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Bronze-mean hexagonal quasicrystal


The most striking feature of conventional quasicrystals is their non-traditional symmetry characterized by icosahedral, dodecagonal, decagonal or octagonal axes1,2,3,4,5,6. The symmetry and the aperiodicity of these materials stem from an irrational ratio of two or more length scales controlling their structure, the best-known examples being the Penrose7,8 and the Ammann–Beenker9,10 tiling as two-dimensional models related to the golden and the silver mean, respectively. Surprisingly, no other metallic-mean tilings have been discovered so far. Here we propose a self-similar bronze-mean hexagonal pattern, which may be viewed as a projection of a higher-dimensional periodic lattice with a Koch-like snowflake projection window. We use numerical simulations to demonstrate that a disordered variant11 of this quasicrystal can be materialized in soft polymeric colloidal particles with a core–shell architecture12,13,14,15,16,17. Moreover, by varying the geometry of the pattern we generate a continuous sequence of structures, which provide an alternative interpretation of quasicrystalline approximants observed in several metal–silicon alloys18.

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Figure 1: Bronze tiling.
Figure 2: Hard-core/square-shoulder random bronze quasicrystal.
Figure 3: Tile frequencies.
Figure 4: Higher-dimensional analysis.
Figure 5: Phason defects.


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This project has been supported by the Japan Society for the Promotion of Science through Grant-in-Aid for Scientific (B) (No. 16H04037), and has received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No. 642774. The authors acknowledge the financial support from the Slovenian Research Agency (research core funding No. P1-0055).

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T.D. and P.Z. conceived the project and proposed the tiling theory. S.B. and T.D. performed simulations and higher-dimensional analysis, and T.D. and P.Z. wrote the manuscript.

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Correspondence to Tomonari Dotera.

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The authors declare no competing financial interests.

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Dotera, T., Bekku, S. & Ziherl, P. Bronze-mean hexagonal quasicrystal. Nature Mater 16, 987–992 (2017).

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