##
**Tilings and patterns.**
*(English)*
Zbl 0601.05001

New York: W. H. Freeman and Company. IX, 700 p. £54.95 (1987).

Tilings and patterns have attracted artists and scientists for thousands of years, leading to a vast literature on this subject. However, most of these publications consist of little more than collections of examples, and often they are only concerned with the ’art’ of designing tilings and patterns rather than with an attempt to a general mathematical theory.

’Tilings and Patterns’ is the first truly comprehensive, and systematic mathematical treatement of the theory of (plane) tilings and patterns. This fascinating book of 700 pages is highly recommended to everyone who is interested in the subject. It summarizes our present knowledge on tilings and patterns, including the authors’ outstanding contributions in the past 10 or 15 years (partially published before in research articles).

One of the major achievements is that the book unifies the old, and sets up new, notation and concepts for the theory of tilings and patterns, which not only provided the framework for the book itself but will also serve as the ’language’ for future research on the subject. In fact, there is no doubt that this important book will be \(the\) definite reference on tilings and patterns for a long time. It is one of the most fascinating aspects of the book that, in spite of the rigor of the mathematical discussion, it still keeps the artistic flavour and the aesthetic and intellectual enjoyment of the subject. Among other things the huge variety of figures and diagrams is responsible for this.

The book falls into two parts. As pointed out by the authors, the first part (Chapters 1 to 7) can be used as the text book for a geometry course at the undergraduate level. Chapter 1 is introductory and sets up basic notions. Chapter 2 deals with tilings of the plane, in which the tiles are regular polygons or star polygons. General aspects in the theory of tilings are presented in Chapters 3 and 4. Chapter 5 is concerned with plane patterns and their classification. In Chapter 6 the authors give a detailed description of the classification of tilings with certain transitivity properties of their symmetry group. Chapter 7 discusses the classification problems of Chapters 5 and 6 from the more general point of view of classifying certain geometric objects with respect to symmetries (homeomeric classification).

The second part of the book (Chapters 8 to 12) presents detailed surveys of various aspects of the theory of tilings and patterns. Chapter 8 deals with colored patterns and tilings and groups of color symmetry. Plane tilings by polygons are the topic of Chapter 9; in particular, a survey of the classification problem for prototiles of monohedral tilings is given. Chapter 10 of the book is the first detailed account of the intriguing topic of aperiodic tilings available in the literature. Chapter 11 discusses a special class of tiles called Wang tiles. The final Chapter 12 contains several results on tilings with unusual kinds of tiles.

To facilitate the use of the book as a textbook the authors have included many exercises, ranging from standard exercises to unsolved research problems. Each chapter ends with detailed additional notes and references on the special topic under discussion; this is particularly helpful for the reader interested in research. Particularly impressive is also the extensive list of references, which covers almost 50 pages. This is in accordance with the detailed description of historical developments, which the authors have given for all topics of the book.

I hope that this review conveys my impression that ’Tilings and Patterns’ is an excellent book on one of the oldest mathematical disciplines. Most certainly this book will be the ’bible’ for this kind of geometry.

’Tilings and Patterns’ is the first truly comprehensive, and systematic mathematical treatement of the theory of (plane) tilings and patterns. This fascinating book of 700 pages is highly recommended to everyone who is interested in the subject. It summarizes our present knowledge on tilings and patterns, including the authors’ outstanding contributions in the past 10 or 15 years (partially published before in research articles).

One of the major achievements is that the book unifies the old, and sets up new, notation and concepts for the theory of tilings and patterns, which not only provided the framework for the book itself but will also serve as the ’language’ for future research on the subject. In fact, there is no doubt that this important book will be \(the\) definite reference on tilings and patterns for a long time. It is one of the most fascinating aspects of the book that, in spite of the rigor of the mathematical discussion, it still keeps the artistic flavour and the aesthetic and intellectual enjoyment of the subject. Among other things the huge variety of figures and diagrams is responsible for this.

