This paper studies common unfoldings of various classes of polycubes, as well as a new type of unfolding of polyominoes. Previously, Knuth and Miller found a common unfolding of all tree-like tetracubes. By contrast, we show here that all 23 tree-like pentacubes have no such common unfolding, although 22 of them have a common unfolding. On the positive side, we show that there is an unfolding common to all "non-spiraling" k-ominoes, a result that extends to planar non-spiraling k-cubes.

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Aloupis, G, Bose, P, Collette, S. (Sébastien), Demaine, E.D. (Erik D.), Demaine, M.L. (Martin L.), Douïeb, K. (Karim), … Morin, P. (2011). Common unfoldings of polyominoes and polycubes. doi:10.1007/978-3-642-24983-9_5