What Is a UV map to a higher dimensional space that enables tiling continuity of 3D cubes?

This is expanding on the Idea of This Blender Stack Exchange post Describing tiling 2D noise using a projection into a higher-dimensional space. As a continuation of this method, would it be possible to have a UV projection function or combination of multiple UV projection functions, $P_{roject}(p), Dom(P_{roject}) = \{p\in \mathbb R^2\}, Range(P_{roject}) \in \mathbb R^N, N \in \mathbb N$ such that when provided a continuous Higher dimensional function, $f(x), Dom(f) = \{x\in \mathbb R^N\}, Range(f) = \{f \in \mathbb R, f \ge -1, f \le 1\}, f(P_{roject}(uv))$ provides a brightness UV texture for a cube that is continuous for all tiling configurations. For example, this image, would look completely continuously textured, even at non-planar edges.