A Topologically Convex Vertex-Ununfoldable Polyhedron

Transcription

1 A Topologically Convex Vertex-Ununfoldable Polyhedron Zachary Abel 1 Erik D. Demaine 2 Martin L. Demaine 2 1 MIT Department of Mathematics 2 MIT CSAIL CCCG 2011 Abel, Demaine, and Demaine (MIT) A Vertex-Ununfoldable Polyhedron CCCG / 8

2 Vertex-Unfolding Fundamentals Vertex-unfolding is like edge-unfolding Cut some edges Leave it connected Fold it flat Abel, Demaine, and Demaine (MIT) A Vertex-Ununfoldable Polyhedron CCCG / 8

3 Vertex-Unfolding Fundamentals Vertex-unfolding is like edge-unfolding Cut some edges: Cut all edges! Leave it connected Fold it flat Abel, Demaine, and Demaine (MIT) A Vertex-Ununfoldable Polyhedron CCCG / 8

4 Vertex-Unfolding Fundamentals Vertex-unfolding is like edge-unfolding Cut some edges: Cut all edges! Leave it connected with vertex hinges Fold it flat Abel, Demaine, and Demaine (MIT) A Vertex-Ununfoldable Polyhedron CCCG / 8

5 Vertex-Unfolding Fundamentals Vertex-unfolding is like edge-unfolding Cut some edges: Cut all edges! Leave it connected with vertex hinges Fold it flat: folds like a hinged figure Abel, Demaine, and Demaine (MIT) A Vertex-Ununfoldable Polyhedron CCCG / 8

6 Vertex-Unfolding Fundamentals Vertex-unfolding is like edge-unfolding Cut some edges: Cut all edges! Leave it connected with vertex hinges Fold it flat: folds like a hinged figure Crossing hinges are not allowed. Abel, Demaine, and Demaine (MIT) A Vertex-Ununfoldable Polyhedron CCCG / 8

7 Vertex-Unfolding Fundamentals Vertex-unfolding is like edge-unfolding Cut some edges: Cut all edges! Leave it connected with vertex hinges Fold it flat: folds like a hinged figure Crossing hinges are not allowed. Introduced in [DEEHO, SoCG 02] to be easier than edge-unfolding. Any edge-unfolding is a vertex-unfolding. Abel, Demaine, and Demaine (MIT) A Vertex-Ununfoldable Polyhedron CCCG / 8

8 Vertex-Unfolding Fundamentals Vertex-unfolding is like edge-unfolding Cut some edges: Cut all edges! Leave it connected with vertex hinges Fold it flat: folds like a hinged figure Crossing hinges are not allowed. Introduced in [DEEHO, SoCG 02] to be easier than edge-unfolding. Any edge-unfolding is a vertex-unfolding. Open Question Can every convex polyhedron be edge-unfolded? Abel, Demaine, and Demaine (MIT) A Vertex-Ununfoldable Polyhedron CCCG / 8

9 Vertex-Unfolding Fundamentals Vertex-unfolding is like edge-unfolding Cut some edges: Cut all edges! Leave it connected with vertex hinges Fold it flat: folds like a hinged figure Crossing hinges are not allowed. Introduced in [DEEHO, SoCG 02] to be easier than edge-unfolding. Any edge-unfolding is a vertex-unfolding. Open Question (Weaker) Can every convex polyhedron be vertex-unfolded? Abel, Demaine, and Demaine (MIT) A Vertex-Ununfoldable Polyhedron CCCG / 8

10 Previous Work: Positive Results Theorem (DEE+02) Any triangulated manifold can be vertex-unfolded. So the Witch s Hat Tetrahedron has a vertex unfolding (but no edge unfolding). Abel, Demaine, and Demaine (MIT) A Vertex-Ununfoldable Polyhedron CCCG / 8

11 Previous Work: Positive Results Theorem (DEE+02) triangle shares one vertex with the previous triangle in the graphics pipeline. This result is in some sense best possible: an ideal rendering for a 2-vertex cache in which every adjacent pair of triangles shares two vertices is not always achievable, because there are triangulations whose dual graphs have no Hamiltonian path [1]. Figure 2(a) shows a vertex-unfolding of the triangulated surface of a cube, obtained from a facet path by our algorithm. Figure 2(b) shows a less regular vertex-unfolding. Note that the vertices do not necessarily lie on a line. Several more complex examples are shown in Figure 3. In our examples, we permit the triangles to touch along segments at the strip boundaries (as in Any triangulated manifold can be vertex-unfolded into a chain. So the Witch s Hat Tetrahedron has a vertex unfolding (but no edge (a) of the figure), but this could easily be avoided if desired so that each strip unfolding). boundary contains just the one vertex shared between the adjacent triangles. (a) (b) Figure 2. Laying out facet paths in vertical strips: (a) cube; Abel, Demaine, and Demaine (b) (MIT) 16-facet convex A Vertex-Ununfoldable polyhedron. Polyhedron CCCG / 8

12 Previous Work: Negative Results Not every convex polyhedron has a chain vertex-unfolding [DEE+02]: [Image source: Wikipedia] Abel, Demaine, and Demaine (MIT) A Vertex-Ununfoldable Polyhedron CCCG / 8

13 Previous Work: Negative Results Not every convex polyhedron has a chain vertex-unfolding [DEE+02]: Not every polyhedron has a vertex-unfolding [BDDLOORW, CCCG 98]: [Image source: Wikipedia] Abel, Demaine, and Demaine (MIT) A Vertex-Ununfoldable Polyhedron CCCG / 8

14 Local Obstructions to Vertex-Unfolding Suppose we have two vertices v 1, v 2 of different polygons with angles α 1, α 2 respectively. Observation 1 If α 1 + α 2 > 360, these vertices cannot be hinged in the plane without overlap. Observation 2 If α 1 + α 2 = 360, and the polygons are hinged at these vertices without overlap, then they must be oriented to exactly cover the 360 surrounding the hinge. Abel, Demaine, and Demaine (MIT) A Vertex-Ununfoldable Polyhedron CCCG / 8

15 A New Vertex-Ununfoldable Polyhedron Topologically Convex B S 1 C S 2 β γ A α G D E S 3 F Polyhedron P A Abel, Demaine, and Demaine (MIT) A Vertex-Ununfoldable Polyhedron CCCG / 8

16 A New Vertex-Ununfoldable Polyhedron Topologically Convex B S 1 C S 2 β γ A α G D E S 3 F Polyhedron P A Abel, Demaine, and Demaine (MIT) A Vertex-Ununfoldable Polyhedron CCCG / 8

17 A More Local Example Topologically Convex C B D β γ E A α G F Polyhedron P Abel, Demaine, and Demaine (MIT) A Vertex-Ununfoldable Polyhedron CCCG / 8

18 Looking Forward Which families of polyhedra have vertex unfoldings? Not always: All Polyhedra Topologically convex (and star-shaped) Open: Convex faces (and topologically convex) Convex Always: Triangulated Complexity of vertex-unfolding? Abel, Demaine, and Demaine (MIT) A Vertex-Ununfoldable Polyhedron CCCG / 8