# Branko Grünbaum's books

Branko Grünbaum wrote four books, three of which were single authored and one was written in collaboration with G C Shephard. He received the Leroy P Steele Prize from the American Mathematical Society in 2005 for his book

For information about four papers co-authored with Geoffrey C Shephard, see 5., 6., 8., and 9. at THIS LINK.

Convex Polytopes (1967)

Arrangements and Spreads (1972)

Tilings and Patterns (1986) with G C Shephard

Tilings and Patterns, An Introduction (1989) with G C Shephard

Convex Polytopes (Second Edition) (2002)

Configurations of Points and Lines (2009)

Tilings and Patterns (Dover reprint) (2016) with G C Shephard

*Complex polytopes*(1967, Second Edition 2002). Because of various editions of the four books, we have in fact listed seven books below. We give extracts from reviews and some other information on each work.For information about four papers co-authored with Geoffrey C Shephard, see 5., 6., 8., and 9. at THIS LINK.

**Click on a link below to go to information on that book**Convex Polytopes (1967)

Arrangements and Spreads (1972)

Tilings and Patterns (1986) with G C Shephard

Tilings and Patterns, An Introduction (1989) with G C Shephard

Convex Polytopes (Second Edition) (2002)

Configurations of Points and Lines (2009)

Tilings and Patterns (Dover reprint) (2016) with G C Shephard

**1. Convex Polytopes (1967), by Branko Grünbaum.**

**1.1. From the Preface.**

Convex polytopes - as exemplified by convex polygons and some three-dimensional solids - have been with us since antiquity. However, hardly any results worth mentioning and dealing specifically with the combinatorial properties of convex polytopes were discovered prior to Euler's famous theorem concerning the number of vertices, edges, and faces of three-dimensional polytopes. Euler's relation, hailed by Klee as "the first landmark" in the theory of convex polytopes, served as the starting point of a multitude of investigations which led to the determination of its limits of validity, and helped focus attention on the notion of convexity. Additional ideas and results came from such mathematicians as Cauchy, Steiner, Sylvester, Cayley, Möbius, Kirkman, Schläfli, and Tait. Since the middle of the last century, polytopes of four or more dimensions attracted interest; crystallography, generalisations of Euler's theorem, the search for polytopes exhibiting regularity features, and many other fields provided added impetus to the investigation of convex polytopes.

About the turn of the century, however, a steep decline in the interest in convex polytopes was produced by two causes working in the same direction. Efforts at enumerating the different combinatorial types of polytopes, started by Euler and pursued with much patience and ingenuity during the second half of the XIXth century, failed to produce any significant results even in the three-dimensional case; this lead to a widespread feeling that the interesting problems concerning polytopes are hopelessly hard. Simultaneously, the ascendance of Klein's "Erlanger Program" and the spread of its normative influence tended to cast the preoccupation with the combinatorial theory of convex polytopes into a rather disreputable role - and that at a time when such "legitimate" fields as algebraic geometry and in particular topology started their spectacular development.

Due to this combination of circumstances and pressures it is probably not too surprising that only few specialised directions of research in polytopes remained active during the first half of the present century. Stretching slightly the time limits, the most prominent examples of those efforts were: Minkowski's fundamental contributions, related to his work on convexity in general, and applications to number theory in particular; Coxeter's work on regular polytopes; A D Aleksandrov's investigations in the metric theory of polytopes.

Nevertheless, as far as "main-line mathematics" is concerned, the combinatorial theory of convex polytopes was "out". Despite the appreciable number of published papers dealing with isolated (mostly extremal) problems, the whole area was relegated to the borderline between serious research and amateurish curiosity. The one notable exception in this respect among first-rank mathematicians was Ernst Steinitz, who devoted a sizable part of his life and efforts to the combinatorial theory of polytopes. Unfortunately, his beautiful results did not become as well known as they deserve, and till very recently did not stimulate additional research.

It was mainly under the influence of computational techniques (in particular, linear programming) that a renewed interest in the combinatorial theory of convex polytopes became evident slightly more than ten years ago. The phenomenon of "neighbourly polytopes" was rediscovered by Gale in 1955 (the rather involved history of this concept is related in Section 7.4). Neighbourly polytopes, and Motzkin's "upper-bound conjecture" (1957) served as focal points for many investigations (see Chapters 9 and 10). Despite many scattered results on the upper-bound conjecture and other combinatorial problems about convex polytopes, obtained by different authors in the first few years of the 1960's, the emergence of a theory proper began only with Klee's work, starting in 1962. Klee's results on the Dehn-Sommerville equations (the interesting history of this topic is given in Section 9.8) and his almost complete solution of the upper-bound conjecture were the source and basis for many of the subsequent developments.

During the last three years, research into the combinatorial structure of convex polytopes has grown at an astonishing rate. It would be premature to attempt to give here even the briefest historic outline of this period. Instead, detailed bibliographic references are supplied with each topic discussed in the book.

The present book grew out of lecture notes prepared by the author for a course on the combinatorial theory of convex polytopes given at the Hebrew University of Jerusalem in 1964/65. The main part of the final version was written while the author was lecturing on the same topic at the Michigan State University in East Lansing during 1965/66.