The book falls into two parts. As pointed out by the authors, the first part (Chapters 1 to 7) can be used as the text book for a geometry course at the undergraduate level. Chapter 1 is introductory and sets up basic notions. Chapter 2 deals with tilings of the plane, in which the tiles are regular polygons or star polygons. General aspects in the theory of tilings are presented in Chapters 3 and 4. Chapter 5 is concerned with plane patterns and their classification. In Chapter 6 the authors give a detailed description of the classification of tilings with certain transitivity properties of their symmetry group. Chapter 7 discusses the classification problems of Chapters 5 and 6 from the more general point of view of classifying certain geometric objects with respect to symmetries (homeomeric classification).

The second part of the book (Chapters 8 to 12) presents detailed surveys of various aspects of the theory of tilings and patterns. Chapter 8 deals with colored patterns and tilings and groups of color symmetry. Plane tilings by polygons are the topic of Chapter 9; in particular, a survey of the classification problem for prototiles of monohedral tilings is given. Chapter 10 of the book is the first detailed account of the intriguing topic of aperiodic tilings available in the literature. Chapter 11 discusses a special class of tiles called Wang tiles. The final Chapter 12 contains several results on tilings with unusual kinds of tiles.

To facilitate the use of the book as a textbook the authors have included many exercises, ranging from standard exercises to unsolved research problems. Each chapter ends with detailed additional notes and references on the special topic under discussion; this is particularly helpful for the reader interested in research. Particularly impressive is also the extensive list of references, which covers almost 50 pages. This is in accordance with the detailed description of historical developments, which the authors have given for all topics of the book.

I hope that this review conveys my impression that ’Tilings and Patterns’ is an excellent book on one of the oldest mathematical disciplines. Most certainly this book will be the ’bible’ for this kind of geometry.

Reviewer: E.Schulte

### MSC:

05-02 | Research exposition (monographs, survey articles) pertaining to combinatorics |

05B45 | Combinatorial aspects of tessellation and tiling problems |

52C17 | Packing and covering in \(n\) dimensions (aspects of discrete geometry) |

52-02 | Research exposition (monographs, survey articles) pertaining to convex and discrete geometry |

### Keywords:

colored tilings; Tilings and patterns; plane patterns; classification; symmetry group; surveys; colored patterns; Plane tilings by polygons; prototiles; monohedral tilings; aperiodic tilings; Wang tiles; historical developments### Online Encyclopedia of Integer Sequences:

Number of poly-IH73-tiles (holes allowed) with n cells.Number of n-uniform tilings having n different arrangements of polygons about their vertices.

Number of polyominoes with n cells that tile the plane by translation.

Number of polyominoes with n cells that tile the plane by translation but not by 180-degree rotation (Conway criterion).

Number of polyominoes with n cells that tile the plane both by translation and by 180-degree rotation (Conway criterion).

Number of polyominoes with n cells that tile the plane by 180-degree rotation (Conway criterion) but not by translation.

Number of polyominoes with n cells that tile the plane by 180-degree rotation (Conway criterion).

Number of polyominoes with n cells that tile the plane by translation or by 180-degree rotation (Conway criterion).

Number of polyominoes with n cells that tile the plane isohedrally but not by translation or by 180-degree rotation (Conway criterion).

Number of polyominoes with n cells that tile the plane isohedrally.

Number of anisohedral polyominoes with n cells.

Number of polyhexes with n cells that tile the plane by translation.

Number of polyhexes with n cells that tile the plane by translation but not by 180-degree rotation (Conway criterion).

Number of polyhexes with n cells that tile the plane both by translation and by 180-degree rotation (Conway criterion).

Number of polyhexes with n cells that tile the plane by 180-degree rotation (Conway criterion) but not by translation.

Number of polyhexes with n cells that tile the plane by 180-degree rotation (Conway criterion).

Number of polyhexes with n cells that tile the plane by translation or by 180-degree rotation (Conway criterion).

Number of polyhexes with n cells that tile the plane isohedrally but not by translation or by 180-degree rotation (Conway criterion).

Number of polyhexes with n cells that tile the plane isohedrally.

