Geodesic
How many faces can the polycubes of lattice tilings by translation of \(\mathbb R^3\) have?
The electronic journal of combinatorics, Tome 18 (2011) no. 1
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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.
DOI : 10.37236/686
Classification : 05B45
Mots-clés : tilings of \(\mathbb R^3\), tilings by translation, lattice periodic tilings, space-fillers 
@article{10_37236_686,
     author = {I. Gambini and L. Vuillon},
     title = {How many faces can the polycubes of lattice tilings by translation of \(\mathbb {R^3\)} have?},
     journal = {The electronic journal of combinatorics},
     year = {2011},
     volume = {18},
     number = {1},
     doi = {10.37236/686},
     zbl = {1230.05094},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/686/}
}
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I. Gambini; L. Vuillon. How many faces can the polycubes of lattice tilings by translation of \(\mathbb R^3\) have?. The electronic journal of combinatorics, Tome 18 (2011) no. 1. doi: 10.37236/686

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