**1.2. Contents.**

Preface

**1 Notation and prerequisites**

1.1 Algebra

1.2 Topology

**2 Convex sets**

2.1 Definition and elementary properties

2.2 Support and separation

2.3 Convex hulls

2.4 Extreme and exposed points; faces and poonems

2.5 Unbounded convex sets

2.6 Polyhedral sets

2.7 Remarks

**3 Polytopes**

3.1 Definition and fundamental properties

3.2 Combinatorial types of polytopes; complexes

3.3 Diagrams and Schlegel diagrams

3.4 Duality of polytopes

3.5 Remarks

**4 Examples**

4.1 The $d$-simplex

4.2 Pyramids

4.3 Bipyramids

4.4 Prisms

4.5 Simplicial and simple polytopes

4.6 Cubical polytopes

4.7 Cyclic polytopes

4.8 Exercises

**5 Fundamental properties and constructions**

5.1 Representations of polytopes as sections or projections

5.2 The inductive construction of polytopes

5.3 Lower semicontinuity of the functions $f_{k}(P)$

5.4 Gale-transforms and Gale-diagrams

5.5 Existence of combinatorial types

**6 Polytopes with few vertices**

6.1 $d$-Polytopes with $d + 2$ vertices

6.2 $d$-Polytopes with $d + 3$ vertices

6.3 Gale diagrams of polytopes with few vertices

6.4 Centrally symmetric polytopes

6.5 Exercises

6.6 Remarks

**7 Neighborly polytopes**

7.1 Definition and general properties

7.2 [$\large\frac{1}{2}\normalsize d$]-Neighborly $d$-polytopes

7.3 Exercises

7.4 Remarks

**8 Euler's relation**

8.1 Euler's theorem

8.2 Proof of Euler's theorem

8.3 A generalization of Euler's relation

8.4 The Euler characteristic of complexes

8.5 Exercises

8.6 Remarks

**9 Analogues of Euler's relation**

9.1. The incidence equation

9.2 The Dehn-Sommerville equations

9.3 Quasi-simplicial polytopes

9.4 Cubical polytopes

9.5 Solutions of the Dehn-Sommerville equations

9.6. The $f$-vectors of neighborly $d$-polytopes

9.7 Exercises

9.8 Remarks

**10 Extremal problems concerning numbers of faces**

10.1 Upper bounds for $f_{i}, i ≥ 1$, in terms of $f_{0}$

10.2 Lower bounds for $f_{i}, i ≥ 1$, in terms of $f_{0}$

10.3 The sets $f(P^{3})$ and $f(P_{s}^{3})$

10.4 The set $f(P^{4})$

10.5 Exercises

**11 Properties of boundary complexes**

11.1 Skeletons of simplices contained in $B(P)$

11.2 A proof of the van Kampen-Flores theorem

11.3 $d$-Connectedness of the graphs of $d$-polytopes

11.4 Degree of total separability

11.5 $d$-Diagrams

**12 $k$-Equivalence of polytopes**

12.1 $k$-Equivalence and ambiguity

12.2 Dimensional ambiguity

12.3 Strong and weak ambiguity

**13 3-Polytopes**

13.1 Steinitz's theorem

13.2 Consequences and analogues of Steinitz's theorem

13.3 Eberhard's theorem

13.4 Additional results on 3-realizable sequences

13.5 3-Polytopes with circumspheres and circumcircles

13.6 Remarks

**14 Angle-sums relations; the Steiner point**

14.1 Gram's relation for angle-sums

14.2 Angle-sums relations for simplicial polytopes

14.3 The Steiner point of a polytope (by G C Shephard)

14.4 Remarks

**15 Addition and decomposition of polytopes (by G C Shephard)**

15.1 Vector addition

15.2 Approximation of polytopes by vector sums

15.3 Blaschke addition

15.4 Remarks

**16 Diameters of polytopes (by Victor Klee)**

16.1 Extremal diameters of $d$-polytopes

16.2 The functions $\Delta$ and $\Delta _{b}$

16.3 $W_{v}$ Paths

**17 Long paths and circuits on polytopes (by Victor Klee)**

17.1 Hamiltonian paths and circuits

17.2 Extremal path-lengths of polytopes

17.3 Heights of polytopes

17.4 Circuit codes

**18 Arrangements of hyperplanes**

18.1 $d$-Arrangements

18.2 2-Arrangements

18.3 Generalizations

**19 Concluding remarks**

19.1 Regular polytopes and related notions

19.2 $k$-Content of polytopes

19.3 Antipodality and related notions

Tables

Addendum

Bibliography

Index of Terms

Index of Symbols

**1.3. Review by: Werner Fenchel.**

*American Scientist*

**56**(4) (1968), 476A-477A.

In recent years the attention of many mathematicians has been drawn to the study of convex polytopes (the analogues of convex polygons and convex polyhedra in spaces of arbitrary dimensions), not least because of the close relations of this topic with other fields of current interest such as graph theory and linear programming. The book under review contains a very comprehensive account of old and recent investigations primarily about combinatorial properties of convex polytopes. This branch of geometry was initiated by Euler, more than 200 years ago, in the three-dimensional case, his most important contribution being the well-known relation between the numbers of vertices, edges and faces of a convex polyhedron. Even this case still presents challenging and difficult problems, the generalisations of the results to higher dimensions are far from trivial, and surprising new aspects have turned up. Many more or less related topics concerning metrical properties of polytopes are also included. In particular, the interesting relations between angle-sums may be mentioned. The book is a remarkable achievement. A wealth of material is well organised and presented in a lucid style. Problems added to the various sections contain further information mostly about recent results. There are many original contributions by the author, Klee, Perles, and Shephard. This monograph is an unrivalled source of in formation and will undoubtedly stimulate further research in this field.

**1.4. Review by: P R Baxandall.**

*The Mathematical Gazette*

**53**(385) (1969), 342-343.

Branko Grünbaum is one of the leading contributors to a subject which has developed very rapidly since 1950. One can hardly ask more of a book on advanced mathematics than that it is a readable text, can be used for reference over a wide field, provides a complete guide to the literature and serves as a stimulant to research. The author has achieved all this. The first ten of the nineteen chapters would provide an excellent grounding for anyone interested in the field. There are a large number of exercises, many of them linking material with recently published papers. A welcome feature is the remarks at the end of each chapter which put the contents in their historical setting. The book includes recent results over a wide range and many previously unpublished results due to Perles. An adequate bibliography is an essential part of a book such as this - here it fills nineteen pages. In the later chapters there are fewer exercises and more unsolved problems. The author's command of the whole field enables him to give, with the help of Klee and Shephard, a balanced account of current research activity.

There are particular difficulties in reading combinatorial mathematics. The necessarily elaborate notation with a multitude of suffixes obscures the often simple ideas behind the proofs. In addition it is difficult to identify a uniform method of attack, each result seems to require a different ingenious trick. Nevertheless diligence is soon rewarded with some remarkable results. The existence, for example, of a polytope (conjectured by Klee and discovered by Perles) which has no combinatorial copy with vertices at rational points.

The prerequisites for reading most of this book, apart from time, patience, maturity and a taste for combinatorial mathematics, are hardly more than a little linear algebra and point set topology.

...

Professor G C Shephard (of the University of East Anglia) has contributed a chapter on the expression of a polytope as the sum of other polytopes. Victor Klee has written two chapters on the graph formed by the edges and vertices of a polytope. The book finishes with an account of the geometry of hyperplanes in projective space (a topic which often seems close in the previous work) and a final collection of remarks and open questions. It is a great pleasure to acknowledge a book which in itself and in its effect upon others makes a considerable contribution to mathematics.

**1.5. Review by: G Thomas Sallee.**

*Mathematical Reviews*MR0226496

**(37 #2085)**.

The author has produced an excellent and much-needed book about this recently resurgent area of geometry. Stimulated by the development of linear programming, interest in the field of convex polytopes has been renewed and a great many results have been discovered or re-discovered in recent years. However, prior to this book, the theory was scattered about in various research papers or out-of-print volumes. The author has done an extraordinarily thorough job of collecting these numerous results and organising them into readable form. The bibliography is as complete and up-to-date as publication pressures would permit (there is even a short list of results at the end added during the proof stage).

...

Near the end of each chapter there are problems for the student which often contain other important results. In addition, most chapters are concluded by a section of remarks which sketches the history of the topic and gives additional references. The writing style and notation make for good readability as does the high quality of printing.

...

It must also be considered an indispensable reference for the researcher or anyone else interested in the modern developments of geometry.

**2. Arrangements and Spreads (1972), by Branko Grünbaum.**

**2.1. From the Introduction.**

The present survey deals mostly with rather elementary mathematics, -- so elementary in fact that most of its results and problems are (or at least should be) understandable to undergraduates. It was written out of the conviction that many neglected aspects of elementary geometry deserve a wider dissemination, - because of their inherent beauty and interest, and for the inspiration and understanding they can impart to our students and to ourselves.

During the last decade or two, it has become an article of faith with many of us that research in mathematics and its teaching are worthwhile only if they deal with very general and abstract topics. The elements of algebra, set theory, logic, topology, measure theory, etc., -- that is what advanced undergraduates and starting graduate students are being taught in good schools, it would be foolish to say that this should be changed. What is unfortunate, and what cripples most of the students for the rest of their professional lives, is that other topics and points of view are only rarely presented. The students are made to understand that research in, say, finite groups is "important" - no matter how esoteric the problem investigated, - while topics in the Euclidean plane, for example, are certainly old-fashioned since the Euclidean plane is a very special structure. Many aspects of geometry fare particularly badly from this attitude, they are even denied the right to exist unless they fit nice algebraic patterns. In the most ridiculous and extreme aspects of such tendencies geometry is equated with a subdiscipline of linear algebra.

As a counterthrust to that trend I would like to recall just a few of the many historical instances in which the geometric patterns of thought or points of view led not only to another theorem or two but were crucial in gaining new and revolutionary insights:

(i) The discovery of irrational numbers;

(ii) The invention of calculus;

(iii) The development of the axiomatic method;

(iv) The emergence of algebraic topology, and of functional analysis.

