Rob Suzzi Valli: Manhattan College
Unfoldings of the Cube
The matrix-tree theorem can be used to show that there are 384 ways to cut open a cubical surface and unfold it into a planar "net." The number of distinct planar shapes obtained is bounded by the number of orbits of cut patterns under the action of the isometry group of the cube. Each cut pattern is a spanning tree of the vertex-edge graph, so we obtain the number of orbits from Burnside's lemma after finding the number of spanning trees fixed by each isometry. To find this number efficiently, we first prove a fixed-edge lemma that reduces the number of types of non-identity isometries that need to be considered to two; then we use Burnside's Lemma to obtain an upper bound on the number of distinct planar shapes. Finally, we describe a combinatorial process on the unfolded shapes themselves that generates enough unfolded shapes to attain the upper bound.
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