chair tiling

{{Short description|Nonperiodic substitution tiling}}

File:L_substitution_tiling.svg

In geometry, a chair tiling (or L tiling) is a nonperiodic substitution tiling created from L-tromino prototiles. These prototiles are examples of rep-tiles and so an iterative process of decomposing the L tiles into smaller copies and then rescaling them to their original size can be used to cover patches of the plane.{{cite journal |last1=Robinson Jr. |first1=E. Arthur |title=On the table and the chair |journal=Indagationes Mathematicae |date=1999-12-20 |volume=10 |issue=4 |pages=581–599 |doi=10.1016/S0019-3577(00)87911-2|doi-access=free }}{{rp|581}} Chair tilings do not possess translational symmetry, i.e., they are examples of nonperiodic tilings, but the chair tiles are not aperiodic tiles since they are not forced to tile nonperiodically by themselves.{{Citation| last = Goodman-Strauss| first = Chaim| author-link = Chaim Goodman-Strauss| editor-last = Sadoc| editor-first = J. F.| editor2-last = Rivier| editor2-first = N.| year = 1999| title = Foams and Emulsions| chapter = Aperiodic Hierarchical Tilings| chapter-url = https://strauss.hosted.uark.edu/papers/AHT.pdf| publisher = Springer| place = Dordrecht| pages = 481–496| isbn = 978-90-481-5180-6| doi = 10.1007/978-94-015-9157-7_28}}{{rp|482}} The trilobite and cross tiles are aperiodic tiles that enforce the chair tiling substitution structure{{cite journal |last1=Goodman-Strauss |first1=Chaim |title=A Small Aperiodic Set of Planar Tiles |journal=European Journal of Combinatorics |date=1999 |volume=20 |issue=5 |pages=375–384 |doi=10.1006/eujc.1998.0281|doi-access=free }} and these tiles have been modified to a simple aperiodic set of tiles using matching rules enforcing the same structure.{{cite journal |last1=Goodman-Strauss |first1=Chaim |title=Lots of aperiodic sets of tiles |journal=Journal of Combinatorial Theory | series=Series A |date=2018 |volume=160 |pages=409–445 |doi=10.1016/j.jcta.2018.07.002 |arxiv=1608.07165 }} Barge et al. have computed the Čech cohomology of the chair tiling{{cite journal |last1=Barge |first1=Marcy |last2=Diamond |first2=Beverly |last3=Hunton |first3=John |last4=Sadun |first4=Lorenzo |title=Cohomology of substitution tiling spaces |journal=Ergodic Theory and Dynamical Systems |date=2010 |volume=30 |issue=6 |pages=1607–1627 |doi=10.1017/S0143385709000777 |arxiv=0811.2507 }} and it has been shown that chair tilings can also be obtained via a cut-and-project scheme.{{cite journal |last1=Baake |first1=Michael |last2=Moody |first2=Robert V. |last3=Schlottmann |first3=Martin |title=Limit-(quasi)periodic point sets as quasicrystals with p-adic internal spaces |journal=Journal of Physics A: Mathematical and General |date=1998 |volume=31 |issue=27 |pages=5755–5766 |doi=10.1088/0305-4470/31/27/006 |arxiv=math-ph/9901008 |bibcode=1998JPhA...31.5755B }}