In most of these examples (and in many other cases) it was the spirit of geometric thinking more than any specific fact that led to the breakthrough. I believe that the geometric point of view will provide in the future similarly important impulses and inspirations, and lead to still unexplored disciplines and insights. Therefore, I am convinced that geometric thinking ought to be cultivated among students much more than is the case at the present.

In the following pages I have surveyed a number of topics considered to be not-quite-legitimate mathematics by some of our colleagues - though the bibliography shows that others, of no lesser stature, were willing to think about them. One of my aims was to point out the wealth of open problems in many areas in which some partial answers are already known. The main thrust, however, was to show how this elementary and intuitively appealing material exhibits a far-reaching analogy to the subtle differences encountered in topology between the piecewise linear and the topological situations. In our case, the main dichotomy is between the rectilinear and the curvilinear arrangements, though in each type additional subclasses are worth exploring.

An additional objective was the juxtaposition of the discrete and the continuous variants. This is again analogous to the situation in topology, but in the present context it seems so far to have escaped attention.

For simplicity the whole exposition is restricted to the two-dimensional case. There is no difficulty to extend many of the definitions and results to higher dimensions. but only in a few places have I found it worthwhile to mention such material. (For a recent survey of the results known about arrangements of hyperplanes in higher-dimensional spaces see the first part of Grünbaum [1971].) Much of the beauty of the subject derives, in my opinion, from the fact that already the two-dimensional case is interesting and non-trivial.

The exposition is organised as follows:

Arrangements of lines, that is the structures determined by finite families of straight lines in the real projective plane, are examined in detail in Chapter 2. Isolated problems in this area have been investigated for almost 150 years, but the only previous attempt at a systematic exposition seems to be a short section in Grünbaum [1967]. Possibly part of the reason for this neglect is to be found in the fact that quite a few of the problems and results extend naturally to various more general settings-some to the theory of geometric configurations, to finite projective planes and to different kinds of combinatorial designs, some to convex polyhedra in 3·space or to planar graphs, and so on. However, it is my firm conviction that there are great advantages to be derived from the consideration of all those problems in a unified setting, although that implies some loss of generality for each individual result. (This situation seems to be rather analogous to that in the theory of convex sets in Euclidean spaces: There are very few results about such sets that could not be extended to more general sets or spaces, the generalisation varying from result to result. But the mutual relationship among the various properties is best understood when they are considered in their common core of validity.) Hence the present exposition is limited almost exclusively to the real projective plane, with only a few hints at other possible settings. For all assertions that are not self-evident or verifiable by exhaustive checking we provide either proofs or references to the literature.

**2.2. Review by: E Jucovic.**

*zbMATH*,

**Zbl 0249.50011**.

The author had again done a remarkable job. He has collected in survey from results of about 150 years work dealing with discrete and continuous families of lines and curves. An arrangement of lines means simply a finite family of lines in the real projective plane. An arrangement of pseudolines or curves means a finite family of simply closed curves such that every two curves have precisely one or two points, respectively, in common in which they intersect each other. Isomorphism types, relations between the number of vertices, lines and cells, multiplicity of vertices, kinds of cells of arrangements, the problem of stretchability of arrangements of pseudolines and many other results and questions are mentioned.

**2.3. Review by: H S M Coxeter.**

*Mathematical Reviews*MR0307027

**(46 #6148)**.

This unusual monograph, on what Hilbert would have called "anschauliche Geometrie", is packed with interesting theorems and conjectures, enough to occupy a whole generation of future geometers. For the sake of simplicity, the author limits his discussion almost entirely to the real projective plane.

...

Some new features appear when lines are replaced by pseudolines, which are not necessarily straight but still have the property that any two of them have just one common point and cross each other there. For instance, in the Pappus configuration $9_{3}$, one of the nine lines can be replaced by a curve that bends away to avoid concurring with two others. Two of the most remarkable arrangements each consist of 19 pseudolines forming 108 cells, all of which are triangles. In Chapter 4 the author considers spreads, which are continuous families of curves that behave like pseudolines. He motivates his final conjecture: that every arrangement of pseudolines can be extended to a topological projective plane. These and many other ideas are set forth in a commendably clear style, with references to books and papers by more than two hundred authors.

**3. Tilings and Patterns (1986), by Branko Grünbaum and G C Shephard.**

**3.1. From the publisher.**

"Remarkable ... It will surely remain the unique reference in this area for many years to come," Roger Penrose, Nature; "... an outstanding achievement in mathematical education," Bulletin of The London Mathematical Society; "I am enormously impressed ... Will be the definitive reference on tiling theory for many decades. Not only does the book bring together older results that have not been brought together before, but it contains a wealth of new material ... I know of no comparable book," Martin Gardner.

**3.2. Review by: Joseph Malkevitch.**

*Science, New Series*

**236**(4804) (1987), 996-997.

Throughout history people have filled floors, stained-glass windows, and fabrics with shapes that occupied the plane without holes or overlaps. These shapes, or tiles, often endow plane surfaces with patterns of remarkable symmetry and beauty. Although tilings of the plane have been found in varied contexts and cultures, the systematic study of their types and properties is surprisingly new. Except for a modest beginning in the 17th century by Johannes Kepler, there were few studies of tilings until the 20th century, and not until the research of this book's authors did the study of tilings become a full-fledged subbranch of geometry. To develop a theory of tilings, one must adhere to the rules concerning which shapes are allowed for tiles and the rules about how shapes can be placed next to one another. For example, if one starts with a supply of identical rectangular tiles, one may place the tiles so that the edges of the rectangles match, producing the familiar tiling of floorings, where four tiles meet at a point. But other tilings by rectangles can occur if we allow the tiles to adjoin in additional ways. Infinitely many types of tiling are possible when the tiles are not laid down edge to edge. Patterns, the other word sharing the book's title with tilings, involve symmetry considerations that arise when motifs of various kinds are systematically located in the plane. Although beautiful patterns have been created and examined by mathematicians in addition to craftsmen and artists for thousands of years, a coherent theory of patterns was not developed until that presented in this book.

What Grünbaum and Shephard have done, in a dazzling display of scholarship, erudition, and research, is collect in one volume a compendium of the accumulated knowledge about tilings and patterns developed by a wide range of individuals including artisans and craftsmen, mathematicians, crystallographers, and physicists. In doing so they were forced to take a fresh look at what was known, what was thought to be known, and what had yet to be investigated, and to provide a framework in which all of this information could be succinctly put down. The project, which was started about 10 years ago and has only now been brought to (partial) completion, is well worth the wait. To present this material Grünbaum and Shephard needed to develop a standardised terminology. They have chosen terminology that is clear yet flexible and thus well suited to the rich range of phenomena encountered. Since this book begins with the simplest of concepts, it can be started and read with enjoyment by a high-school student. However, the reader needs staying power. As the book progresses, relatively elementary concepts and definitions are developed, but in the building-block style typical of a mathematical theory. Furthermore, the authors have felt free to use elementary ideas from topology, group theory, and number theory, defining the necessary terminology as they go along. Thus, although the treatment is elementary in the sense that all concepts and ideas that are not well known by persons with modest mathematics backgrounds are fully, clearly, and carefully explained, one cannot hope to understand the statements in the middle of the book without retaining earlier concepts. Furthermore, although the ideas presented are related to a variety of applicable situations, the material is not developed with a view to demonstrating its "usefulness." The discussion, for example, of aperiodic tiles - sets of prototiles that admit tilings of the plane but for which there is no possible periodic tiling - is pursued for its own interest. Yet this beautiful and exciting new field of aperiodic tilings is setting crystallography and physics topsy turvy by providing a framework for the controversial and exciting developments in the area of what have come to be called quasicrystals. These are a form of matter that appears to violate the so-called crystallographic restriction, which states that the only types of rotational symmetry that can be present in physical lattice systems are two-fold, three-fold, four-fold, and six-fold. In particular, the high level of icosahedral symmetry, seemingly confirmable for these new materials from x-ray crystallographic techniques, cannot be explained by classical methods.

