A Topologically Convex Vertex-Ununfoldable Polyhedron
A Topologically Convex Vertex-Ununfoldable Polyhedron Zachary Abel1
Erik D. Demaine2 1 MIT
Martin L. Demaine2
Department of Mathematics 2 MIT
CSAIL
CCCG 2011
Abel, Demaine, and Demaine (MIT)
A Vertex-Ununfoldable Polyhedron
CCCG 2011
1/8
Vertex-Unfolding Fundamentals
Vertex-unfolding is like edge-unfolding I I I
Cut some edges Leave it connected Fold it flat
Abel, Demaine, and Demaine (MIT)
A Vertex-Ununfoldable Polyhedron
CCCG 2011
2/8
Vertex-Unfolding Fundamentals
Vertex-unfolding is like edge-unfolding I I I
Cut some edges: Cut all edges! Leave it connected Fold it flat
Abel, Demaine, and Demaine (MIT)
A Vertex-Ununfoldable Polyhedron
CCCG 2011
2/8
Vertex-Unfolding Fundamentals
Vertex-unfolding is like edge-unfolding I I I
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 2011
2/8
Vertex-Unfolding Fundamentals
Vertex-unfolding is like edge-unfolding I I I
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 2011
2/8
Vertex-Unfolding Fundamentals
Vertex-unfolding is like edge-unfolding I I I
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 2011
2/8
Vertex-Unfolding Fundamentals
Vertex-unfolding is like edge-unfolding I I I
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. I
Any edge-unfolding is a vertex-unfolding.
Abel, Demaine, and Demaine (MIT)
A Vertex-Ununfoldable Polyhedron
CCCG 2011
2/8
Vertex-Unfolding Fundamentals
Vertex-unfolding is like edge-unfolding I I I
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. I
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 2011
2/8
Vertex-Unfolding Fundamentals
Vertex-unfolding is like edge-unfolding I I I
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. I
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 2011
2/8
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 2011
3/8
triangle shares one vertex with the previous triangle in the graphics pipeline. This result isResults in some sense best possible: an ideal rendering Previous Work: Positive 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].
Theorem (DEE+02) 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 Any triangulated manifold can be vertex-unfolded into a chain. 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 the triangles to touch along segments at the strip boundaries (as in So thepermit 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; convex polyhedron. A Vertex-Ununfoldable Polyhedron
(b) (MIT) 16-facet Abel, Demaine, and Demaine
CCCG 2011
3/8
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 2011
4/8
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 2011
4/8
Local Obstructions to Vertex-Unfolding
Suppose we have two vertices v1 , v2 of different polygons with angles α1 , α2 respectively.
Observation 1 If α1 + α2 > 360◦ , these vertices cannot be hinged in the plane without overlap.
Observation 2 If α1 + α2 = 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 2011
5/8
A New Vertex-Ununfoldable Polyhedron Topologically Convex
B C
S2 D
S1 G
β A γ α E
S3 F
A
Polyhedron P
Abel, Demaine, and Demaine (MIT)
A Vertex-Ununfoldable Polyhedron
CCCG 2011
6/8
A New Vertex-Ununfoldable Polyhedron Topologically Convex
B C
S2 D
S1 G
β A γ α E
S3 F
A
Polyhedron P
Abel, Demaine, and Demaine (MIT)
A Vertex-Ununfoldable Polyhedron
CCCG 2011
6/8
A More Local Example Topologically Convex
C′
B
D
′
β′ A γ′ α
G
E
F′
Polyhedron P 0 Abel, Demaine, and Demaine (MIT)
A Vertex-Ununfoldable Polyhedron
CCCG 2011
7/8
Looking Forward
Which families of polyhedra have vertex unfoldings? Not always: I I
All Polyhedra Topologically convex (and star-shaped)
Open: I I
Convex faces (and topologically convex) Convex
Always: I
Triangulated
Complexity of vertex-unfolding?
