1.1. Preface.

A system of discrete points is called regular if any two points of the system can be transformed into each other by a single movement such that the entire system coincides with itself; in short, if no point of the system is superior to any other. Such a point system is associated with other regularly designed figures, for example, polygons, polysurfaces, or spatial partitions. Regular arrangements of points or figures have continually occupied the human imagination and have particularly captivated the interest of mathematicians. Of the numerous names that could be mentioned here, only Plato, Archimedes, Kepler, Bravais, and Schläfli will be noted.

In three-dimensional space, group-theoretical considerations made it possible to gain an overview of regular point systems and thereby provide a natural explanation for the crystal forms found in nature. These investigations culminated in the famous discovery of the 230 crystallographic space groups by Fedorov (1885), Schoenflies (1891), and Barlow (1894).

Later, attention turned to certain extremal problems concerning regular point systems, as attempts were made to explain various physical and chemical properties of crystals. One such problem is that of the closest regular sphere packing. Let us imagine the molecules of a certain substance as equally sized spheres that touch each other but cannot overlap. The goal is to find the regular molecular arrangement that contains the greatest possible number of molecules per unit volume.

Minkowski gave a tremendous impetus to the investigation of such extremal problems. He recognised the connection between certain number-theoretical questions and storage problems concerning figure lattices, thereby establishing a field of mathematics that is still intensively cultivated today: the geometry of numbers.

In physical chemistry and the geometry of numbers, the problems mainly involve extremal issues, whereby the arrangements permitted for comparison are subject to certain regularity conditions from the outset. In contrast, the present work is dedicated to storage problems, in which arbitrary irregular arrangements are also considered. The regular shape of the extremal figure is often a consequence of the extremal requirement.

We mention two typical problems.

1. In what arrangement can the most Heller pieces fit on a "large" table? The answer is that each Heller must touch six others. The best arrangement is therefore lattice-like.

2. Consider twelve points of a solid sphere. In what storage arrangement is the volume of the convex hull of the points maximised? This problem leads us to the regular icosahedron.

The first problem, i.e., the problem of the densest planar circular storage, was solved by the great Norwegian number theorist A Thue in an early work (1892). Then there was a considerable pause in development in this direction, so that most of the results we want to discuss here are fruits of about the last 10 to 12 years. This set of problems has not yet been addressed in a textbook.

No prior knowledge is necessary to understand the problems raised here. They are simple, natural, and intuitive questions, but the typical difficulties they contain often make them serious problems. In most cases, however, the solutions require nothing "higher," so that almost the entire book could be kept in a way that is easy to understand. Yet this relatively elementary set of questions contains a wealth of unsolved problems. One of the main purposes of our book is to draw attention to these issues in order to attract more contributors to this appealing field of geometry.

I owe special thanks to Professors H Hadwiger, G Hajos, and B L van der Waerden, who read the manuscript and supported me with a number of valuable comments. I am indebted to my collaborator J Molnár for most of the numerical calculations and several remarks. I am also indebted to Mr M Kneser for his help with the proofreading.

Veszprém, March 1953.

L Toth Fejes.

1.2. Review by: W Fenchel.
Mathematica Scandinavica 2 (1) (1954), 173-176.

The subject of this book is extremal problems for figures in Euclidean geometry, in which discontinuities, such as integer parameters, play a crucial role, either by definition or due to imposed conditions. However, while regularity (lattice structure, invariance under a set of motions) is assumed for the discrete structures under consideration in the geometry of numbers and in geometric crystallography, such assumptions are not made here from the outset. The general questions to which the main content of the book is subordinate can be described as follows: Within a given region, regions of a prescribed type are to be placed such that they do not overlap and that the largest possible part of the region is covered. The "density" of the arrangement, that is, the ratio of the sum of the contents of the incorporated regions to the content of the region, should therefore be maximised. For an infinite region, e.g., the entire plane or all of space, the density is naturally defined in an obvious way as a limit. These problems of densest packing are contrasted with dual problems of thinnest covering. The latter require completely covering the given region with regions of a prescribed type in the most economical way. The goal is to minimise the density, which is defined similarly and is therefore at least 1. Closely related to these problems, especially for the surface of a sphere, are extremal problems for polyhedra with a given number of vertices or faces, and the like. These general problems can be specialised and varied in numerous ways. The results usually take the form of inequalities, e.g. estimates of the aforementioned densities.

In cases where it is possible to obtain the best possible estimates, the question then arises as to the configurations of the type in question for which the extreme values ​​are achieved. Of particular interest are storage and covering problems in the plane, on the surface of a sphere, and in space with congruent circles, spherical caps, or spheres. These problems allow for a second, very intuitive formulation: In a given region, a given number $N$ of points are to be placed such that the minimum distance between any two points is as large as possible, or such that the maximum distance of a point in the region from the nearest of the $N$ points is as small as possible. For an infinite region, the number $N$ is to be replaced by the number density, which is to be defined as the limiting value. For the plane, these problems have been solved; the best possible values ​​are achieved in both cases when the points are arranged in a grid, specifically in an equilateral triangular grid. For the surface of a sphere, the best arrangements have only been found for some small values ​​of $N$; In general, they are necessarily irregular, and a new problem arises for each value of $N$. The problems of the densest sphere arrangement and the thinnest covering of space by congruent spheres present considerable difficulties that have not yet been overcome. Various conjectures and estimations exist. In addition to those mentioned, numerous other special cases and variants of the general problems above are treated. Apart from the isoperimetric problem for polyhedra, which S A J Lhuilier addressed in the 18th century, and J Steiner and L Lindelöf in the 19th century, the oldest significant result relevant here appears to be A Thue's determination of the densest circular arrangement in the plane in 1892. Since H Minkowski's foundation of the geometry of numbers around the turn of the century, many fine contributions to the topic of this book have been made by representatives of this field. The majority of the content, however, dates from the last 15 years, and the author himself has made a very significant contribution to this.