**3.3. Review by: H S M Coxeter.**

*Mathematical Reviews*MR0857454

**(88k:52018)**.

Tilings and patterns have been made and enjoyed for thousands of years. Their mathematical treatment was begun by J Kepler but was then forgotten until the nineteenth-century development of crystallography. In this unique book, with its abundant illustrations, the authors explain exactly what one means by "tiling" and "pattern", restricting the treatment to two dimensions. There are many surprises ...

...

The most exciting developments are reserved for Chapters 10 and 11, on "Aperiodic tilings". Before 1966, nobody could imagine the existence of a set of prototiles which would admit infinitely many tilings of the plane although no such tiling is periodic. Obviously, even such a simple prototile as a domino admits a nonperiodic tiling; but the exciting new idea, embodied in the term "aperiodic", is a set of $n$ prototiles which cannot possibly be arranged in a periodic fashion. R Berger discovered the first aperiodic tiling, with $n$ = 20426. Berger himself soon reduced this fantastic number to 104, D E Knuth to 92, H Läuchli to 40, R M Robinson to 35, R Penrose to 34, Robinson (again) to 32, and later to 24, R Ammann to 16, and later to 6, then Penrose (again) to 5, and ultimately to 2! Many of the amazing ramifications of this theory, including some by J H Conway, are published here for the first time. Chapter 11 deals with "Wang tiles": square tiles having coloured edges which must match with their neighbours, only translations being allowed. These aperiodic tilings are relevant to questions of logic and computing, because it is possible to find sets of 16 Wang tiles which mimic the behaviour of any Turing machine.

**3.4. Review by: Marjorie Senechal.**

*American Scientist*

**75**(5) (1987), 521-522.

Repeating patterns, in the plane and in space, are found in the decorative art of every culture and in the fine and gross structures of many natural and synthetic materials. On close inspection, these de signs turn out to be variations on a set of themes. The search for these themes has inspired a wide variety of artists and scientists to develop methods for classifying patterns, and these classification schemes in turn have suggested the possibilities of new designs. There is perhaps no better example of the symbiotic relation between mathematics and art.

Despite its intuitive accessibility and visual appeal, the mathematical theory of repeating patterns is decidedly nontrivial. "The art of ornament," wrote Hermann Weyl (in Symmetry), "contains in implicit form the oldest piece of higher mathematics known to us." Johann Kepler seems to have been the first to make the subject explicit; in

*Harmonices Mundi*(1619) he investigated the ways in which regular polygons could be fitted together to tile the plane, showing that there are 11 types of patterns in which the arrangement of polygons at every vertex is uniform throughout. Since then, designers, artists, mathematicians, crystallographers, and other scientists have developed an incoherent body of literature, as their work led them to ask questions such as What shapes can be fitted together to tile a plane or space? What are the symmetries of the resulting pattern? Which patterns should be considered equivalent, and which ones distinct? How can physically significant characteristics of motifs be indicated in a geometric pattern? Can I be sure that a finite pattern can be extended forever in all directions? Will the infinite pattern necessarily be a repeating pattern? Most recently, attention has focused on nonperiodic patterns, such as the famous Penrose tilings of the plane by "kites" and "darts," which have properties analogous to those found recently in the aluminium-magnesium alloys popularly known as "quasicrystals." Thus the ancient problem of designing and classifying patterns is very much alive today.

There has long been a need for a definitive monograph in which the subject of patterns and, in particular, tilings is critically surveyed and a research program is mapped out. When Grünbaum and Shephard decided to write such a book, they found that although the literature was abundant, it was scattered in books and journals in many languages, at widely varying levels of sophistication. Worse, many of the published results were poorly formulated, incomplete, or incorrect. Some fundamental questions? - for example, What is a pattern? - seem never to have been asked. Thus in the course of writing the book, the authors found it necessary to reconstruct the subject as a whole. (Their bibliography lists many of the research articles that they wrote along the way.) Although the authors restrict themselves to tilings and patterns in the plane, many of their results lay the foundation for the further development of the theory of tilings and patterns in higher dimensional spaces.

I have had the pleasure of watching their book evolve over a decade, as basic concepts were defined and refined, the boundaries of the subject were sketched out, and critical details were filled in. The result is a definitive encyclopaedic work with comprehensive, critical historical notes and a superb bibliography. Yet it is written in a clear informal style and requires no more than a good background, in high school mathematics. Even the casual reader will be rewarded with a grasp of the development of the theory from Kepler through Penrose (and beyond) and an understanding of the role of patterns and tilings in diverse fields. But who could remain a casual reader of the book, which is filled with challenging problems, including many that are still unsolved?

Every once in a while a book comes along that is required reading for the scientifically literate.

**3.5. Review by: Solomon W Golomb.**

*The American Mathematical Monthly*

**95**(1) (1988), 63-64.

This is a marvellous book. I am tempted to say "I wish I had written it," but the comprehensiveness and attention to detail exceed anything I would have ever contemplated, let alone executed. The book is profusely illustrated, with an average of two figures per page, and each picture is meticulously drawn. It must have been a difficult decision for the publisher to confine all illustrations to black-and-white, thereby foregoing the art gift book market, though the production cost would no-doubt have been exorbitant. However, for the mathematician, all the necessary distinctions are amply illustrated with shadings, letters, and numbers, and an excellent aesthetic effect is achieved.

One major achievement is the organisation of the vast literature on planar tilings into a systematic structure, arranged into twelve chapters.

...

As extensive as this book is, it obviously cannot present every result ever obtained about tiling. However, the authors have excellent taste in selecting their material, and make an admirable effort to cite references to the topics which they do not cover. In a first edition of a book of this length there are inevitably some typographical errors, but they are surprisingly few and far between. I recommend this book enthusiastically to anyone interested in problems of tiling the plane.

**3.6. Review by: H C Williams.**

*The Mathematical Gazette*

**71**(458) (1987), 347-348.

Do not be put off by the price before reading further. The book is good value for money for an institution particularly as it includes about two and a half thousand examples of tiling and patterns. This is a very significant book and no University or College library should be without one and many mathematicians will desire a personal copy. It is a great pity Freemans have abandoned the paperback version indicated in one of their catalogues.

I thought I knew a fair amount about tessellations until I started reading this book but the field is so much wider than I realised that my knowledge was included in the first hundred pages leaving 550 pages of new ideas and results. This means I may not be a fair judge of the significance of this book so let me repeat the quotation from Martin Gardner on the book's cover. "I am enormously impressed. ... Will be the definitive reference on tiling theory for many decades. Not only does the book bring together older results that have not been brought together before, but it contains a wealth of new material. ... I know of no comparable book."

The book is aimed at students, professional mathematicians and non-mathematicians, such as artists, architects and crystallographers, whose interests include patterns and shapes. By their skilful presentation and style the authors have made their material available to this wide audience. There are sections where the ideas are difficult but with the next section or chapter the pressure is off for a while. Many proofs, particularly of the enumeration variety, are not done in full but the strategies are explained in detail; reference given to where to find the complete proof; and results given as diagrams and tables.

