Presentation on theme: "Part III: Polyhedra b: Unfolding Joseph ORourke Smith College."— Presentation transcript:
Part III: Polyhedra b: Unfolding Joseph ORourke Smith College
Outline: Edge-Unfolding Polyhedra zHistory (Dürer) ; Open Problem; Applications zEvidence For zEvidence Against
Unfolding Polyhedra zCut along the surface of a polyhedron zUnfold into a simple planar polygon without overlap
Edge Unfoldings zTwo types of unfoldings: yEdge unfoldings: Cut only along edges yGeneral unfoldings: Cut through faces too
Commercial Software Lundström Design,
Albrecht Dürer, 1425
Albrecht Dürer, 1425 Snub Cube
Open: Edge-Unfolding Convex Polyhedra Does every convex polyhedron have an edge- unfolding to a simple, nonoverlapping polygon? [Shephard, 1975]
Open Problem Can every simple polygon be folded (via perimeter self-gluing) to a simple, closed polyhedron? How about via perimeter-halving?
Symposium on Computational Geometry Cover Image
Cut Edges form Spanning Tree Lemma: The cut edges of an edge unfolding of a convex polyhedron to a simple polygon form a spanning tree of the 1-skeleton of the polyhedron. o spanning: to flatten every vertex o forest: cycle would isolate a surface piece o tree: connected by boundary of polygon
Unfolding the Platonic Solids Some nets:
Unfolding the Archimedean Solids
Nets for Archimedian Solids
Unfolding the Globe: Unfolding the Globe: Fullers map
Open: Edge-Unfolding Convex Polyhedra (revisited) Does every convex polyhedron have an edge- unfolding to a net (a simple, nonoverlapping polygon)?
Open: Fewest Nets For a convex polyhedron of n vertices and F faces, what is the fewest number of nets (simple, nonoverlapping polygons) into which it may be cut along edges? F F (2/3)F [for large F] (1/2)F [for large F] < ½?; o(F)?