Number of anisohedral polyhexes with n cells.

Number of polyiamonds with 2n cells that tile the plane by translation.

Number of polyiamonds with 2n cells that tile the plane by translation but not by 180-degree rotation (Conway criterion).

Number of polyiamonds with 2n cells that tile the plane both by translation and by 180-degree rotation (Conway criterion).

Number of polyiamonds with n cells that tile the plane by 180-degree rotation (Conway criterion) but not by translation.

Number of polyiamonds with n cells that tile the plane by 180-degree rotation (Conway criterion).

Number of polyiamonds with n cells that tile the plane by translation or by 180-degree rotation (Conway criterion).

Number of polyiamonds with n cells that tile the plane isohedrally but not by translation or by 180-degree rotation (Conway criterion).

Number of polyiamonds with n cells that tile the plane isohedrally.

Number of anisohedral polyiamonds with n cells.

Consider the tiling of the plane with squares of two different sizes as in Fig. 2.4.2(g) of Grünbaum and Shephard, p. 74. a(n) is the number of connected figures that can be formed on this tiling, from n big squares and n small squares.

Consider the tiling of the plane with squares of two different sizes as in Fig. 2.4.2(g) of Grünbaum and Shephard, p. 74. a(n) is the number of connected figures that can be formed on this tiling, from n tiles, each composed of a big square and an adjacent little square.

Consider the tiling of the plane with squares of two different sizes as in Fig. 2.4.2(g) of Grünbaum and Shephard, p. 74. a(n) is the number of connected figures that can be formed on this tiling, from any n squares.

Number of one-sided chessboard polyominoes with n cells (similar to but different from A001071).

Number of poly-IH64-tiles (holes allowed) with n cells.

Number of strip poly-IH64-tiles (holes allowed) with n cells.

Number of 2-sided n-polycairos.

Number of poly-IH68-tiles (holes allowed) with n cells.

Number of strip poly-IH68-tiles (holes allowed) with n cells.

Number of free poly-[3^3.4^2]-tiles (polyhouses) (holes allowed) with n cells.

Number of one-sided poly-[3^3.4^2]-tiles (polyhouses) (holes allowed) with n cells.

Number of fixed poly-[3^3.4^2]-tiles (polyhouses) (holes allowed) with n cells.

Number of free poly-[3^4.6]-tiles (holes allowed) with n cells.

Number of fixed poly-[3^4.6]-tiles (holes allowed) with n cells.

Number of free poly-[3.6.3.6]-tiles (holes allowed) with n cells (division into rhombi is significant).

Number of one-sided poly-[3.6.3.6]-tiles (holes allowed) with n cells (division into rhombi is significant).

Number of free tetrakis polyaboloes (poly-[4.8^2]-tiles) with n cells, allowing holes, where division into tetrakis cells (triangular quarters of square grid cells) is significant.

Number of one-sided poly-[4.8^2]-tiles (holes allowed) with n cells (division into triangles is significant).

Number of fixed poly-[4.8^2]-tiles (holes allowed) with n cells (division into triangles is significant).

Number of free poly-IH10-tiles (holes allowed) with n cells.

Number of free poly-IH8-tiles (holes allowed) with n cells.

Number of free poly-IH18-tiles (holes allowed) with n cells.

Number of free poly-IH19-tiles (holes allowed) with n cells.

Number of free poly-IH12-tiles (holes allowed) with n cells.

Number of free poly-IH14-tiles (holes allowed) with n cells.

Number of free poly-IH4-tiles (holes allowed) with n cells.

Number of free poly-IH13-tiles (holes allowed) with n cells.

Number of free poly-IH5-tiles (holes allowed) with n cells

Number of free poly-IH22-tiles (holes allowed) with n cells.

Number of free poly-IH9-tiles (holes allowed) with n cells.

Coordination sequence for 3.3.4.3.4 Archimedean tiling.

Coordination sequence for planar net 3.3.3.3.6 (also called the fsz net).

