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The icosahedral quasiperiodic tiling and its self-similarity

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Abstract

We investigate a 3-dimensional analogue of the Penrose tiling, a class of 3-dimensional aperiodic tilings whose edge vectors are the vertex vectors of a regular icosahedron. It arises by an equivariant orthogonal projection of the unit lattice in euclidean 6-space with its natural representation of the icosahedral group, given by its action on the 6 icosahedral diagonals (with orientation). The tiling has a canonical subdivision by a similar tiling (“deflation”). We give an essentially local construction of this subdivision, independent of the actual position inside the tiling. In particular we show that the subdivisions of the edges, faces and tiles (with some restriction) are unique.

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References

  1. Baake, M., Kramer, P., Schlottmann, M., Zeidler, D.: Planar patterns with fivefold symmetry as sections of periodic structures in 4-space. Int. J. Modern Phys. B 4, 2217–2268 (1990)

  2. Bohr, H.: Zur Theorie der fastperiodischen Funktionen. Acta Math. 45, 29–127 (1924) (46, 101–214 1924)

  3. Coxeter, H.S.M.: The Beauty of Geometry: Twelve Essays. Dover Publications, Mineola 1999. Unveränderte Neuveröffentlichung von Twelve Geometric Essays. Southern Illinois University Press, Carbondale (1968)

  4. De Bruijn, N.G.: Algebraic Theory of Penrose’s Non-Periodic Tilings of the Plane. Indag. Math. 43, 39–66 (1981). http://www.math.brown.edu/~res/M272/pentagrid

  5. Dietl, R.: Penrose-Muster: Unterteilung und Projektionsmethode. Schriftliche Hausarbeit zur 1. Staatsprüfung für das Lehramt an Gymnasien. Augsburg (2008). http://myweb.rz.uni-augsburg.de/~eschenbu/ZulassungsarbeitScan.PDF

  6. Dietl, R.: Dreidimensionale Penrose-Muster und Selbstähnlichkeit. Dissertation. Augsburg (2011). http://myweb.rz.uni-augsburg.de/~eschenbu/diss_dietl

  7. Eschenburg, J.-H.: Die Zahl Fünf und die Quasikristalle. Augsburg (2004). http://myweb.rz.uni-augsburg.de/~eschenbu/penrose

  8. Eschenburg J.-H., Rivertz H.J.: Self similar symmetric planar tilings. J. Geom. 87, 55–75 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. Eschenburg, J.-H., Rivertz, H.J.: The Penrose Decagon (2016) (preprint)

  10. Katz A., Duneau M.: Quasiperiodic patterns and icosahedral symmetry. Phys. Rev. Lett. 47, 181–196 (1986)

    MathSciNet  MATH  Google Scholar 

  11. Kepler, J.: Strena Seu de Nive Sexangula, Frankfurt 1511. In: Caspar, M., Hammer, F. (eds.) Gesammelte Werke, vol. IV, pp. 259–280. München (1941). http://www.thelatinlibrary.com/kepler/strena.html

  12. Kepler, J.: Harmoniae Mundi, Linz (1619). http://archive.org/stream/ioanniskepplerih00kepl#page/

  13. Kramer P., Neri R.: On periodic and non-periodic space fillings of E m obtained by projection. Acta Crystallogr. Sect. A 40, 580–587 (1984)

    Article  MATH  Google Scholar 

  14. Kowalewski, G.: Der Keplersche Körper und andere Bauspiele. K.F. Koehlers Antiquarium, Leipzig (1938)

  15. Mackay A.L.: De Nive Quinquangula: On the Pentagonal Snowflake, Soviet Physics. Crystallography 26, 517–522 (1981)

    Google Scholar 

  16. Mackay A.L.: Crystallography and the Penrose Pattern. Phys. A 114, 609–613 (1982)

    Article  MathSciNet  Google Scholar 

  17. Ogawa T.: On the Structure of a Quasicrystal—Three-Dimensional Penrose Transformation. J. Phys. Soc. Jpn. 54, 3205–3208 (1985)

    Article  Google Scholar 

  18. Ogawa, T. Symmetry of three-dimensional quasicrystals. Mater. Sci. Forum 22–24, 187–200 (1987).

  19. Penrose R.: Pentaplexity—a class of non-periodic tilings of the plane. Math. Intell. 2, 32–37 (1979)

    Article  MATH  Google Scholar 

  20. Senechal M.: The Mysterious Mr. Ammann. Math. Intell. 26, 10–21 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  21. Senechal, M.: Donald and the Golden Rhombohedra. In: Davis, C., Ellers E.W. (eds.) The Coxeter Legacy. AMS Fields, Providence (2006)

  22. Shechtman D., Blech I., Gratias D., Cahn J.W.: Metallic phase with long-range orientational order and no translational symmetry. Phys. Rev. Lett. 53, 1951–1953 (1984)

    Article  Google Scholar 

  23. Stern, D.: Penrose type tilings. Diplomarbeit, Augsburg (2002). http://myweb.rz.uni-augsburg.de/~eschenbu/diplomarbeit

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Authors and Affiliations

  1. Universitätsstrasse 14, 86135, Augsburg, Germany

    Ruth Maria Katharina Dietl & Jost-Hinrich Eschenburg

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  1. Ruth Maria Katharina Dietl
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  2. Jost-Hinrich Eschenburg
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Correspondence to Ruth Maria Katharina Dietl.

Additional information

To Paul Hildebrandt, Zometool Inc.

This work was completed with the support of Deutsche Forschungsgemeinschaft. We thank the referee for several valuable hints and suggestions.

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Dietl, R.M.K., Eschenburg, JH. The icosahedral quasiperiodic tiling and its self-similarity. J. Geom. 108, 319–354 (2017). https://doi.org/10.1007/s00022-016-0342-2

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  • DOI: https://doi.org/10.1007/s00022-016-0342-2

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