Their way of explaining the meaning of results and particularly definitions is excellent. In preparing the book they have discovered many erroneous results in the literature and the two most common causes are lack of proper definitions and incorrect enumeration. They give clear definitions and the reasons for them are well explained, often with examples of the nasties you let in if a condition is relaxed.

...

This is a book I am delighted to have on my book shelf and one advantage of completing this review is I can now go back and get to work on some of the situations it has introduced me to.

**3.7. Review by: John A Wenzel.**

*The Mathematics Teacher*

**80**(6) (1987), 497-498.

The authors have set high goals: (1) "to write a book on 'visual geometry' - a rigorous book, but one that would also encourage geo metric appreciation by the use of 'pure' geometric reasoning"; (2) to write "with three main groups of readers in mind - students, professional mathematicians and non mathematicians whose interests include patterns and shapes"; and (3) to write so that "[t]he whole book should be accessible to any reader attracted to geometry, regardless of (or even in spite of) his previous mathematical education."

To achieve these goals they have presented an abundance of clear, informative illustrations and an exhaustive, authoritative bibliography, with entries ranging from Kepler's

*Harmonie Mundi*of 1619 to many articles appearing in 1985. One particularly helpful feature is that for each bibliographic entry, the sections where it is referenced are given. Each section contains an ample collection of exercises, ranging from those that test under standing of definitions and basic concepts to those that pose unsolved problems. "Notes and References," the last section of each chapter, provides historical perspectives and indications for future developments.

The standard material that one would expect to find (M C Escher,

*Islamic art, and Archimedean tilings*) appears but with an eye to accuracy, cohesion, and correct attribution of credit. Tools of group theory are developed as appropriate and used as necessary. The notation follows international conventions and is at times over whelming to the uninitiated.

Are Grünbaum and Shephard successful? The book is rigorous and has much to intrigue students. It will be helpful to those interested in shapes and patterns. Mathematicians will find challenging problems. Each user group will dis cover its favourite sections. A must for your reference library, this book is envisioned as "the first step in carrying out a programme" to "encourage geometric appreciation." Let us hope that the next steps will be equally impressive and soon to come.

**3.8. Review by: L Fejes Tóth.**

*Bulletin of the American Mathematical Society*

**17**(1987), 369-372.

From time immemorial artisans and artists have constructed ingenious tilings and ornaments using repeated motives. This is demonstrated in the introduction of the beautiful volume under review by numerous examples from widely separated cultures. However, the importance of tilings and patterns in crystallography and some related branches of science was recognised only towards the end of the last century. From this time on many crystallographers, chemists, physicists, architects, engineers, and mathematicians have been working in this field. Although they accumulated a vast literature in books and periodicals, "much effort has been wasted duplicating previously known results." When the authors started collecting material for this book, they were surprised to find "how little about tilings and patterns is known," and how many errors were made because of "badly formulated definitions and lack of rigour."

For more than a decade the authors were busy critically revising the earlier results and making significant contributions to the theory of tilings and ornaments in a series of papers of their own. Their effort is crowned by the unique comprehensive monograph

*Tilings and patterns*, which lays a solid foundation for one of the most attractive fields in geometry.

The book gives evidence of the sound didactic sense of the authors. The introduction of new concepts is carefully prepared, often supported by convincing intuitive arguments, and most formal definitions are richly illustrated by figures or some other means. The exposition is informal, but always clear and exact. Most sections contain carefully selected exercises, which often assign the completion of some arguments in the text to the reader, and many promising research problems. The "Notes and References" at the end of the chapters point out many interesting connections with other disciplines in mathematics, art, and science, and give an exhaustive survey of the related results. Consequently, the book turns out to be much more than "just" a fundamental scientific work. It is also an excellent textbook, which is meant not only for students and professional mathematicians, but for anyone interested, by profession or hobby, in geometrical configurations. The intriguing topic, the beautiful figures, and the profusion of the challenging open problems will be an inexhaustible source of pleasure and inspiration for generations to come. In addition, the book will certainly contribute to the restoration of the balance between abstract and intuitive trends in our anti-geometrical age.

**3.9. Review by: R L E Schwarzenberger.**

*Bulletin of the London Mathematical Society*20 (1988), 167-192.

References to this book have been appearing, with various conjectured dates of publication, since 1980. Now that it has arrived it is seen to be more impressive than anyone who saw draft chapters can have imagined. It is a rich collection of illustrations, examples, references and historical data on tilings of the plane. It is the first rigorous and authoritative account of the classification of various natural kinds of tiling (here synonymous with tessellation, mosaic or paving), and of the classification of discrete patterns which is used to achieve this. It contains many recent, and previously unpublished, research results together with a long list of unsolved problems. It would be a good investment at twice the price since, quite aside from its high quality throughout, it contains three books in one.

**4. Tilings and Patterns, An Introduction (1989), by Branko Grünbaum and G C Shephard.**

**4.1. From the Preface.**

This volume is a brief edition, unchanged except for correction of typographical errors, comprising the first seven chapters of our earlier book of the same title. These seven chapters were described in our original preface as the first part of that book. The present paperback version contains all the material from the original text that deals with tilings by regular polygons, the topological and symmetry properties of tilings, the motif-transitive patterns in general, and the special cases where the motif is a circular or elliptical disk or a straight-line segment. It also includes several classifications of very symmetric tilings. The full list of references from the original version is reproduced in the present volume, although not all of the references are cited in the text. Several mentions of material in Chapters 8 through 12 of the earlier book have also been left in place.

**4.2. Review by: Joe Donegan.**

*The Mathematics Teacher*

**83**(2) (1990), 167.

*Tilings and Patterns: An Introduction*by Grünbaum and Shephard may be a misleading title for this book. It is far more than an introduction to this concept; the authors have pulled a voluminous amount of information on the subject. Their in-depth treatment of tilings and patterns has resulted in a book that would be most valuable as a reference for serious study and research. The scope of information covered includes not only a historical analysis but also a projection of the use of these concepts into other fields. The authors begin with a commentary on basic notations that requires more than a casual acquaintance with mathematical terminology.

The text contains chapters on topology of tilings and patterns, as well as two chapters on classifications of tilings: first, those tilings with the transitive property; the other, classifications with respect to symmetry. My feeling that this work is a unique tool for reference and research is further supported by the fact that the authors have included over forty pages of additional references.

The extensive illustrations and tables help the reader to better understand different types of tilings, from those made by regular polygons to tilings in which neither the tiles nor the vertices are assumed to be regular. Understanding theorems and their proofs requires more than the understanding of mathematics required for teacher certification.

My final evaluation would be that

*Tilings and Patterns: An Introduction*is an excellent, well thought-out, and detailed book that targets a selective population of readers: the serious research student, the mathematician, and those who use mathematical theory and concepts in special fields.

**4.3. Review by: Paul Garcia.**

*The Mathematical Gazette*

**74**( 468) (1990), 207-209.

This is a paperback version of the first seven chapters of

*Tilings and patterns*, reviewed in the

*Gazette*in December 1987. The material is unchanged, apart from correcting some of the typographical errors. All the references to chapters after 7 have been left in (which might prove very frustrating if you haven't got the original work), and the list of references at the back is the same as in the original book.