Coordination sequence of point of type 3.3.4.3.4 in 4-uniform tiling {3.3.4.3.4; 3.3.4.12; 3.3.12.4; 3.4.3.12}.

Coordination sequence of point of type 3.3.12.4 in 4-uniform tiling {3.3.4.3.4; 3.3.4.12; 3.3.12.4; 3.4.3.12}.

Coordination sequence of point of type 3.4.3.12 in 4-uniform tiling {3.3.4.3.4; 3.3.4.12; 3.3.12.4; 3.4.3.12}.

Coordination sequence of point of type 3.3.4.12 in 4-uniform tiling {3.3.4.3.4; 3.3.4.12; 3.3.12.4; 3.4.3.12}.

Coordination sequence of 2-uniform tiling {3.4.6.4, 4.6.12} with respect to a point of type 4.6.12.

Coordination sequence of 2-uniform tiling {3.4.6.4, 4.6.12} with respect to a point of type 3.4.6.4.

Number of n-isohedral edge-to-edge colorings of regular polygons.

Coordination sequence for the Cairo or dual-3.3.4.3.4 tiling with respect to a trivalent point.

G.f.: (x^4+3*x^3+6*x^2+3*x+1)/((1-x)*(1-x^3)).

Coordination sequence for node of type 3.12.12 in ”cph” 2-D tiling (or net).

Coordination sequence for a tetravalent node of type 3.4.3.12 in ”cph” 2-D tiling (or net).

Expansion of (x^4+3*x^3+x^2+3*x+1) / ((x^2+1)*(x-1)^2).

Expansion of (x^2+x+1)^2 / ((x^2+1)*(x-1)^2).

Expansion of (1 + 4*x + 4*x^2 + 4*x^3 + x^4)/((1 - x)*(1 - x^3)).

Coordination sequence for node of type V1 in ”krq” 2-D tiling (or net).

Coordination sequence for node of type V2 in ”krq” 2-D tiling (or net).

Coordination sequence for node of type V1 in ”krr” 2-D tiling (or net).

Coordination sequence for node of type V2 in ”krr” 2-D tiling (or net).

Coordination sequence for node of type V1 in ”krs” 2-D tiling (or net).

Coordination sequence for node of type V2 in ”krs” 2-D tiling (or net).

Coordination sequence for node of type V1 in ”krn” 2-D tiling (or net).

Coordination sequence for node of type V2 in ”krn” 2-D tiling (or net).

Coordination sequence for node of type V1 in ”krg” 2-D tiling (or net).

Coordination sequence for node of type V2 in ”krg” 2-D tiling (or net).

Coordination sequence for node of type V1 in ”krh” 2-D tiling (or net).

Coordination sequence for node of type V2 in ”krh” 2-D tiling (or net).

Coordination sequence for node of type V1 in ”krf” 2-D tiling (or net).

Coordination sequence for node of type V2 in ”krf” 2-D tiling (or net).

Expansion of (1 + 5*x + 4*x^2 + 5*x^3 + x^4)/((1 - x)*(1 - x^3)).

Coordination sequence for node of type V2 in ”krj” 2-D tiling (or net).

Coordination sequence for node of type V1 in ”krc” 2-D tiling (or net).

Coordination sequence for node of type V2 in ”krc” 2-D tiling (or net).

Coordination sequence for node of type V1 in ”usm” 2-D tiling (or net).

Coordination sequence for node of type V2 in ”usm” 2-D tiling (or net).

Coordination sequence for node of type V1 in ”kre” 2-D tiling (or net).

Coordination sequence for node of type V2 in ”kre” 2-D tiling (or net).

Coordination sequence for node of type V1 in ”krb” 2-D tiling (or net).

Coordination sequence for node of type V2 in ”krb” 2-D tiling (or net).

Coordination sequence for node of type V1 in ”kra” 2-D tiling (or net).

Coordination sequence for node of type V2 in ”kra” 2-D tiling (or net).

Coordination sequence with respect to the central vertex of a dodecagon-based tiling of the plane by copies of a certain Goldberg quadrilateral tile.