Cyclotomic Aperiodic Substitution Tilingsaverkov/geotag... · Aperiodic substitution tilings based...

Cyclotomic Aperiodic Substitution TilingsStefan Pautze

Cyclotomic Aperiodic Substitution Tilings

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Stefan PautzeVisualien der Breitbandkatze

[email protected]://ww.pautze.de

Geometrietag 2017Otto-von-Guericke-Universität Mageburg, Germany

2017-12-08 - 2017-12-09

Open access article available at:http://www.mdpi.com/2073-8994/9/2/19

mailto:[email protected]

http://ww.pautze.de/

http://www.mdpi.com/2073-8994/9/2/19


Motivation

● The Nobel Prize in Chemistry 2011awarded to Dan Shechtmanfor the discovery of quasicrystals

● Penrose-Tiling

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Dan Shechtman – Facts, Nobelprize.org, 2011http://www.nobelprize.org/nobel_prizes/chemistry/laureates/2011/shechtman-facts.html,

Photo: U. Montan

http://www.nobelprize.org/nobel_prizes/chemistry/laureates/2011/shechtman-facts.html




Motivation

● Aperiodic substitution tilings based on Girih Tiles(Islamic Architecture)

Makovicky 1992 + Lu & Steinhardt 2007

● My own version:

Lu, P.J.; Steinhardt, P.J.Decagonal and Quasi-Crystalline Tilings in Medieval Islamic Architecture. Science 2007, 315, 1106–1110.

Makovicky, E. Hargittai, I. (Ed.)800-year-old pentagonal tiling from Maragha, Iran, and the new varieties of a periodic tiling it inspired Fivefold symmetry, World Scientific: Singapore, 1992, 67-86.

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Motivation

● Do aperiodic substitution tilings with other symmetries exist?

Yes!

– 8-fold: Ammann-Beenker-Tiling

– 12-fold: Shield-Tiling Socolar-Tiling Stampfli-Gähler / Ship-Tiling

– 7 and 14-fold: Nischke & Danzer

– 11-fold: Maloney

D. Frettlöh, F. Gähler, and E. O. Harris.Tilings encyclopedia. available at http://tilings.math.uni-bielefeld.de/.

M. Baake and U. Grimm. Aperiodic Order. Vol 1. A Mathematical Invitation, volume 149 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 2013.

… and references therein ...

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Motivation

● Do generalizations for aperiodic substitution tilings with anyn-fold symmetries exist?

Yes!

– Warrington 1988

– Nischke & Danzer 1996

– Harris 2005 … generalization of a rhomb tiling …

– Hibma 2015 … generalization of Lançon-Billard / Binary-tiling and more …

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Motivation

● … with focus on individual n-fold symmetry?

Yes!

– Gähler, Kwan, Maloney 2015

– Maloney 2015

– Kari & Rissanen 2016

... formal proof that aperiodic substitution tilings with any individual n-fold symmetry exist.

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Motivation

Warrington, D.H.Two Dimensional Inflation Patterns and Corresponding Diffraction Patterns. In Quasicrystalline Materials: Proceedings of the I.L.L./CODEST Workshop; Janot, C., Dubois, J., Eds.; World Scientific: Singapore, 1988; pp. 243–254.

Nischke, K.P.; Danzer, L.A construction of inflation rules based on n-fold symmetry. Discret. Comput. Geom. 1996, 15, 221–236.

Harriss, E.O.Non-Periodic Rhomb Substitution Tilings that Admit Order n Rotational Symmetry. Discret. Comput. Geom. 2005, 34, 523–536.

Hibma, T.Generalization of Non-periodic Rhomb Substitution Tilings. arXiv 2015, arXiv:1509.02053.

Hibma, T. Aperiodic Rhomb Tiling. Available online: http://www.hibma.org/wpaperiodictiling/

Gähler, F.; Kwan, E.E.; Maloney, G.R.A computer search for planar substitution tilings with n-fold rotational symmetry. Discret. Comput. Geom. 2015, 53, 445–465.

Maloney, G.R.On substitution tilings of the plane with n-fold rotational symmetry. Discret. Math. Theor. Comput. Sci. 2015, 17, 395–412.

Kari, J.; Rissanen, M.Sub Rosa, A System of Quasiperiodic Rhombic Substitution Tilings with n-Fold Rotational Symmetry. Discret. Comput. Geom. 2016, 55, 972–996.

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Motivation

● But is there a convinient way to describe all those substitution tilings?

In detail …

– Substitution matrix

– Minimal inflation multiplier

Well …

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Definitions

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Diagonals or regular n-gons vs. roots of unity

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Diagonal product formula

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● Nischke & Danzer 1996

Steinbach 1997

➔ Positive sums of µn,k form a commutative semiring ...

Nischke, K.P.; Danzer, L.A construction of inflation rules based on n-fold symmetry. Discret. Comput. Geom. 1996, 15, 221–236.

Steinbach, P. Golden fields: A case for the heptagon. Math. Mag. 1997, 70, 22–31.


Areas vs. diagonals or regular n-gons

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● Areas of proto tiles and super tiles with vertices supported by or can be described as sums of triangles spanned by roots of unity.

● The areas of triangles spanned by roots of unity and diagonals of regular n-gons have similar proportions:


Eigenwert problem ...

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Eigenwert problem ...

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● With we can conclude:

… and derive a general approach for the substitution matrices.Examples for n=8 and n=9:


Minimal inflation multiplier ...

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● For n ≥ 4 we can derive ...


Thank you ...

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… for your attention ...

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CAST ∆7-1.1.1Installation – Enhanced version of aaperiodic substitution tiling by L. DanzerShown 2017-11 at “Drei Dimensionen der Ebene”Kunstpavillon Munich, GermanyInstallation + Photo by Stefan Pautze


Backup / Literature

● B. Grünbaum and G. C. Shephard.Tilings and Patterns. W. H. Freeman & Co.: New York, NY, USA, 1987.

● D. Frettlöh, F. Gähler, and E. O. Harris.Tilings encyclopedia. available at http://tilings.math.uni-bielefeld.de/.

● M. Baake and U. Grimm. Aperiodic Order. Vol 1. A Mathematical Invitation, volume 149 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 2013.

● M. Baake and U. Grimm. Aperiodic Order. Vol 2. Crystallography and Almost Periodicity, volume 166 of Encyclopedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 2017.

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