Abel, Demaine, and Demaine (MIT)
A Vertex-Ununfoldable Polyhedron
CCCG 2011
8/8
Erik D. Demaine2 1 MIT
Martin L. Demaine2
Department of Mathematics 2 MIT
CSAIL
CCCG 2011
Abel, Demaine, and Demaine (MIT)
A Vertex-Ununfoldable Polyhedron
CCCG 2011
1/8
Vertex-Unfolding Fundamentals
Vertex-unfolding is like edge-unfolding I I I
Cut some edges Leave it connected Fold it flat
Abel, Demaine, and Demaine (MIT)
A Vertex-Ununfoldable Polyhedron
CCCG 2011
2/8
Vertex-Unfolding Fundamentals
Vertex-unfolding is like edge-unfolding I I I
Cut some edges: Cut all edges! Leave it connected Fold it flat
Abel, Demaine, and Demaine (MIT)
A Vertex-Ununfoldable Polyhedron
CCCG 2011
2/8
Vertex-Unfolding Fundamentals
Vertex-unfolding is like edge-unfolding I I I
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 2011
2/8
Vertex-Unfolding Fundamentals
Vertex-unfolding is like edge-unfolding I I I
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 2011
2/8
Vertex-Unfolding Fundamentals
Vertex-unfolding is like edge-unfolding I I I
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 2011
2/8
Vertex-Unfolding Fundamentals
Vertex-unfolding is like edge-unfolding I I I
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. I
Any edge-unfolding is a vertex-unfolding.
Abel, Demaine, and Demaine (MIT)
A Vertex-Ununfoldable Polyhedron
CCCG 2011
2/8
Vertex-Unfolding Fundamentals
Vertex-unfolding is like edge-unfolding I I I
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. I
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 2011
2/8
Vertex-Unfolding Fundamentals
Vertex-unfolding is like edge-unfolding I I I
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. I
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 2011
2/8
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 2011
3/8
triangle shares one vertex with the previous triangle in the graphics pipeline. This result isResults in some sense best possible: an ideal rendering Previous Work: Positive 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].
Theorem (DEE+02) 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 Any triangulated manifold can be vertex-unfolded into a chain. 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 the triangles to touch along segments at the strip boundaries (as in So thepermit 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; convex polyhedron. A Vertex-Ununfoldable Polyhedron
(b) (MIT) 16-facet Abel, Demaine, and Demaine
CCCG 2011
3/8
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 2011
4/8
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 2011
4/8
Local Obstructions to Vertex-Unfolding
Suppose we have two vertices v1 , v2 of different polygons with angles α1 , α2 respectively.
Observation 1 If α1 + α2 > 360◦ , these vertices cannot be hinged in the plane without overlap.
Observation 2 If α1 + α2 = 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 2011
5/8
A New Vertex-Ununfoldable Polyhedron Topologically Convex
B C
S2 D
S1 G
β A γ α E
S3 F
A
Polyhedron P
Abel, Demaine, and Demaine (MIT)
A Vertex-Ununfoldable Polyhedron
CCCG 2011
6/8
A New Vertex-Ununfoldable Polyhedron Topologically Convex
B C
S2 D
S1 G
β A γ α E
S3 F
A
Polyhedron P
Abel, Demaine, and Demaine (MIT)
A Vertex-Ununfoldable Polyhedron
CCCG 2011
6/8
A More Local Example Topologically Convex
C′
B
D
′
β′ A γ′ α
G
E
F′
Polyhedron P 0 Abel, Demaine, and Demaine (MIT)
A Vertex-Ununfoldable Polyhedron
CCCG 2011
7/8
Looking Forward
Which families of polyhedra have vertex unfoldings? Not always: I I
All Polyhedra Topologically convex (and star-shaped)
Open: I I
Convex faces (and topologically convex) Convex
Always: I
Triangulated
Complexity of vertex-unfolding?
Abel, Demaine, and Demaine (MIT)
A Vertex-Ununfoldable Polyhedron
CCCG 2011
8/8
