Can Every Simplicial Polyhedron Be Unfolded?

Title: Can Every Simplicial Polyhedron Be Unfolded?

Abstract: This research investigates the possibility of unfolding the surface of a simplicial polyhedron, where all faces are triangles, into a flat, connected planar layout without overlap. The unfolding is done by connecting the edges of the polyhedron, creating a layout that may have a disconnected interior. The main question addressed is whether every simplicial polyhedron, of any genus, can be unfolded in this way.

Methodology: The study begins by defining a lattice graph, which represents the face lattice of the polyhedron. The graph consists of nodes representing the facets, edges, and vertices of the polyhedron, with an arc for each incidence. The researchers then introduce the concept of a facet path, which alternates between vertices and facets and includes each facet exactly once. They show that if the graph contains a facet path, then a vertex-unfolding exists.

However, the researchers cannot guarantee the existence of a facet path for every graph. Therefore, they introduce an "unfolding path," which is a path in the graph that alternates between vertices and nonvertices, covers each facet exactly once, and never repeats the same vertex twice in a row. They prove that every simplicial polyhedron has an unfolding path, allowing them to conclude that every simplicial polyhedron, of any genus, can be vertex-unfolded.

Results: The main result of this research is that every simplicial polyhedron, of any genus, can be cut along edges and unfolded to a planar, nonoverlapping, connected layout. This is achieved by creating a vertex-unfolding, where the faces of the polyhedron are joined at vertices (and sometimes edges).

Implications: This research provides a positive answer to the long-standing open problem of whether every convex polyhedron can be edge-unfolded. It also relaxes the definition of "in one piece" to allow for a nonoverlapping connected region, which leads to a simpler proof. The researchers note that the problem remains open for nonsimplicial polyhedra with simply connected faces.

In conclusion, this study shows that every simplicial polyhedron, of any genus, can be unfolded to a planar layout by connecting the edges of the polyhedron and joining the faces at vertices. This result provides valuable insights into the properties of polyhedra and the possibilities of unfolding them.

Link to Article: https://arxiv.org/abs/0107023v1 Authors: arXiv ID: 0107023v1