Skip to main content Accessibility help

We use cookies to distinguish you from other users and to provide you with a better experience on our websites. Close this message to accept cookies or find out how to manage your cookie settings.

Close cookie message
×
Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-03T05:16:22.361Z Has data issue: false hasContentIssue false

Chapter 1 - More Inflation Tilings

Published online by Cambridge University Press:  26 October 2017

By
Dirk Frettlöh
Edited by
Michael Baake  and
Uwe Grimm
Dirk Frettlöh
Affiliation:
Technische Fakultät Universität Bielefeld, Germany
Michael Baake
Affiliation:
Universität Bielefeld, Germany
Uwe Grimm
Affiliation:
The Open University, Milton Keynes
Get access

Summary

A summary is not available for this content so a preview has been provided. Please use the Get access link above for information on how to access this content.
Image of the first page of this content. For PDF version, please use the ‘Save PDF’ preceeding this image.'
Type
Chapter
Information
Aperiodic Order , pp. 1 - 38
Publisher: Cambridge University Press
Print publication year: 2017

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[AO1] Baake, M. and Grimm, U. (2013) Aperiodic Order. Vol. 1: A Mathematical Invitation (Cambridge University Press, Cambridge).
[1] Aigner, M. and Ziegler, G.M. (2004) Proofs from THE BOOK, 5th ed. (Springer, Heidelberg).
[2] Arnoux, P., Harriss, E.O., Ito, S. and Furukado, M. (2011) Algebraic numbers,group automorphisms and substitution rules on the plane, Trans. Amer. Math. Soc. 363, 4651–4699.CrossRefGoogle Scholar
[3] Baake, M., Spindeler, T. and Strungaru, N. (2017) Diffraction of compatible random substitutions in one dimension, in preparation.
[4] Berthè, V. and Siegel, A. (2005) Tilings associated with beta-numeration and substitutions, Integers 5, A2 (46 pages).Google Scholar
[5] Cromwell, P.R. (2009) The search for quasi-periodicity in Islamic 5-fold ornament, Math. Intelligencer 31, 36–56.CrossRefGoogle Scholar
[6] Danzer, L. (2001) An inflation-species of planar triangular tilings which is not repetitive, Ferroelectrics 250, 163.Google Scholar
[7] Danzer, L. (2004) Inflation species of planar tilings which are not of locally finite complexity, Proc. Steklov Inst. Math. 239, 108–116.Google Scholar
[8] Danzer, L., Grünbaum, B. and Klee, V. (1982) Can all tiles of a tiling have five-fold symmetry? Amer. Math. Monthly 89, 568–570 and 583–585.
[9] Durand, F. (2000) Linearly recurrent subshifts have a finite number of non-periodic subshift factors, Ergodic Th. & Dynam. Syst. 20, 1061–1078. arXiv:0807.4430.Google Scholar
[10] Ferenczi, S. (1996) Rank and symbolic complexity, Ergodic Th. & Dynam. Syst. 16, 663–682.Google Scholar
[11] Frank, N.P. (2008) A primer of substitution tilings of the Euclidean plane, Expos. Math. 26, 295–326. arXiv:0705.1142.Google Scholar
[12] Frank, N. (2015) Tilings with infinite local complexity. In Mathematics of Aperiodic Order, Kellendonk, J., Lenz, D. and Savinien|J. eds., pp. 223–257 (Birkhäuser, Basel). arXiv:1312.4987.CrossRef
[13] Frank, N. and Sadun, L. (2014) Fusion: A general framework for hierarchical tilings of Rd, Geom. Dedicata 171, 149–186. arXiv:1101.4930.Google Scholar
[14] Frank, N. and Robinson, E.A. (2008) Generalized β-expansions,substitution tilings,and local finiteness, Trans. Amer. Math. Soc. 360, 1163–1177. arXiv:math.DS/0506098.Google Scholar
