Thursday, December 4, 2014

Voxel-based surfaces are completely foldable

The boundary surface (vox surface) of any vox solid (i.e., a voxel-based solid) is a quadrangulation. Any such quadrangulation is bipartite because the edges and vertices in a three-dimensional packing of cubes form a bipartite graph and our quadrangulation is merely a subgraph of this.

Since the quadrangulation is already bipartite, we can convert it into a tripartite triangulation by placing a vertex of a third color in the center of each quadrangular (actually square) face, and connecting an edge to each of the four surrounding vertices.

Once triangulated in this way, a voxel-based surface is completely foldable because its tripartite.

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