How many faces can polycubes of lattice tilings by translation of R3 have?

Abstract : We construct a class of polycubes that tile the space by translation in a lattice- periodic way and show that for this class the number of surrounding tiles cannot be bounded. The first construction is based on polycubes with an L-shape but with many distinct tilings of the space. Nevertheless, we are able to construct a class of more complicated polycubes such that each polycube tiles the space in a unique way and such that the number of faces is 4k + 8 where 2k + 1 is the volume of the polycube. This shows that the number of tiles that surround the surface of a space-filler cannot be bounded.
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The Electronic Journal of Combinatorics, Open Journal Systems, 2011, 18, pp.#P199
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Ian Gambini, Laurent Vuillon. How many faces can polycubes of lattice tilings by translation of R3 have?. The Electronic Journal of Combinatorics, Open Journal Systems, 2011, 18, pp.#P199. 〈hal-00943569〉

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