[15] Frettlöh, D. (1998) Inflationäre Pflasterungen der Ebene mit minimaler Musterfamilie und D2m+1-Symmetrie, Diploma thesis (Dortmund University).
[16] Frettlöh, D. (2002) Nichtperiodische Pflasterungen mit ganzzahligem Inflationsfaktor, Ph.D. thesis (Dortmund University), available at https://hdl.handle.net/2003/2309.
[17] Frettlöh, D. (2005) Duality of model sets generated by substitutions, Rev. Roumaine Math. Pures Appl. 50, 619–639. arXiv:math/0601064.Google Scholar
[18] Frettlöh, D. (2008) Self-dual tilings with respect to star-duality, Theoret. Comput. Sci. 391, 39–50. arXiv:0704.2528.Google Scholar
[19] Frettlöh, D. (2008) Substitution tilings with statistical circular symmetry, European J. Combin. 29, 1881–1893. arXiv:0803.2172.Google Scholar
[20] Frettlöh, D. (2008) About substitution tilings with statistical circular symmetry, Philos. Mag. 88, 2033–2039. arXiv:0704.2521.Google Scholar
[21] Frettlöh, D., Gähler, F. and Harriss E. Tilings Encyclopedia, available at https://tilings.math.uni-bielefeld.de.
[22] Frettlöh, D. and Hofstetter, K. (2015) Inductive rotation tilings, Proc. Steklov Inst. Math. 288, 269–280. arXiv:1410.0592.Google Scholar
[23] Gähler, F. (2010) MLD relations of Pisot substitution tilings, J. Phys.: Conf. Ser. 226, 012020: 1–6. arXiv:1001.2744.CrossRefGoogle Scholar
[24] Gähler, F. and Maloney, G.R. (2013) Cohomology of one-dimensional mixed substitution tiling spaces, Topol. Appl. 160, 703–719. arXiv:1112.1475.Google Scholar
[25] Gähler, F., Kwan, E.E. and Maloney, G.R. (2015) A computer search for planar substitution tilings with n-fold rotational symmetry, Discr. Comput. Geom. 53, 445–465. arXiv:1404.5193.Google Scholar
[26] Godrèche, C. and Luck, J.M. (1989) Quasiperiodicity and randomness in tilings of the plane, J. Stat. Phys. 55, 1–28.Google Scholar
[27] Goodman-Strauss, C. (1996) A non-periodic self-similar tiling with non-unique decomposition, Technical Report UofA-R-126 (University of Arkansas).
[28] Goodman-Strauss, C. (1998) Matching rules and substitution tilings, Ann. of Math. 147, 181–223.Google Scholar
[29] Gummelt, P. (2006) Private communication.
[30] Harriss, E.O. (2005) Non-periodic rhomb substitution tilings that admit order n rotational symmetry, Discr. Comput. Geom. 34, 523–536.CrossRefGoogle Scholar
[31] Kannan, S. and Soroker, D. (1992) Tiling polygons with parallelograms, Discr. Comput. Geom. 7, 175–188.CrossRefGoogle Scholar
[32] Kari, J. and Rissanen, M. (2016) Sub Rosa,a system of quasiperiodic rhombic substitution tilings with n-fold rotational symmetry, Discr. Comput. Geom. 55, 972–996. arXiv:1512.01402.Google Scholar
[33] Kenyon, R. (1992) Self-replicating tilings. In Symbolic Dynamics and its Applications, Walters, P. eds., CONM 135, pp. 239–263 (AMS, Providence, RI).
[34] Kenyon, R. (1993) Tiling a polygon with parallelograms, Algorithmica 9, 382–397.Google Scholar
[35] Kenyon, R. (1996) The construction of self-similar tilings, Geom. Funct. Anal. (GAFA) 6, 417–488. arXiv:math.MG/9505210.Google Scholar
[36] Maloney, G.R. (2015) On substitution tilings of the plane with n-fold rotational symmetry, Discr. Math. Theor. Comput. Sci. 17, 395–412. arXiv:1409.1828.Google Scholar
[37] Moll, M. (2014) Diffraction of random noble means words, J. Stat. Phys. 156, 1221–1236. arXiv:1404.7411.CrossRefGoogle Scholar
