Nonperiodic tiling

Nonperiodic tiling
Name	Nonperiodic tiling

Nonperiodic tiling is a class of plane and space coverings made from one or more prototiles that admit no translational symmetry, producing arrangements that never repeat periodically. These tilings link deep results in Mathematics with visual phenomena in Art and Architecture, and have influenced research at institutions such as the Institute for Advanced Study, the Max Planck Society, and the Clay Mathematics Institute. Developments in nonperiodic tiling have engaged figures and bodies including Roger Penrose, the Royal Society, the American Mathematical Society, the Hermann Weyl tradition, and exhibitions at the Museum of Modern Art.

Definition and characteristics

A nonperiodic tiling is defined by local tile shapes and matching rules that prevent any nontrivial translational symmetry, a concept examined in the work of Emil Post, Hassler Whitney, John Conway, Branko Grünbaum, and Gunnar Berg. The distinction between an aperiodic prototile set and a nonperiodic tiling was clarified in writings by Berger, Penrose, Michael Freedman, C. P. Rourke, and researchers associated with Princeton University. Researchers at the Courant Institute and the University of Cambridge have emphasized combinatorial local rules, while groups at MIT and the University of California, Berkeley have emphasized geometric constraints and matching conditions. Classification problems reference earlier work by David Hilbert, Andrei Kolmogorov, Kurt Gödel, and later computational perspectives from Stephen Cook and the Alan Turing school.

Historical development

Early instances connecting tiling and nonperiodicity trace to ornamentation studies in Alhambra scholarship and to mathematical puzzles discussed by Johannes Kepler and Marston Morse, with formal mathematical formulations appearing in the 20th century. The undecidability result of Berger linked the topic to Hilbert's Entscheidungsproblem and to developments at Princeton University and the Institute for Advanced Study, while later explicit aperiodic sets emerged from work by Berger, Penrose, John Conway, R. Berger, and Raphael M. Robinson. Breakthroughs involving single aperiodic tiles and the notion of an "einstein" were pursued by investigators at University of Cambridge, the University of Manchester, and independent researchers associated with the Mathematical Association of America. Innovations were publicized through conferences held by the American Mathematical Society, the London Mathematical Society, and venues such as Guggenheim Fellowship-supported projects.

Examples and notable tilings

Famous constructions include the Penrose tiling developed by Roger Penrose and elaborated by N. G. de Bruijn, the hierarchical Robinson tiling by Raphael Robinson, and Berger's original sets discovered by Robert Berger. Other significant examples are the Ammann–Beenker tiling associated with Robert Ammann and Frans Beenker, the Socolar–Taylor tile from Joshua Socolar and Joan Taylor, and tilings connected to quasicrystals studied by researchers at Ammann Laboratories and in diffraction experiments led by Dan Shechtman at institutions like the National Institute of Standards and Technology. Variants and extensions include constructions by Marcel Berger, Chaim Goodman-Strauss, Heinz-Otto Peitgen, Michael Baake, and Uwe Grimm, as well as novel examples by teams at University of Illinois, the University of Vienna, and the Australian National University.

Mathematical properties and theory

Theoretical foundations draw on concepts developed by David Hilbert, Emmy Noether, Hermann Weyl, and later formalized using methods from Topology and Ergodic theory as advanced by scholars at Princeton University and the University of California, Los Angeles. Spectral properties and diffraction patterns relate to studies by Jean Bellissard, Marcelo Baake, and Peter Kramer, while cohomological and K-theory approaches have been pursued by researchers associated with the Max Planck Institute for Mathematics and the Centre National de la Recherche Scientifique. Growth rates, local complexity, and substitution dynamics are topics in the work of Michael Baake, F. Gähler, Thomas Ward, and Andrew Clark, with computational complexity perspectives influenced by Richard Karp and Leslie Valiant. Connections to tiling spaces and C*-algebras appear in publications linked to the American Mathematical Society and the European Mathematical Society.

Methods of construction

Construction techniques include substitution systems developed by N. G. de Bruijn and M. Gardner-inspired explorations, matching-rule enforcement used by Roger Penrose and Raphael Robinson, cut-and-project methods related to the work of Yves Meyer and Nikolaas Kuiper, and algorithmic generation tied to automata theory as advanced by Emil Post and Alan Turing. Combinatorial and geometric methods were refined by Branko Grünbaum, G. C. Shephard, and Chaim Goodman-Strauss, while computational search approaches have been deployed by teams at Google Research, Microsoft Research, and university labs including ETH Zurich and Carnegie Mellon University.

Applications and related fields

Nonperiodic tilings inform the understanding of physical quasicrystals discovered by Dan Shechtman and examined at facilities like the Lawrence Berkeley National Laboratory and the Los Alamos National Laboratory, and influence materials research at the Max Planck Society and industrial work at IBM Research. They inspire artistic practice in exhibits at the Museum of Modern Art, Tate Modern, and design projects by architects affiliated with Zaha Hadid Architects, Frank Gehry, and Norman Foster-linked studios. Interdisciplinary links extend to theoretical computer science communities around ACM SIGACT, to condensed matter groups at Harvard University and Massachusetts Institute of Technology, and to crystallography networks including the International Union of Crystallography.

Open problems and research directions

Active questions include the search for a single connected "einstein" tile explored by researchers at University of Cambridge, University of Minnesota, and independent investigators publishing through arXiv, the classification of aperiodic prototile sets pursued by teams at the University of Vienna and University of Washington, and algorithmic decidability inquiries tied to complexity theory advanced by Stephen Cook and Richard Karp. Further avenues involve links between tiling dynamics and number theory studied by Enrico Bombieri and Peter Sarnak, the role of nonperiodic order in topological phases of matter pursued at Caltech and the Perimeter Institute, and computational discovery efforts supported by initiatives from the Simons Foundation and the European Research Council.

Category:Tessellations