Publications with Erik Demaine
A disk-packing algorithm for an origami magic trick.
M. Bern, E. Demaine, D. Eppstein, and B. Hayes.
Int. Conf. Fun with Algorithms, Elba, June 1998.
Proceedings in Informatics 4, Carleton Scientific, Waterloo, Canada, 1999, pp. 32–42.
Origami3: Proc. 3rd Int. Mtg. Origami Science, Math, and Education (Asilomar, California, 2001), A K Peters, 2002, pp. 17–28.We apply techniques from "Quadrilateral meshing by circle packing" to a magic trick of Houdini: fold a piece of paper so that with one straight cut, you can form your favorite polygon.
Ununfoldable polyhedra.
M. Bern, E. Demaine, D. Eppstein, E. Kuo, A. Mantler, and J. Snoeyink.
arXiv:cs.CG/9908003.
Tech. rep. CS-99-04, Univ. of Waterloo, Dept. of Computer Science, Aug. 1999.
11th Canad. Conf. Comp. Geom., 1999.
4th CGC Worksh. Computational Geometry, Johns Hopkins Univ., 1999.
Comp. Geom. Theory & Applications (special issue for 4th CGC Worksh.) 24 (2): 51–62, 2003.We prove the existence of polyhedra in which all faces are convex, but which can not be cut along edges and folded flat.
Note variations in different versions: the CCCG one was only Bern, Demain, Eppstein, and Kuo, and the WCG one had the title "Ununfoldable polyhedra with triangular faces". The journal version uses the title "Ununfoldable polyhedra with convex faces" and the combined results from both conference versions.
Hinged dissections of polyominos and polyforms.
E. Demaine, M. Demaine, D. Eppstein, G. Frederickson, and E. Friedman.
arXiv:cs.CG/9907018.
11th Canad. Conf. Comp. Geom., 1999.
Computational Geometry: Theory and Applications 31 (3): 237–262, 2005 (special issue for 11th CCCG).We show that, for any n, there exists a mechanism formed by connecting polygons with hinges that can be folded into all possible n-ominos. Similar results hold as well for n-iamonds, n-hexes, and n-abolos.
Phutball endgames are hard.
E. Demaine, M. Demaine, and D. Eppstein.
arXiv:cs.CC/0008025.
More Games of No Chance, R. J. Nowakowski, ed., MSRI Publications 42, pp. 351–360.We show that, in John Conway's board game Phutball (or Philosopher's Football), it is NP-complete to determine whether the current player has a move that immediately wins the game. In contrast, the similar problems of determining whether there is an immediately winning move in checkers, or a move that kings a man, are both solvable in polynomial time.
Vertex-unfoldings of simplicial manifolds.
E. Demaine, D. Eppstein, J. Erickson, G. Hart, and J. O'Rourke.
Tech. Reps. 071 and 072, Smith College, 2001.
arXiv:cs.CG/0107023 and cs.CG/0110054.
18th ACM Symp. Comp. Geom., Barcelona, 2002, pp. 237–243.
Discrete Geometry: In honor of W. Kuperberg's 60th birthday, Pure and Appl. Math. 253, Marcel Dekker, pp. 215–228, 2003.We unfold any polyhedron with triangular faces into a planar layout in which the triangles are disjoint and are connected in a sequence from vertex to vertex
Flat foldings of plane graphs with prescribed angles and edge lengths.
Z. Abel, E. Demaine, M. Demaine, D. Eppstein, A. Lubiw, and R. Uehara.
arXiv:1408.6771.
22nd Int. Symp. Graph Drawing, Würzburg, Germany, 2014.
Springer, Lecture Notes in Comp. Sci. 8871, 2014, pp. 272–283.
J. Computational Geometry 9 (1): 74–93, 2018.Given a plane graph with fixed edge lengths, and an assignment of the angles 0, 180, and 360 to the angles between adjacent edges, we show how to test whether the angle assignment can be realized by an embedding of the graph as a flat folding on a line. As a consequence, we can determine whether two-dimensional cell complexes with one vertex can be flattened. The main idea behind the result is to show that each face of the graph can be folded independently of the other faces.
Folding a paper strip to minimize thickness.
E. Demaine, D. Eppstein, A. Hesterberg, H. Ito, A. Lubiw, R. Uehara, and Y. Uno.
arXiv:1411.6371.
9th International Workshop on Algorithms and Computation (WALCOM 2015), Dhaka, Bangladesh.
Springer, Lecture Notes in Comp. Sci. 8973 (2015), pp. 113–124.
Journal of Discrete Algorithms 36: 18–26, 2016 (special issue for WALCOM).If a folding pattern for a flat origami is given, together with a mountain-valley assignment, there might still be multiple ways of folding it, depending on how some flaps of the pattern are arranged within pockets formed by folds elsewhere in the pattern. It turns out to be hard (but fixed-parameter tractable) to determine which of these ways is best with respect to minimizing the thickness of the folded pattern.
Folding polyominoes into (poly)cubes.
O. Aichholzer, M. Biro, E. Demaine, M. Demaine, D. Eppstein, S. P. Fekete, A. Hesterberg, I. Kostitsyna, and C. Schmidt.
27th Canadian Conference on Computational Geometry, Kingston, Ontario, Canada, 2015, pp. 101–106.
arXiv:1712.09317.
Int. J. Comp. Geom. & Appl. 28 (3): 197–226, 2018.We classify the polyominoes that can be folded to form the surface of a cube or polycube, in multiple different folding models that incorporate the type of fold (mountain or valley), the location of a fold (edges of the polycube only, or elsewhere such as along diagonals), and whether the folded polyomino is allowed to pass through the interior of the polycube or must stay on its surface.
Rigid origami vertices: Conditions and forcing sets.
