Tilings and patterns

Tilings and patterns is a book by mathematicians Branko Grünbaum and Geoffrey Colin Shephard published in 1987 by W.H. Freeman. The book was 10 years in development, and upon publication it was widely reviewed and highly acclaimed.

Structure and topics

The book is concerned with tilings—a partition of the plane into regions —and patterns—repetitions of a motif in the plane in a regular manner.
The book is divided into two parts. The first seven chapters define concepts and terminology, establish the general theory of tilings, survey tilings by regular polygons, review the theory of patterns, and discuss tilings in which all the tiles, or all the edges, or all the vertices, play the same role.
The last five chapters survey a variety of advanced topics in tiling theory: colored patterns and tilings, polygonal tilings, aperiodic tilings, Wang tiles, and tilings with unusual kinds of tiles.
Each chapter open with an introduction to the topic, this is followed by the detailed material of the chapter, much previously unpublished, which is always profusely illustrated, and normally includes examples and proofs. Chapters close with exercises, and a section of notes and references which detail the historical development of the topic. These notes sections are interesting and entertaining, as they discuss the efforts of the previous workers in the field and detail the good approaches to the topic. The notes also identify unsolved problems, point out areas of potential application, and provide connections to other disciplines in mathematics, science, and the arts.
The book has 700 pages, including a 40-page, 800-entry bibliography, and an index. The book is used as a source on numerous Wikipedia pages.

Audience

In their preface the authors state "We have written this book with three main groups of readers in mind—students, professional mathematicians and non-mathematicians whose interests include patterns and shapes.
Other reviewers commented as follows:

"The most striking feature of the book is its extensive collection of figures, including hundreds of examples of tilings and patterns. The sheer abundance is perhaps one reason why artists and designers have been drawn to it over the years."
"Their idea was that the book should be accessible to any reader who is attracted to geometry."

Reception

Contemporary reviews of the book were overwhelming positive. The book was reviewed by 15 journals in the fields of crystallography, mathematics, and the sciences. Quotations from major reviews:

Influence

The book was praised in later journal articles by multiple authors:
The book was also praised in later books by other authors:

Editions

The hardback original Tilings and patterns was published in 1987.
Tilings and patterns - an introduction, a paperback reprint of the first seven chapters of the 1987 original, was published in 1989.
In 2016 a second edition of the full text was published by Dover in paperback, with a new preface and an appendix describing progress in the subject since the first edition. The reviewer at MAA Reviews commented "Dover has once again done the mathematical community a service in bringing back such a notable volume."

#	Chapter Title	Relevant articles in Wikipedia by section §
1	Basic notions	§1.1 Tiling, Euclidean plane, packing, covering, toplogical disk, §1.2 prototile, regular tiling, monohedral tiling, k-isohedral tiling, §1.3 symmetry, isometry, rotation, translation, reflection, glide reflection, group, transitivity, k-isogonal tiling, §1.4 symmetry element, isomorphism, affine transformation, frieze group, wallpaper group, §1.6 fundamental domain, space group, rod group
2	Tilings by regular polygons and star polygons	§2.1 Uniform tiling, Archimedean tiling, elongated triangular tiling, snub square tiling, truncated square tiling, truncated hexagonal tiling, trihexagonal tiling, snub trihexagonal tiling, rhombitrihexagonal tiling, §2.2 list of k-uniform tilings, demiregular tiling, 3-4-3-12 tiling, 3-4-6-12 tiling, 33344-33434 tiling, §2.3 k-isotoxal tiling, §2.4 tilings that are not edge-to-edge, squaring the square, §2.5 star polygon, regular star polygon, polygram, tilings using star polygons, Kepler's star tiling, pentagram, pentacle, §2.6 dissection tiling, §2.7 regular polygon, Laves tiling, tetrakis square tiling, rhombille tiling, §2.9 uniform coloring, list of uniform colorings, Archimedean and uniform coloring, §2.10 Johannes Kepler's Harmonices Mundi
3	Well-behaved tilings	§3.1 Well-behaved, singular point, locally finite, §3.2 normal tiling, Euler's theorem for tilings, §3.7 periodic tiling, §3.8 Heesch's problem, §3.9 Eberhard's theorem, §3.10 Karl Reinhardt
4	The topology of tilings	§4.1 Homeomorphism (topological equivalence), combinatorial equivalence, isotopy, Metamorphosis III, §4.2 duality, Pythagorean tiling
5	Patterns	§5.1 Pattern, motif, §5.2 group theory, symmetry group, subgroup, §5.3 2-D lattice, §5.4 Dirichlet tiling, §5.5 continuous group, §5.6 Islamic geometric patterns
6	Classification of tilings with transitivity properties	§6.2 Isohedral tiling, §6.3 isogonal tiling, §6.4 isotoxal tiling, list of isotoxal tilings, §6.5 striped pattern, §6.6 Evgraf Fedorov, Alexei Vasilievich Shubnikov, planigon, Boris Delone
7	Classification with respect to symmetries	§7.1 Conjugate element, §7.7 arrangement of lines, §7.8 Circle packing
8	Colored patterns and tilings	§§8.1-8.7 Dichromatic symmetry, polychromatic symmetry, perfect coloring, §8.8 Truchet tiles, M.C. Escher
9	Tilings by polygons	§9.1 Tilings by polygons, triangular tiling, quadrilteral tiling, pentagonal tiling, hexagonal tiling, parallelogon, §9.2 non-convex polygon tilings, §9.3 anisohedral tiling, §9.4 polyomino, heptomino, polyiamond, polyhex, §9.5 Voderberg tiling, §9.6 Marjorie Rice
10	Aperiodic tilings	§10.1 Similarity, §10.2 aperiodic tiling, Raphael M. Robinson, list of aperiodic sets of tiles, Ammann A1 tilings, §10.3 Penrose tiling, golden ratio, §10.4 Ammann–Beenker tiling, aperiodic set of prototiles, §10.7 Roger Penrose, Robert Ammann, John H. Conway, Alan Lindsay Mackay, Dan Shechtman, Einstein problem
11	Wang tiles	§11.1 Wang tile, §11.2 Hao Wang, §11.3 decidability, §11.4 Turing machine
12	Tilings with unusual kinds of tiles	§12.1 Cut point, §12.2 disconnected tiles, §12.3 hollow tiling, vertex figure, §12.4 Riemann surface, Harold [Scott MacDonald Coxeter\|H.S.M. Coxeter]