Tiling the Space by Polycube Analogues of Fedorov's Polyhedra

• Autorzy: Gambini I. , Vuillon L.

Identyfikatory

DOI

10.3233/FI-2016-1381

• Języki publikacji: EN

Abstrakty

EN

We investígate minimal polycubes in terms of volume that tile the R³ space like the Fedorov's polyhedra. In fact the 5 Fedorov's polyhedra are convex polyhedra that tile the space by translation and we construct geometrical discrete objects formed by union of cubes with the same number of faces than the Fedorov's polyhedra.

Słowa kluczowe

EN

Fedorov's polyhedra; lattice periodic tilings; polycubes; tilings by translation; tilings of R3

• Wydawca: IOS Press
• Czasopismo: Fundamenta Informaticae
• Rocznik: 2016
• Tom: Vol. 146, nr 2
• Strony: 197--209
• Opis fizyczny: Bibliogr. 12 poz., rys.. tab.

Twórcy

autor

Gambini I.

• Laboratoire d’Informatique Fondamentale (LIF), Aix-Marseille Université, 13288 Marseille Cedex 9, France

autor

Vuillon L.

• Laboratoire de mathématiques, CNRS UMR 5127, Université de Savoie Mont Blanc, 73376 Le Bourget-du-lac Cedex, France

Bibliografia

• [1] Grunbaum B, and Shephard GC. Tilings with congruent tiles, Bulletin of American Mathematical Society, 1980;3(3):951–974.
• [2] Beauquier D, and Nivat M. On translating one polyomino to tile the plane, Discrete & Computational Geometry, 1991;(6)4:575–592. doi:10.1007/BF02574705.
• [3] Stein SK, and Szabo S. Algebra and Tiling: Homomorphisms In The Service Of Geometry. Mathematical Association of America, Washington, DC 1994. ISBN-10:0883850281, 13:978-0883850282.
• [4] Guttmann AJ. (ed.), Polygons, Polyominoes and Polycubes, Lecture Notes in Physics, no. 775, Springer, Berlin, 2009. doi:10.1007/978-1-4020-9927-4.
• [5] Grunbaum B. Tilings by some nonconvex parallelohedra, Geombinatorics, 2010;19(3):100–107.
• [6] Grunbaum B. The Bilinski Dodecahedron and Assorted Parallelohedra, Zonohedra, Monohedra, Isozonohedra, and Otherhedra, The Mathematical Intelligencer, 2010;32(4):5–15. doi:10.1007/s00283-010-9138-7.
• [7] Gambini I, and Vuillon L. How many faces can polycubes of lattice tilings by translation of R have?, The electronic journal of combinatorics, 2011;18(1): #P199.
• [8] Gambini I, and Vuillon L. Non lattice periodic tilings of R3 by single polycubes, Theoretical Computer Science, 2012;(432):52–57. doi:10.1016/j.tcs.2012.01.014.
• [9] Ziegler GM. Lectures on Polytopes. Springer, New York 1995. doi: 10.1007/978-1-4613-8431-1.
• [10] Fedorov ES. Nachala Ucheniya o Figurah. [In Russian] (=Elements of the theory of figures) Notices of the Imperial Mineralogical Society (St. Petersburg), 2nd ser., 1885;(24):1–279. Republished by the Russian Academy of Science, Moscow 1953.
• [11] Vallentin F. Sphere coverings, lattices, and tilings (in low dimensions), Ph.D. thesis, Technical University Munich, Germany, 2003, 128 pages.
• [12] Socolar JES, and Taylor JM. Forcing Nonperiodicity with a Single Tile, The Mathematical Intelligencer, 2011;34(1):18–28.

Uwagi

Opracowanie ze środków MNiSW w ramach umowy 812/P-DUN/2016 na działalność upowszechniającą naukę (zadania 2017).