Z. Abel, J. Cantarella, E. Demaine, D. Eppstein, T. Hull, J. Ku, R. Lang, and T. Tachi.
arXiv:1507.01644.
J. Computational Geometry 7 (1): 171–184, 2016.We give an exact characterization of the one-vertex origami folding patterns that can be folded rigidly, without bending the parts of the paper between the folds.
Reconfiguration of satisfying assignments and subset sums: Easy to find, hard to connect.
J. Cardinal, E. Demaine, D. Eppstein, R. Hearn, and A. Winslow.
arXiv:1805.04055
Proc. 24th International Computing and Combinatorics Conference (COCOON 2018), Qingdao, China.
Springer, Lecture Notes in Comp. Sci. 10976 (2018), pp. 365–377.
Theor. Comput. Sci. 806: 332–343, 2020.For several problems with polynomial-time solutions, we show that finding a sequence of steps that converts one solution into another (maintaining a valid solution at each step) is hard. These include planar monotone not-all-equal 3-sat, subset sum on integers of polynomial magnitude, and 0-1 integer programming with a constant number of linear inequality constraints.
Reconfiguring undirected paths.
E. Demaine, D. Eppstein, A. Hesterberg, K. Jain, A. Lubiw, R. Uehara, and Y. Uno.
arXiv:1905.00518.
16th Algorithms and Data Structures Symp. (WADS 2019), Edmonton, Alberta.
Springer, Lecture Notes in Comp. Sci. 11646 (2019), pp. 353–365.A motion that slides an undirected path along its length from one configuration to another in a graph (allowing moves in both directions) can be found in time that is fixed-parameter tractable in the path length. However the problem is PSPACE-complete for paths of unbounded length, even when the host graph has bounded bandwidth.
Ununfoldable Polyhedra with 6 Vertices or 6 Faces.
H. A. Akitaya, E. Demaine, D. Eppstein, T. Tachi, and R. Uehara.
22nd Japan Conference on Discrete and Computational Geometry, Graphs, and Games (JCDCG3), Tokyo, Japan, 2019, pp. 27–28.
Comp. Geom. Theory & Applications 103: 101857, 2022.We find a (nonconvex, but topologically equivalent to convex) polyhedron with seven vertices and six faces that cannot be unfolded to a flat polygon by cutting along its edges. Both the number of vertices and the number of faces are the minimum possible. The JCDCG3 version used the title "Minimal ununfoldable polyhedron".
Some polycubes have no edge-unzipping.
E. Demaine, M. Demaine, D. Eppstein, and J. O'Rourke.
arXiv:1907.08433.
Proc. 32nd Canadian Conference on Computational Geometry, 2020, pp. 101–105.
Geombinatorics 31 (3): 101–109, 2022.We find polycubes that cannot be cut along a simple path through their vertices and edges and unfolded to form a flat polygon in the plane.
Existence and hardness of conveyor belts.
M. Baird, S. C. Billey, E. Demaine, M. Demaine, D. Eppstein, S. P. Fekete, G. Gordon, S. Griffin, J. S. B. Mitchell, and J. P. Swanson.
arXiv:1908.07668.
Electronic J. Combinatorics 27 (4), Paper P4.25, 2020.A conveyor belt is a tight simple closed curve that touches each of a given set of disjoint disks. We show that certain special families of disks always have a conveyor belt, but that it is NP-complete to tell whether one exists for arbitrary families of disks. It is always possible to add a linear number of "guide disks" to a given set of disks to allow them to be connected by a conveyor belt.
Acutely triangulated, stacked, and very ununfoldable polyhedra.
E. Demaine, M. Demaine, and D. Eppstein.
arXiv:2007.14525.
Proc. 32nd Canadian Conference on Computational Geometry, 2020, pp. 106–113.We construct non-convex but topologically spherical polyhedra in which all faces are acute triangles, with no unfolded net.
New results in sona drawing: hardness and TSP separation.
M.-W. Chiu, E. Demaine, M. Demaine, D. Eppstein, R. Hearn, A. Hesterberg, M. Korman, I. Parada, and M. Rudoy.
arXiv:2007.15784.
Proc. 32nd Canadian Conference on Computational Geometry, 2020, pp. 63–72.A sona drawing is a self-crossing curve in the plane that separates each of a given system of points into its own region. We show that it is hard to find the shortest such curve, and we explore how much shorter than the TSP it can be.
Multifold tiles of polyominoes and convex lattice polygons.
K. Chida, E. Demaine, M. Demaine, D. Eppstein, A. Hesterberg, T. Horiyama, J. Iacono, H. Ito, S. Langerman, R. Uehara, and Y. Uno.
23rd Thailand-Japan Conference on Discrete and Computational Geometry, Graphs, and Games, 2021, pp. 28–29.
Thai Journal of Mathematics 21 (4): 957–978, 2023.We investigate shapes whose congruent copies can cover the plane exactly \(k\) times per point, but not a number of times that is a nonzero integer smaller than \(k\). We find polyominoes with this property for all \(k\ge 2\), and convex integer-coordinate polygons with this property for \(k=5\) and for all \(k\ge 7\).
Geodesic paths passing through all faces on a polyhedron.
E. Demaine, M. Demaine, D. Eppstein, H. Ito, Y. Katayama, W. Maruyama, and Y. Uno.
24th Japan Conference on Discrete and Computational Geometry, Graphs, and Games, September 9–11, 2022.
Springer, Lecture Notes in Comp. Sci. 14364 (2026), pp. 184–209.Which convex polyhedra have the property that there exist two points on the surface of the polyhedron whose shortest path passes through all of the faces of the polyhedron? The answer is yes for the tetrahedron, and for certain prisms, but no for all other regular polyhedra.