The book starts with a lavishly illustrated historical introduction to tilings and patterns, describing the use of geometric patterns from antiquity to the present day, with examples from across the globe, from Ely to Mongolia. The final paragraphs list a number of non-mathematical texts which deal in some way with the subject (how many art writers would include a list of relevant material from the sciences, I wonder. ...). The main business then starts with a chapter called "Basic notions", which provides rigorous definitions of tile, tiling and a patch (a finite number of tiles whose union is a topological disk). It is in this chapter that the incredibly beautiful spiral tilings are introduced (but the reader is referred to Chapter 9 (not in this book) for further examples). The use of symmetry as a means of classification is introduced here and forms a continuous thread throughout the rest of the book. I found parts of this a bit difficult to follow, because I am not familiar with the international symbols for symmetry groups, and I do not have ready access to the references wherein they are described. A short appendix with this information would have been very helpful (this is my only criticism of the book-in all other respects it is wonderful).

...

The authors' style is clear and very robust. No punches are pulled in the "Notes and references" sections at the end of each chapter, which present an overview of previous work done on the subject under consideration. For example, at the end of chapter 6 on the "Classification of tilings with transitivity properties", the attempt by Fedotov in 1978 to classify isogonal tilings is described and the paragraph concludes, "Whichever way one tries to interpret his statements, the enumeration is woefully inadequate".

In the Preface, the authors state: "It is curious that almost all aspects of geometry relevant to the 'man in the street' (and woman, I hope) are ignored by our educational systems. Geometry has been almost squeezed out of school and university syllabuses and what little remains is rarely of any use ... the essence of the subject - its visual appeal - has been completely submerged in technicalities and abstractions." They go on to observe that other branches of mathematics (e.g. topology and analysis) have suffered from this neglect of geometry wherein lie their roots.

The original book was a first step towards redressing the situation, and appears to have succeeded. Many new results have been discovered since the book was first published in 1987 (and the authors offer to send a brief survey of developments to interested readers); I hope that this paperback edition will inspire more teachers to introduce geometrical ideas into their lessons. So often I hear teachers deriding work on tilings as "mumbo-jumbo" or "not proper mathematics"; the erudition of this book would soon disabuse them of such feelings, if only they could be persuaded to read it and use it!

**5. Convex Polytopes (Second Edition) (2002), by Branko Grünbaum.**

**5.1. From the Preface.**

There is no such thing as an "updated classic" - so this is not what you have in hand.

In his 1966 preface, Branko Grünbaum expressed confidence "that the current surge of interest and research in the combinatorial properties of convex polytopes will continue and will render the book obsolete in a few years." He also stated his "hope that the book itself will contribute to the revitalisation of the field and act as a stimulant to further research."

This hope has been realised. The combinatorial study of convex polytopes is today an extremely active and healthy area of mathematical research, and the number and depth of its relationships to other parts of mathematics have grown astonishingly. To some extent, Branko's confidence in the obsolescence of his book was also justified, for some of the most important open problems mentioned in it have by now been solved. However, the book is still an outstanding compendium of interesting and useful information about convex polytopes, containing many facts not found elsewhere.

Major topics, from Gale diagrams to cubical polytopes, have their beginnings in this book. The book is comprehensive in a sense that was never achieved (or even attempted) again. So it is still a major reference for polytope theory (without needing any changes).

Unfortunately, the book went out of print as early as 1970, and some of our colleagues have been looking for "their own copy" since then. Thus, responding to "popular demand", there have been continued efforts to make the book accessible again. Now we are happy to say: Here it is!

The present new edition contains the full text of the original, in the original typesetting, and with the original page numbering - except for the table of contents and the index, which have been expanded. You will see yourself all that has been added: The notes that we provide are meant to help to bridge the thirty-five years of intensive research on polytopes that were to a large extent initiated, guided, motivated, and fuelled by this book. However, to make this edition feasible, we had to restrict these notes severely, and there is no claim or even attempt for any complete coverage. The notes that we provide for the individual chapters try to summarise a few important developments with respect to the topics treated by Grünbaum, quite a remarkable number of them triggered by his exposition. Nevertheless, the selection of topics for these notes is clearly biased by our own interests.

The material that we have added provides a direct guide to more than 400 papers and books that have appeared since 1967; thus references like "Grünbaum [a]" refer to the additional bibliography which starts on page 448a. Many of those publications are themselves surveys, so there is also much work to which the reader is guided indirectly. However, there remain many gaps that we would have liked to fill if space permitted, and we apologise to fellow researchers whose favourite polytopal papers are not mentioned here.

...

We have taken advantage of some tools available in 2002 (but not in 1967), in order to compute and to visualise examples.

**5.2. Review by: Nick Lord.**

*The Mathematical Gazette*

**89**(514) (2005), 164-166.

Several reasons may be adduced for the staccato nature of the development of the theory of convex polytopes. Firstly, the fundamental enumeration problems in this area are intrinsically hard. For example, although the existence of all the regular Platonic solids and semiregular Archimedean solids have been known since antiquity, the enumeration of all regular-faced convex polyhedra was completed by Johnson and Zalgaller as recently as the 1960s. Equally, it is striking how many of the key examples of higher dimensional polytopes have been discovered relatively recently: familiarity with the full range of possibilities has been slow in coming and intuition correspondingly slow to build. Secondly, 'big theorems' are hard to find and hard to prove. Many Gazette readers will be familiar with Lakatos' account of the trials and tribulations of constructing a proof of Euler's relation for polyhedra: such problems were replicated by those 19th century mathematicians seeking to generalise Euler's relation to $d$-polytopes (in $\mathbb{R}^{d}$). Thirdly, until about 1950 polytope theory tended to fall between the two stools of being too abstract to be of concrete use and too specialised to be of abstract use. All this changed dramatically with the birth of linear programming where the construction of efficient algorithms depended on an intimate knowledge of the geometry of the polytope defined by the constraints of the problem. Since then, there has been an explosion of activity with deep, mutual connections made with areas of mathematics as diverse as computational geometry, functional analysis and algebraic geometry.

The first edition of Grünbaum's Convex polytopes appeared in 1967 on the crest of the wave of renewed interest in this area. It immediately established itself as a comprehensive, authoritative reference work on the combinatorial properties of convex polytopes. It collated a wealth of detailed information (both in the text and in the copious collection of exercises) that was previously scattered throughout the literature and included historical asides that attempted to straighten-out the convoluted history of the subject - a history replete with discovery and re-discovery. Grünbaum also provided two major services to his readers: one in presenting fully- watertight proofs of all the main theorems, the other by highlighting open problems for future investigations - many, but not all, of these have now been resolved. Ironically then, the introductory chapters (1-7) on convexity and specific types of polyhedra still serve as such, but some of the later chapters - which were state of the art in 1967 - have dated noticeably. A significant point here (and one which serves to make the book so readable) is that Grünbaum focuses throughout on elementary proofs whereas some of the more recent developments have involved the introduction of new techniques and the use of sophisticated algebraic machinery.

For the second edition Kaibel, Klee and Ziegler have retained the whole of the text and page-numbering of the first edition but have augmented each chapter with an alphabetically 'numbered' 'Additional notes and comments' section with cross-references to and from an extensive additional bibliography and an updated index. This minimal intervention approach preserves the spirit and vigour of the first edition but, to my mind, worked a little unevenly in the later chapters.

**5.3. Review by: Alexander Zvonkin.**

*Mathematical Reviews*MR1976856

**(2004b:52001)**.

Branko Grünbaum's book is a classical monograph on convex polytopes first published by John Wiley & Sons in 1967. As was noted by many researchers, for many years the book provided a central reference for work in the field and inspired a whole generation of specialists in polytope theory. Since 1970, when the book went out of print, there was a constant demand for a new edition. The main problem was, however, to find an appropriate form of reflecting the spectacular progress of the theory of convex polytopes since then. It seems that the editors have found such a form. Every chapter of the book is supplied with a section entitled "Additional notes and comments", prepared by Volker Kaibel, Victor Klee, and Günter Ziegler; these notes summarise the most important developments with respect to the topics treated by Grünbaum. Together with additional bibliographical references, this material represents 74 pages of the book. The reader is thus provided both with the source text and with very valuable information on the subsequent development generated by the book. Each of the 19 chapters of the monograph contains, besides the main material, historical notes and an unusually large number of exercises, which may also serve as an additional source of interesting results.