[38] Nilsson, J. (2012) On the entropy of a family of random substitutions, Monatsh. Math. 168, 563–577. arXiv:1103.4777.Google Scholar
[39] Nilsson, J. (2013) On the entropy of a two step random Fibonacci substitution, Entropy 15, 3312–3324. arXiv:1303.2526.Google Scholar
[40] Pautze, S. (2017) Cyclotomic aperiodic substitution tilings, Symmetry 9, 19: 1–41. arXiv:1606.06858.Google Scholar
[41] Pytheas Fogg, N. (2002) Substitutions in Dynamics, Arithmetics and Combinatorics, LNM 1794 (Springer, Berlin).
[42] Radin, C. (1997) Aperiodic tilings, ergodic theory, and rotations. In The Mathematics of Long-Range Aperiodic Order, NATO ASI Series C 489, MoodyR.V. eds., pp. 403–441 (Kluwer, Dordrecht).
[43] Robinson, E.A. (1999) On the table and the chair, Indag. Math. 10, 581–599.Google Scholar
[44] Sadun, L. (1998) Some generalizations of the pinwheel tiling, Discr. Comput. Geom. 20, 79–110. arXiv:math.GR/9712263.Google Scholar
[45] Sadun, L. (2008) Private communication.
[46] Senechal, M. (2004) The mysterious Mr. Ammann, Math. Intelligencer 26, 10–21.CrossRefGoogle Scholar
[47] Sing, B. (2006) Pisot Substitutions and Beyond, PhD thesis (Univ. Bielefeld).
[48] Sing, B. (2016) Private communication.
[49] Sloane, N.J.A. eds.. The On-Line Encyclopedia of Integer Sequences, available at https://oeis.org/.
[50] Socolar, J.E.S. and Taylor, J.M. (2011) An aperiodic hexagonal tile, J. Combin. Theory A 118, 2207–2231. arXiv:1003.4279.CrossRefGoogle Scholar
[51] Solomyak, B. (1998) Non-periodicity implies unique composition property for selfsimilar translationally finite tilings, Discr. Comput. Geom. 20, 265–279.CrossRefGoogle Scholar
[52] Spindeler, T. (2017) Diffraction intensities of a class of binary Pisot substitutions via exponential sums, Monatsh. Math. 182, 143–153. arXiv:1608.01969.Google Scholar
[53] Spindeler, T. (2018) On the Spectral Theory of Random Inflation Systems, PhD thesis (Bielefeld University, in preparation).
[54] Thurston, W. (1989) Groups, tilings and finite state automata, unpublished lecture notes, available at http://timo.jolivet.free.fr/docs/ThurstonLectNotes.pdf.
[55] Washington, L.C. (1997) Introduction to Cyclotomic Fields, 2nd ed. (Springer, New York).

Save book to Kindle

To save this book to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.

Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.

Find out more about the Kindle Personal Document Service.

  • More Inflation Tilings
  • Edited by Michael Baake, Universität Bielefeld, Germany, Uwe Grimm, The Open University, Milton Keynes
  • Book: Aperiodic Order
  • Online publication: 26 October 2017
  • Chapter DOI: https://doi.org/10.1017/9781139033862.003
Available formats
×

Save book to Dropbox

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.

  • More Inflation Tilings
  • Edited by Michael Baake, Universität Bielefeld, Germany, Uwe Grimm, The Open University, Milton Keynes
  • Book: Aperiodic Order
  • Online publication: 26 October 2017
  • Chapter DOI: https://doi.org/10.1017/9781139033862.003
Available formats
×

Save book to Google Drive

To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.

  • More Inflation Tilings
  • Edited by Michael Baake, Universität Bielefeld, Germany, Uwe Grimm, The Open University, Milton Keynes
  • Book: Aperiodic Order
  • Online publication: 26 October 2017
  • Chapter DOI: https://doi.org/10.1017/9781139033862.003
Available formats
×