional grid cube is a subset S of Rⁿ with center c = (c₀, c₁, ..., c_n) and radius r defined by

S = {x = (x₀, x₁, ..., x_n): max_i |x_i - c_i| = r}.

We again denote the interior of S by D₁ and the exterior by D₂.

Definition. A subset A of Rⁿ has the Grid-Cube Two-Piece Property (GTPP) if for every grid cube S the intersection of A with closure(D_i) is path connected, i = 1, 2.

Proposition 3. If A is a closed curve with the GTPP, then A must lie on a grid cube.

proof. Forthcoming.

The next proposition deals with skew hexagons (see this demo), which are non-planar hexagons that are formed by the intersection of two grid cubes, each of which contains exactly one vertex of the other.

Proposition 4. Every skew hexagon whose bounding grid box is either a cube or a rectangular prism whose two largest faces are square has the GTPP.

proof. Forthcoming.

The next section classifies the closed sets in the plane with the GTPP.