...

The new edition, by Springer Verlag, of Grünbaum's book is an excellent gift for all geometry lovers.

**5.4. Review by: Peter McMullen.**

*Combinatorics, Probability and Computing*

**14**(2005), 623-626.

When the first edition of Convex Polytopes was published in 1967, it was very well received. It was not a completely comprehensive account of the subject; for example, it said nothing about equidissectability (Hilbert's Third Problem), and very little about the associated idea of valuations. Nevertheless, it covered a large range of combinatorial topics, giving what was then the most up-to-date information on them. Grünbaum himself expressed the hope - and expectation - that the book would prompt research and further progress on these topics; however, he could hardly have anticipated the torrent of results which was unleashed. As a consequence, in a number of areas

*Convex Polytopes*was almost immediately rendered obsolete. Even so, it remained required background reading for researchers in convex polytopes, and its going out of print was much to be regretted; its reappearance is thus very welcome.

The second edition is not an up-dated version of the first, in the sense that the original material has not been rewritten in any way. Instead, the trio who have prepared it have added a few pages at the end of each chapter; in these, brief reports on progress on the relevant topics are given. To complement these, there are also (towards the end of the book) a list of errata in the first edition, and a valuable extensive additional bibliography; moreover, the indices have been completely rewritten.

...

It would obviously be wrong to castigate those who have prepared this second edition for not doing various things that would have been very desirable. I must again emphasise, however, that even a straight reprinting of

*Convex Polytopes*would have been very welcome. But it also has to be said that the subject has moved on, and many of the most powerful techniques which have been developed since 1967 are, because of the restrictions which the redactors have imposed on themselves, much under-represented. Nevertheless, within the terms of reference which they have set themselves, they have done a fine job, and their efforts should be rewarded by all those with even a marginal interest in the subject placing this volume on their bookshelves.

**6. Configurations of Points and Lines (2009), by Branko Grünbaum.**

**6.1. From the Publisher.**

This is the only book on the topic of geometric configurations of points and lines. It presents in detail the history of the topic, with its surges and declines since its beginning in 1876. It covers all the advances in the field since the revival of interest in geometric configurations some 20 years ago. The author's contributions are central to this revival. In particular, he initiated the study of 4-configurations (that is, those that contain four points on each line, and four lines through each point); the results are fully described in the text. The main novelty in the approach to all geometric configurations is the concentration on their symmetries, which make it possible to deal with configurations of rather large sizes. The book brings the readers to the limits of present knowledge in a leisurely way, enabling them to enjoy the material as well as enticing them to try their hand at expanding it.

**6.2. From the preface.**

It is easy to explain the concept of a configuration of points and lines to any ten-years-old youngster. Why then a book on this topic in a graduate series? There are several good reasons:

(i) First and foremost, configurations are mathematically challenging even though easily accessible.

(ii) The study of configurations leans on many fields: classical geometry, combinatorics, topology, algebraic geometry, computing, and even analysis and number theory.

(iii) There is a visual appeal to many types of configurations.

(iv) There are opportunities for serious innovation that do not rely on long years of preliminary study.

The truly remarkable aspect of configurations is the scarcity of results in a field that was explicitly started well over a century ago, and informally much earlier. One of the foremost aims of the present text is to make available, essentially for the first time ever, a coherent account of the material.

Historical aspects are presented in order to enable the reader to follow the advances (as well as the occasional retreats) of the understanding of configurations. As explained more fully in the text, an initial burst of enthusiasm in the late nineteenth century produced several basic results. For almost a century, these were not matched by any comparably important new achievements. But near the end of the last century it turned out that the early results were incorrect, and this became part of the impetus for a reinvigorated study of configurations.

The recent realisation that symmetries may play an important role in the investigations of configurations provided additional points of view on configurations. Together with the increased ability to actually draw configurations - made possible by advances in computer graphics - the stage was set for renewed efforts in correcting the ancient mistakes and to studying configurations that were never contemplated in the past.

This text relies very heavily on the graphical presentation of configurations. This is practically inevitable considering the topic and greatly simplifies the description of the many types of configurations dealt with. Most of the diagrams have been crafted using Mathematica, Geometers Sketchpad, and ClarisDraw, often in combination.

In many respects this is a "natural history" of configurations - the properties and methods of generation depend to a large extent on the kind of configuration, and we present them in separate chapters and sections. We have avoided insisting on proofs of properties that are visually obvious to such an extent that formal proofs would needlessly lengthen the exposition and make it quite boring. We firmly believe that an appropriate diagram is as much of a valid argument as a pedantic verbal explanation, besides being more readily understandable. It is hoped that the reader will agree!

The text is narrowly restricted to the topic of its title. There are many other kinds of configurations that might have been included. However, the nature of such configurations, for example, of points and planes, or of various higher-dimensional flats, is totally different from our topic. It is well possible that the early attempts to cover all possibilities led to very general definitions followed by very meagre results.

Two exceptions to the restricted character of the presentation concern combinatorial configurations and topological configurations. The former are essential to the theory of geometric configurations, and we present the topic with this aim in mind. We do not enlarge on the various more general aspects of combinatorial designs and finite geometries - there are many excellent texts on these matters. Completely different is the situation regarding topological configurations. Very little is known about them, and the present text collects most of what is available.

One other aspect not covered here is the detailed investigations of the hierarchies of some special configurations. It seems that at one time it was fashionable to start with a simple result, such as the theorem of Pascal, and generate a whole family of objects by permuting the starting elements, then considering all the intersections of the resulting lines and the lines generated by the obtained points, etc. This way one could secure a family of points or lines or whatever to be attached to one's name. The interested reader may gain access to this literature through other means.

For almost all the material covered, we provided as ample and detailed references as we were able to find. However, we did not give details concerning the programs that produced the various computer-generated enumerations. The reason - besides lack of competence - is that the programs and the computers on which they run change too rapidly for any printed information to be of lasting value. The interested reader should contact the authors of these results to obtain the most up-to-date status.

Results for which no reference is given are the author's and appear here for the first time. Also, as noted in appropriate places, several colleagues have been kind enough to allow the inclusion of their unpublished results - I am greatly indebted to them for this courtesy.

My gratitude goes to several other people and institutions. The American Mathematical Society was extremely helpful at all stages of the preparation of this text; in particular, allowing the illustrations to be in colour has greatly increased the appeal of the book, as well as its instructional value.

...

Last - but certainly not least - my thanks go to my wife Zdenka, not only for her patience and forbearance over the long haul of my study of configurations and the preparation of the manuscript of this book, but even more for her love and support during well over half a century.

