Ammann–Beenker tiling

Ammann–Beenker tiling
Chaim Goodman-Strauss · CC BY-SA 4.0 · source
Name	Ammann–Beenker tiling
Type	Aperiodic tiling
Discovered	1977–1982
Discovered by	Robert Ammann; F. P. M. Beenker
Symmetry	Eightfold

Ammann–Beenker tiling is an aperiodic, nonperiodic tiling of the plane exhibiting eightfold rotational order and inflation symmetry. It arises in the study of quasicrystals, aperiodic order, and mathematical tiling theory, and connects to algebraic number theory, Fourier analysis, and discrete geometry. The tiling has been used in models of materials and appears in the literature of mathematical physics, combinatorics, and topology.

Definition and basic properties

The tiling was developed by Robert Ammann and independently by F. P. M. Beenker and is constructed from two prototiles commonly taken as a square and a 45° rhombus; the pattern is nonperiodic and exhibits local isomorphism with unique composition rules. It displays global eightfold rotational symmetry similar to patterns studied by Alan Turing in morphogenesis, and its combinatorial structure relates to substitution systems introduced by Roger Penrose in his eponymous tilings and to inflation rules studied by Heinz-Otto Peitgen. The set of local patches satisfies the local matching rules concept advanced by John Conway and links to the decision problems considered by Harold V. McIntosh and Marjorie Rice in recreational mathematics. Its vertex configurations and matching rules are catalogued analogously to classification projects by William Thurston and John H. Conway on planar tilings.

Construction methods

One method uses inflation and substitution with linear expansion by the silver ratio related to units of the ring of integers in the quadratic field studied by Kurt Gödel-era algebraists; this substitution parallels schemes used by Penrose and substitution tilings formalized by Branko Grünbaum and G. C. Shephard. Another approach is via cut-and-project from an eight-dimensional lattice embedded by projection rules similar to methods developed for quasicrystals by Nicolas G. de Bruijn and formalized in model sets by Yves Meyer; the projection formalism mirrors constructions used in the analysis of the Fibonacci word and model sets studied by Jean-Paul Allouche. A third practical method employs Ammann bars or matching decorations credited in expositions by Michael Baake and Uwe Grimm, adapting techniques that echo the strip projection methods of Peter Kramer and Roberto V. Moody. Implementations exploit combinatorial algorithms from computational geometry like those in the corpus by Herbert Edelsbrunner and utilize algebraic frameworks related to the silver ratio studied by Mahler.

Mathematical properties and symmetry

The tiling admits local isomorphism classes and repetitivity properties proven using substitution matrices similar to Perron–Frobenius theory applied in works by Oskar Perron and Frobenius; eigenvalues relate to the silver mean studied by Hermann Minkowski-influenced number theorists. Its point diffraction and Fourier transform link to methods in harmonic analysis pioneered by Norbert Wiener and to spectral theory explored by Barry Simon in mathematical physics. The eightfold symmetry does not permit lattice periodicity by crystallographic restriction theorem contexts examined by Evgraf Fedorov and Arthur Moritz Schönflies; instead the tiling exhibits noncrystallographic symmetry similar to findings in Dan Shechtman’s quasicrystal experiments and theoretical treatments by Alan Mackay. Topological properties of its tiling space connect to Čech cohomology computations like those worked out by John Conway collaborators and by Edwin Hewitt-era topologists.

Spectral and dynamical properties

The dynamical system associated with the translation action on the tiling hull has pure point spectrum under conditions analogous to those proved by D. L. Baake and Michael Baake for model sets; ergodic properties relate to measure-theoretic frameworks developed by Marian Kupka-style ergodic theorists and to abstract results by Hillel Furstenberg. Diffraction yields a discrete Bragg peak structure reminiscent of analyses by Paul Steinhardt and Dov Levine in quasicrystal theory; quantitative spectral features are studied using trace map techniques similar to those applied by B. Sutherland and R. Penney. The spectral measures and eigenfunctions connect to Schrödinger operator studies carried out by Barry Simon and to aperiodic Schrödinger models investigated by Jean Bellissard and collaborators.

Relations to quasicrystals and applications

The Ammann–Beenker tiling serves as an idealized mathematical model for octagonal quasicrystals observed in electron diffraction studies by Dan Shechtman and analyzed by P. J. Steinhardt; it informs HRTEM image interpretation in materials science groups at institutions like IBM research labs and university materials departments modeled after work at University of California, Berkeley. Applications extend to photonic quasicrystals studied by research groups at MIT and Harvard University, and to metamaterials investigated at ETH Zurich and Max Planck Society institutes. Computational implementations and visualizations have been produced in open-source projects inspired by software developed at Los Alamos National Laboratory and algorithms taught in courses at Princeton University and University of Cambridge.

Variants and decorations

Variants include decorated prototiles with Ammann bars that enforce matching rules, as documented in monographs by Branko Grünbaum and G. C. Shephard and in expository articles by Marjorie Rice-style popularizers. Other variants arise from altering substitution rules akin to approaches by Roger Penrose for kite-and-dart variants and by John Conway for multigrid methods; decorations produce locally derivable patterns studied by Stephen Wolfram in cellular automata contexts and by Solomon Golomb in tiling puzzles. Generalizations to higher dimensions and other nonperiodic symmetries have been pursued by researchers at University of Manchester and University of Vienna following frameworks developed by Roberto V. Moody and Yves Meyer.

Category:Aperiodic tilings