Abstract
This paper presents a general family of 3D hinged dissections for polypolyhedra, i.e., connected 3D solids formed by joining several rigid copies of the same polyhedron along identical faces. (Such joinings are possible only for reflectionally symmetric faces.) Each hinged dissection consists of a linear number of solid polyhedral pieces hinged along their edges to form a flexible closed chain (cycle). For each base polyhedron P and each positive integer n, a single hinged dissection has folded configurations corresponding to all possible polypolyhedra formed by joining n copies of the polyhedron P. In particular, these results settle the open problem posed in [7] about the special case of polycubes (where P is a cube) and extend analogous results from 2D [7].Along the way, we present hinged dissections for polyplatonics (where P is a platonic solid) that are particularly efficient: among a type of hinged dissection, they use the fewest possible pieces.
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Aichholzer, O., Aurenhammer, F.: Straight skeletons for general polygonal figures in the plane. In: Cai, J.-Y., Wong, C.K. (eds.) COCOON 1996. LNCS, vol. 1090, pp. 117–126. Springer, Heidelberg (1996)
Aichholzer, O., Aurenhammer, F., Alberts, D., Gärtner, B.: A novel type of skeleton for polygons. Journal of Universal Computer Science 1(12), 752–761 (1995)
Akiyama, J., Nakamura, G.: Dudeney dissection of polygons. In: Akiyama, J., Kano, M., Urabe, M. (eds.) JCDCG 1998. LNCS, vol. 1763, pp. 14–29. Springer, Heidelberg (2000)
Arkin, E.M., Held, M., Mitchell, J.S.B., Skiena, S.S.: Hamiltonian triangulations for fast rendering. The Visual Computer 12(9), 429–444 (1996)
Boltianskii, V.G.: Hilbert’s Third Problem. V. H. Winston & Sons (1978)
Cheng, S.-W., Vigneron, A.: Motorcycle graphs and straight skeletons. In: Proc. 13th Ann. ACM-SIAM Sympos. Discrete Algorithms, pp. 156–165 (2002)
Demaine, E.D., Demaine, M.L., Eppstein, D., Frederickson, G.N., Friedman, E.: Hinged dissection of polyominoes and polyforms. Computational Geometry: Theory and Applications (to appear) http://arXiv.org/abs/cs.CG/9907018
Dudeney, H.E.: Puzzles and prizes. Weekly Dispatch, April 6 (1902)
Eppstein, D.: Hinged kite mirror dissection (June 2001), http://arXiv.org/abs/cs.CG/0106032
Eppstein, D., Erickson, J.: Raising roofs, crashing cycles, and playing pool: Applications of a data structure for finding pairwise interactions. Discrete & Computational Geometry 22(4), 569–592 (1999)
Erickson, J.: Personal communication (February 2000)
Frederickson, G.N.: Dissections: Plane and Fancy. Cambridge Univ. Press, Cambridge (1997)
Frederickson, G.N.: Hinged Dissections: Swinging & Twisting. Cambridge Univ. Press, Cambridge (2002)
Griffith, S.: Growing Machines. PhD thesis, MIT Media Laboratory, September 2004
Kranakis, E., Krizanc, D., Urrutia, J.: Efficient regular polygon dissections. Geometriae Dedicata 80, 247–262 (2000)
Lowry, M.: Solution to question 269, [proposed] by Mr. W. Wallace. In: T. Leybourn, ed., Mathematical Repository, vol. 3, part 1, pp. 44–46. W. Glendinning (1814)
Mao, C., Thallidi, V.R., Wolfe, D.B., Whitesides, S., Whitesides, G.M.: Dissections: Self-assembled aggregates that spontaneously reconfigure their structures when their environment changes. J. Amer. Chemical Soc. 124, 14508–14509 (2002)
Palmer, L.: The helium stockpile: Under shifting conditions of heat and pressure. In: Installation, Radcliffe College, Cambridge, Massachusetts (April 2004)
Rus, D., Butler, Z., Kotay, K., Vona, M.: Self-reconfiguring robots. Communications of the ACM 45(3), 39–45 (2002)
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Demaine, E.D., Demaine, M.L., Lindy, J.F., Souvaine, D.L. (2005). Hinged Dissection of Polypolyhedra. In: Dehne, F., López-Ortiz, A., Sack, JR. (eds) Algorithms and Data Structures. WADS 2005. Lecture Notes in Computer Science, vol 3608. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11534273_19
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DOI: https://doi.org/10.1007/11534273_19
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