Class of 2002
CSC 274: Computational Geometry
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Understanding Polytope Unfolding
Cutting and unfolding along the edges of a polytope, in a special way, allows a polytope to be unfolded into a flat sheet. The goal is to open the polytope in one flat plane without overlapping edges creating a net. A net is an unfolding of the surface of a polytope into a flat plane, together with edge matching information, without fold lines identified.
Is every unfolding of a convex polytope non-selfoverlapping?
There are many constructions known for the negative answer of, "Is every convex polytope non-selfoverlapping?". Makoto Namiki constructed the smallest example, a skinny tetrahedron, which admits a selfoverlapping unfolding. I created a 3-D polytope out of metal that is constructed so it can be unfolded in multiple different ways along the edge creases. One of these unfoldings yields overlapping edges as seen below. This polytope is Makoto Namiki's skinny tetrahedron:
The polytope I constructed, in a flattened state, mirroring Makoto Namiki's skinny tetrahedron:
This is an example of a "bad" unfolding.
Does every polytope have a net?
Mathematicians, computer scientists, and others have described this question as an "intuitive" geometrical problem. The answer is said to be "obviously" yes. However intuitive this problem may seem, up to now, it has remained unproven. No one has been able to construct a polytope that does not have a net. If even one polytope could be found that does not have a net, or if it could be found that an algorithm without fail finds the net of a polygon of any dimension, it would prove this conjecture. Therefore, all the polytopes I have created have a net.
Is every unfolding of a convex polytope unambiguous?
An unfolding is defined to be unambiguous if the original polytope is uniquely constructable from it. It is easy to find examples of polygons with ambiguous unfoldings.
There are many methods currently being used to find an algorithm that finds the net for all polytopes. One method being explored is that of Hamiltonian unfoldings. A Hamiltonian unfolding is an unfolding along one of the path through the dual graph of the polytope. The dual graph of a polytope has a node for each face of P, and an arc connecting two faces that share an edge. Unofficial results indicate that the chance of overlap is considerably less that that of polytopes unfolded along random trees. As of today, no one has proved that Hamiltonian unfoldings are the answer to the complex net problem.
Shephard proved that overlap is avoidable for certain combinatorial classes of polytopes. These include pyramids, bi-pyramids, prisms, anti-prisms, cyclic polytopes, and wedges. He proved that combinatorial polytopes with six vertices or less and certain stack types of "stack" polytopes have members that can avoid overlap.
Significant progress was make by Sharir and Schorr in 1984. They used shortest paths on the surface of polytopes to create unfoldings. These unfoldings, called "peels", are constructed by picking a point "x" on the surface of a polytope, and calculating shortest paths from this point to every other point on the polytope.
The current race to find the perfect unfolding algorithm still continues. My polytopes are not nearly advanced enough to exemplify either of the above methods of finding the net. The polytopes I have created can only be unfolded along an edge, instead of the line that joins random points. Metal is a poor medium for displaying the above methods. Java applets, similar to many of the ones we have seen in class, are the perfect medium for showing the above conjectures if they are ever proven.
In trying to find classes of polytopes that do not overlap, it is naturally desirable to be able to make a statement such as "all polytopes with less than n vertices can be unfolded without overlap." In order to examine all polytopes of less than n vertices, we first divide those polytopes into combinatorial classes. All polytopes can be represented in terms of their planar maps. The planar map of a polytope is represented by G(n), the set of all combinatorially distinct 1-skeletons of a polytope P with n vertices, embedded in a plane, without edge crossings.
If a polytope is cut open along a Hamiltonian path, such that the surface angle is strictly less than U (pi) at all inner verticies of the path, the resulting angle does not overlap (Corollary).
There were a few conjectures posed by Fukuda: The first is that a minimum-perimeter unfolding of a polytope is always a net. (A minimum perimeter unfolding is an unfolding obtained form a minimum spanning tree of G(P), where the length of an edge taken to be the length of the corresponding edge on P.) Rote constructed counterexamples to another open conjecture of Fukuda that cutting along a shortest path tree of G(P) leads to a net.
Yet another approach to polytope unfolding is not to require that the cuts are spanning tree of the 1-skeleton of P, but rather allow arbitrary (straight line) cuts through the interior of facets. A special kind of unfolding is the star unfolding. It is constructed as follows: Let the source x be a point on the surface of a polytope P, such that there is a unique shortest to every vertex of P (distance measured on the surface of P). The star unfolding is then obtained by cutting the boundary of P along these paths and flattening the resulting surface in the plane.
Counterexamples for the "minimum perimeter" conjecture of Fukuda.
The open questions and conjectures mentioned, except for the shortest-paths tree conjecture, are still open.