Branko Grünbaum

Seattle,

October 23, 2008

**6.3. Contents.**

Preface

**Chapter 1.**Beginnings

§1.1. Introduction

§1.2. An informal history of configurations

§1.3. Basic concepts and definitions

§1.4. Tools for the study of configurations

§1.5. Symmetry

§1.6. Reduced Levi graphs

§1.7. Derived figures and other tools

**Chapter 2.**3-Configurations

§2.0. Overview

§2.1. Existence of 3-configurations

§2.2. Enumeration of 3-configurations (Part 1)

§2.3. Enumeration of 3-configurations (Part 2)

§2.4. General constructions for combinatorial 3-configurations

§2.5. Steinitz's theorem - the combinatorial part

§2.6. Steinitz's theorem - the geometric part

§2.7. Astral 3-configurations with cyclic symmetry group

§2.8. Astral 3-configurations with dihedral symmetry group

§2.9. Multiastral 3-configurations

§2.10. Duality of astral 3-configurations

§2.11. Open problems (and a few exercises)

**Chapter 3.**4-Configurations

§3.0. Overview

§3.1. Combinatorial 4-configurations

§3.2. Existence of topological and geometric 4-configurations

§3.3. Constructions of geometric 4-configurations

§3.4. Existence of geometric 4-configurations

§3.5. Astral 4-configurations

§3.6. 2-Astral 4-configurations

§3.7. 3-Astral 4-configurations

§3.8. $k$-Astral 4-configurations for k ≥ 4

§3.9. Open problems

**Chapter 4.**Other Configurations

§4.0. Overview

§4.1. 5-configurations

§4.2. $k$-configurations for $k ≥ 6$

§4.3. [3,4]- and [4,3]-configurations

§4.4. Unbalanced [$q, k$]-configurations with [$q, k$] ≠ [3,4]

§4.5. Floral configurations

§4.6. Topological configurations

§4.7. Unconventional configurations

§4.8. Open problems

**Chapter 5.**Properties of Configurations

§5.0. Overview

§5.1. Connectivity of configurations

§5.2. Hamiltonian multilaterals

§5.3. Multilateral decompositions

§5.4. Multilateral-free configurations

§5.5. "Density" of trilaterals in configurations

§5.6. The dimension of a decomposition

§5.7. Movable decompositions

§5.8. Automorphisms and duality

§5.9. Open problems

Postscript

Appendix

**6.4. Review by: Rolf Riesinger.**

*zbMATH*, Zbl

**1205.51003.**

The author of this book is one of the leading geometers of our time, he is co-founder of several branches of modern combinatorics and discrete geometry, he published more than 200 research articles and some books, and about 1990 he initiated especially the revival of the theory of configurations. The author's experience gathered in more than 50 years of scientific work is of great benefit for the reader. The text is both interesting and attractive and it is a pleasure to read it because the presentation is of leisurely pace and appeals to geometric intuition. The main novelty in the approach to all geometric configurations is the concentration on their symmetries, which make it possible to deal with configurations of rather large size. The author takes the reader to the limits of present knowledge. For further investigation the reader finds a large number of open problems and 240 references which include also historic aspects. The book contains numerous tables and over 300 very aesthetical coloured figures many of them comprising more than one diagram.

**6.5. Review by: Darren Glass.**

*Mathematical Association of America*: https://www.maa.org/press/maa-reviews/configurations-of-points-and-lines

Is it possible to draw a set of 7 points and 7 lines so that each point lies on exactly three of the lines and each line contains exactly three of the points? What about 10 points and 10 lines? What if you want each line to contain exactly four points? If you are intrigued by these types of questions, then I have a book for you. Branko Grünbaum's new book, Configurations of Points and Lines studies questions about points and lines which intersect in prescribed ways and their generalisations. While many of the questions could be posed to an elementary school student, the answers get very sophisticated very quickly, and there is ample material to fill a book and leave many open questions for the reader to work on.

...

This book is written in a style which is very different from most books you will find in the "AMS Graduate Studies in Mathematics" series. In particular, it is much chattier and friendlier to its readers, with loads of colour illustrations and digressions into historical and philosophical asides. But this is not to say that the mathematics it contains is not interesting and sophisticated: at various times, Grünbaum's techniques involve combinatorics, algebraic geometry, group theory, and topology. But the book is almost entirely self-contained, and the author has a tone that is extremely readable; it is probably the only book in this series that I have considered bedtime reading! There are plenty of topics in mathematics that it is hard to imagine writing about in this style - motivic cohomology or lattice delay equations, for example - but I think many authors could learn a thing or two about readability from Grünbaum, and reading his book was a real pleasure.

**7.**

**Tilings and Patterns (Dover reprint) (2016), by Branko Grünbaum and G C Shephard.**

**7.1. From the Publisher.**

The definitive book on tiling and geometric patterns, this magnificently illustrated volume features 520 figures and more than 100 tables. Accessible to anyone with a grasp of geometry, it offers numerous graphic examples of two-dimensional spaces covered with interlocking figures, in addition to related problems and references. Suitable for geometry courses as well as independent study, this inspiring book is geared toward students, professional mathematicians, and readers interested in patterns and shapes - artists, architects, and crystallographers, among others. Along with helpful examples from mathematics and geometry, it draws upon models from fields as diverse as crystallography, virology, art, philosophy, and quilting. The self-contained chapters need not be read in sequence, and each concludes with an excellent selection of notes and references. The first seven chapters can be used as a classroom text, and the final four contain fascinating browsing material, including detailed surveys of colour patterns, groups of colour symmetry, and tilings by polygons.

**7.2. Review by: William J Satzer.**

*Mathematical Association of America*: https://www.maa.org/press/maa-reviews/tilings-and-patterns

*Tilings and Patterns*, first published in 1987, has recently been re-published by Dover. Although it got outstanding reviews following its appearance, the original publisher chose to discontinue it in 1998. Dover has once again done the mathematical community a service in bringing back such a notable volume.

The authors have left the original unchanged in most respects. Besides correcting minor typographical errors, the authors have added an appendix with references to new material that has appeared in the literature over the intervening decades. Of course, this does not make the book completely up to date. (For instance, some more recently discovered convex pentagonal tilings are not mentioned.) It would be a truly mind-boggling task to update the book completely considering how much is here already.

According to their preface the authors began with the idea of writing a rigorous book on "visual geometry" that would encourage a renewed appreciation of geometry and the use of purely geometrical reasoning. Eventually they decided to focus on tilings and patterns. They envisioned three distinct groups of readers: students, professional mathematicians and a variety of non-mathematicians (artists, architects, crystallographers, and the like.) Their idea was that the book should be accessible to any reader who is attracted to geometry.

The first part of the book focuses on the basic geometry of planar tilings and patterns. A plane tiling is defined broadly as a countable family of closed sets that cover the plane without gaps or overlaps. However, the authors mostly restrict themselves to a class of well-behaved tilings that have conditions designed to eliminate undesirable and pathological examples. Patterns, which can also be defined very broadly, are mostly treated here as repetitions of a motif in the plane in a regular manner with certain natural technical restrictions.

The book starts with an introduction that presents a survey of tilings and patterns from a variety of cultures and historical periods. Other highlights of the first part of the book are an extensive treatment of tilings by regular polygons and star-shaped polygons, topological equivalence of tilings by homeomorphism, and classifications of tilings according to symmetries and transitivity properties. The authors suggest that the first seven chapters of the book could be used for an undergraduate geometry course.

The second part of the book looks at a few advanced topics. Coloured tilings are described at some length. Although coloured tilings have been far and away the most commonly used tilings in the decorative arts, almost no systematic work on them had been done before the current book. (Unfortunately, all the figures in the book are in grayscale). Aperiodic tilings are also treated extensively. Prominent examples include variations due to Penrose and Wang. Wang tilings get special attention because it is possible to find sets of Wang tiles that can be used to mimic the behaviour of any Turing machine, and hence they are relevant to questions of mathematical logic.

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. While portions of the book are accessible to those with limited mathematical backgrounds, a basic knowledge of topology and algebra (and sometime more) is needed to follow the details of the development. Nonetheless, this is a book that almost anyone would enjoy dipping into.

Last Updated February 2023