Abstract
Given a polyhedron, the number of its unfolding is obtained by the Matrix-Tree Theorem. For example, a cube has 384 ways of unfolding (i.e., cutting edges). By omitting mutually isomorphic unfoldings, we have 11 essentially different (i.e., nonisomorphic) unfoldings. In this paper, we address how to count the number of nonisomorphic unfoldings for any (i.e., including nonconvex) polyhedron. By applying this technique, we also give the numbers of nonisomorphic unfoldings of all regular-faced convex polyhedra (i.e., Platonic solids, Archimedean solids, Johnson-Zalgaller solids, Archimedean prisms, and antiprisms), Catalan solids, bipyramids and trapezohedra. For example, while a truncated icosahedron (a Buckminsterfullerene, or a soccer ball fullerene) has 375,291,866,372,898,816, 000 (approximately 3.75 ×1020) ways of unfolding, it has 3,127,432,220, 939,473,920 (approximately 3.13 ×1018) nonisomorphic unfoldings. A truncated icosidodecahedron has 21,789,262,703,685,125,511,464,767,107, 171,876,864,000 (approximately 2.18 ×1040) ways of unfolding, and has 181,577,189,197,376, 045,928,994,520,239,942,164,480 (approximately 1.82 ×1038) nonisomorphic unfoldings.
A preliminary version was presented at EuroCG2013.
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References
Akiyama, J., Kuwata, T., Langerman, S., Okawa, K., Sato, I., Shephard, G.C.: Determination of All Tessellation Polyhedra with Regular Polygonal Faces. In: Akiyama, J., Bo, J., Kano, M., Tan, X. (eds.) CGGA 2010. LNCS, vol. 7033, pp. 1–11. Springer, Heidelberg (2011)
Atkins, P.W., Child, M.S., Phillips, C.S.G.: Tables for Group Theory, Oxford University Press (1970)
Boesch, G.F.T., Bogdanowicz, Z.R.: The Number of Spanning Trees in a Prism, Inter. J. Comput. Math. 21, 229–243 (1987)
Bouzette, S., Vandamme, F.: The regular Dodecahedron and Icosahedron unfold in 43380 ways (unpublished manuscript)
Brown, T.J.N., Mallion, R.B., Pollak, P., de Castro, B.R.M., Gomes, J.A.N.F.: The number of spanning trees in buckminsterfullerene. Journal of Computational Chemistry 12, 1118–1124 (1991)
Brown, T.J.N., Mallion, R.B., Pollak, P., Roth, A.: Some Methods for Counting the Spanning Trees in Labelled Molecular Graphs, examined in Relation to Certain Fullerenes. Discrete Applied Mathematics 67, 51–66 (1996)
Buekenhout, F., Parker, M.: The Number of Nets of the Regular Convex Polytopes in Dimension ≤ 4. Disc. Math. 186, 69–94 (1998)
Burnside, A.: Theory of Groups of Finite Order. Cambridge University Press (1911)
Cromwell, P.R.: Polyhedra. Cambridge University Press (1997)
Coxeter, H.S.M.: Regular and semi-regular polytopes. II. Math. Z. 188, 3–45 (1985)
Demaine, E.D., O’Rourke, J.: Geometric Folding Algorithms: Linkages, Origami, Polyhedra. Cambridge University Press (2007)
Haghigh, M.H.S., Bibaki, K.: Recursive Relations for the Number of Spanning Trees. Applied Mathematical Sciences 3(46), 2263–2269 (2009)
Hippenmeyer, C.: Die Anzahl der inkongruenten ebenen Netze eines regulären Ikosaeders. Elem. Math. 34, 61–63 (1979)
Horiyama, T., Shoji, W.: Edge unfoldings of Platonic solids never overlap. In: Proc. of the 23rd Canadian Conference on Computational Geometry, pp. 65–70 (2011)
Jeger, M.: Über die Anzahl der inkongruenten ebenen Netze des Würfels und des regulären Oktaeders. Elemente der Mathematik 30, 73–83 (1975)
Kleitman, D.J., Golden, B.: Counting trees in a certain class of graphs. Am. Math. Monthly 82, 40–44 (1975)
Kobayashi, M., Suzuki, T.: Data of coordinates of all regular-faced convex polyhedra (1992), http://mitani.cs.tsukuba.ac.jp/polyhedron/
Pandey, S., Ewing, M., Kunas, A., Nguyen, N., Gracias, D.H., Menon, G.: Algorithmic design of self-folding polyhedra. Proc. Natl. Acad. Sci. USA 108(50), 19885–19890 (2011)
Sloane, N.J.A.: Sequence A103535, The On-Line Encyclopedia of Integer Sequences
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Horiyama, T., Shoji, W. (2013). The Number of Different Unfoldings of Polyhedra. In: Cai, L., Cheng, SW., Lam, TW. (eds) Algorithms and Computation. ISAAC 2013. Lecture Notes in Computer Science, vol 8283. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-45030-3_58
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DOI: https://doi.org/10.1007/978-3-642-45030-3_58
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