An *unfolding* of a convex polytope P in R^3 is a planar embedding
of its boundary obtained by cutting the edges of some spanning tree T of
the graph of P and flattening the boundary along the remaining edges. Two
natural (but naive) questions are

- (a) Is every unfolding of a convex polytope non-selfoverlapping?
- (b) Is every unfolding of a convex polytope unambiguous?

Here an unfolding is defined to be *unambiguous* if the original
polytope is uniquely constructible from it.

Both questions have negative answers. There are many constructions known for the negative answer of (a), but Makoto Namiki (namiki@waka.c.u-tokyo.ac.jp) constructed the smallest example, a skinny tetrahedron, which admits a selfoverlapping unfolding:

Fig. 1. Selfoverlapping unfolding by Namiki (Click
here for ps file).

Note that it has a non-selfoverlapping unfolding as well. For the question (b), Tomomi Matsui (tomomi@misojiro.t.u-tokyo.ac.jp) constructed a polytope with 6 facets and 5 vertices which admits an ambiguous unfolding:

Fig. 2. Geometrically ambiguous unfolding by Matsui (Click
here for ps file).

These two examples can be found in the UnfoldPolytope package for Mathematica by Namiki and Fukuda [NF92].

Consequently more intelligent questions are

- (a') Does every convex polytope admit a non-selfoverlapping unfolding?
- (b') Does every convex polytope admit an unambiguous unfolding?

As far as I know, these questions are still open. I conjectured at the Dagstuhl meeting on Computational Geometry (February 1997) that

- (1) Any minimum-length spanning tree of a convex polytope induces a non-selfoverlapping unfolding.

The positive answer to this would resolve the question (a') positively as well.

Recently Günter Rote (rote@opt.math.tu-graz.ac.at) has constructed counterexamples to this conjecture:

Minimum-perimeter selfoverlapping unfolding (Click
here for ps file).

Fig. 3. and its Polytope by Rote (Click
here for ps file).

The smallest among them has 7 facets and 9 vertices, see [Rot97a].

Another minimum-perimeter selfoverlapping unfolding (Click
here for ps file).

Fig. 3'. and its Polytope by Rote (Click
here for ps file).

Rote constructed also a polytope which admits a *combinatorially*
ambiguous unfolding[Rot97b]. One can
construct two combinatorially different polytopes from such an unfolding:

Combinatorially ambiguous unfolding (Click
here for ps file).

Fig. 4. and its two polytopes by Rote (Click
here for ps file).

Matsui's example mentioned above gives rise to two geometrically different polytopes which are combinatorially equivalent.

Note that a question (related to (a')) on the existence of an unfolding without overlaps, where it is allowed to cut any place in the boundary, was answered positively by Aronov and O'Rouke [AO91]. The key idea was to cut through geodesic paths from a fixed vertex to all other vertices. In fact this result motivates us to pose another open problem.

- (2) Does a shortest-path spanning tree
of a convex polytope induce a non-selfoverlapping unfolding?

Recently, Mr. Wolfram Schlickenrieder (schlicke@math.tu-berlin.de) has reported that he found several examples that answered the question above negatively. We shall post some example(s) as soon as we verify his claim.

This page is created bySun Mar 9, 1997, Modified on June 16, 1997