Penrose tilings as model sets

Coordination numbers of the vertex graph of a Penrose tiling

“…Note that this result also explains the replacement of oðnÞ by À10 in remark 4. Also note that an analogue of previous arguments firstly appeared in Shutov & Maleev (2018) for the proof of the theorem on the growth form of a Penrose tiling.…”

“…In this section we briefly discuss the construction of the vertex graph of a Penrose tiling. Details can be found in Shutov & Maleev (2015).…”

“…The choice of the boundaries of pentagons i considered in the paper is not the only possible one. Other possibilities and general rules for choosing boundaries are discussed by Shutov & Maleev (2015). Let eðnÞ and e 0 ðnÞ be the sequences of coordination numbers of two vertex graphs of Penrose tilings, distinguished by the choice of boundaries of pentagons i .…”

An Application of Singular Traces to Crystals and Percolation

“…Firstly, a Penrose tiling. This proposition is entirely based on recent results by A. Shutov and A. Maleev [43,44].…”

An integer representation for periodic tilings of the plane by regular polygons

“…Complex numbers have been used to represent aperiodic tilings. Shutov and Maleev [28] refined the Baake construction of Penrose tilings using 5th roots of the unity. Pautze [29] used 2nth roots of the unity to represent substitution aperiodic tilings with dihedral symmetry.…”