Packing of $\mathbb{R}²$ by crosses

Year, pages: 2015, 935 - 956

A cross in $\mathbb{R}ⁿ$ is a cluster of unit cubes comprising a central one and $2n$ arms. In their monograph Algebra and Tiling, Stein and Szabó suggested that tilings of $\mathbb{R}^{n}$ by crosses should be studied. The question of the existence of such a tiling has been answered by various authors for many special cases. In this paper we completely solve the problem for $\mathbb{R}^{2}$. In fact we do not only characterize crosses for which there exists a tiling of $\mathbb{R}^{2}$ but for each cross we determine its maximum packing density.

Cruz, C., D'azevedo Breda, A., Pinto, M. 2015. Packing of $\mathbb{R}²$ by crosses. In Mathematica Slovaca, vol. 65, no.5, pp. 935-956. 0139-9918. DOI: https://doi.org/10.1515/ms-2015-0063

Cruz, C., D'azevedo Breda, A., Pinto, M. (2015). Packing of $\mathbb{R}²$ by crosses. Mathematica Slovaca, 65(5), 935-956. 0139-9918. DOI: https://doi.org/10.1515/ms-2015-0063