The methods used to solve these problems share the common feature of very elementary tools. Apart from this, however, they are very diverse, and no general solution methods are found. Of closely related problems, one may be almost trivial, a second elegantly solvable by a clever trick, a third may require case distinctions or laborious investigations of elementary functions, while for others, often very obvious ones, there may only be approaches to the solution, conjectures, or even the absence of any such conjectures. The task the author set himself, and which he solved with great skill - namely, to process such material, consisting of many individual results, into a readable book - was undoubtedly not easy.

Each chapter of the book begins with a brief overview of its contents and ends with detailed historical information and references to the literature. In two introductory chapters, tools from elementary geometry and the theory of convex bodies are compiled, often with proofs. Besides familiar material, many interesting details from recent years, which may be less well-known, can already be found here. For example: inequalities between the distances of an arbitrary point from the vertices and sides of a triangle. A proof (originating from B v Sz Nagy and G Bol) of Bonnesen's refinement of the isoperimeter inequality for polygons [In light of a statement by Bol, quoted by the author, that this is the first truly simple proof of this inequality, Bonnesen's own proof (cf. Les problèmes des isopérimètres, Paris, 1929, p. 60) is recalled.] Theorems on the approximation of the area of ​​a convex region by the areas of inscribed and circumscribed polygons of a given number of sides. A very clear treatment of the affine length and related extremal problems. The two following chapters are devoted to plane problems. In addition to those already mentioned, packing and covering problems for regions of various kinds, with circles of different sizes, congruent convex regions, etc., are treated. In the latter case, the shape of the regions can also be varied, which leads to extremal problems for the best densities. The next two chapters, on extremal problems for polyhedra and support and covering problems for the sphere, present, in addition to the already mentioned difficult support problems for the sphere, recent results on extremal properties of regular polyhedra, the progress made in the treatment of the isoperimetric problem, theorems on the approximation of the sphere by polyhedra of a given number of faces or vertices, and much else besides. The last chapter deals with spatial support and covering problems. Here, the difficulties of the problems are primarily demonstrated, partial results are proven, conjectures are formulated, and these are supported by considerations of plausibility.

The readership for this book is very broad. The required prior knowledge is extremely modest; it nowhere exceeds what is taught in introductory courses. In addition, the writing style is almost consistently easy to read and supported by excellent illustrations. The book can therefore be warmly recommended to anyone who enjoys concrete, vivid problem-solving. It doesn't need to be read systematically; one can simply select the most engaging sections. (However, some caution is advised in the later chapters due to the appearance of somewhat imprecise proofs.) Numerous references to unsolved problems are clearly intended to attract new contributors, which will hopefully be successful; for despite the enormous successes of abstract, general theories, specific problems have arguably not yet lost their role as starting points or nuclei for more general insights.

W Fenchel

1.3. Review by: C A Rogers.
The Mathematical Gazette 39 (327) (1955), 73-74.

The title (which can be translated "Packings in the plane on a sphere and in space") describes only one of the various types of problems discussed in this book. While its main aim is to discuss the question of how closely it is possible to pack, that is to arrange without overlapping, congruent convex domains in the plane, equal spherical caps on the surface of a sphere, and equal spheres in space, many other interesting problems are discussed. In particular, the parallel problem of how economically is it possible to cover the plane with domains congruent to a given convex domain is considered. Another problem (which is supposed to have been solved by the honey bee) is to divide the plane by a network of line segments into a system of cells of unit area, the total length of the network to be minimal. As tools for the solution of problems of this type, the author considers the extremal properties of regular polygons and polyhedra, and the approximation of convex domains and spheres by polygons and polyhedra. To quote a single explicit result, it is shown that if a convex polyhedron with $v$ vertices, $f$ faces and $e$ edges contains the unit sphere, then its volume $V$ satisfies

         $V ≥ \large\frac{e}{3}\normalsize \sin (\large\frac{\pi f}{e}\normalsize ) [\tan^{2}(\large\frac{\pi f}{2e}\normalsize ) \tan^{2}(\large\frac{\pi v}{2e}\normalsize ) - 1]$

and that strict inequality holds except in the case of a regular polyhedron circumscribed to the unit sphere.

The book starts with a discussion of some elementary, and some not quite so elementary, results about convex bodies and polygons which are needed later. Then the different problems are discussed in the plane, on the surface of a sphere, and, after a section on polyhedra, in space. As the book proceeds, the author finds questions to which he has not yet succeeded in finding the answer (much of the work described in the book is due to the author); he does not hesitate to supply a conjecture. At the ends of the various sections the author gives collections of remarks containing references to many original papers concerned with the problems he discusses and with many interesting generalisations and variations.

The book is written in a very clear style; the sentences are short, making the task of the dictionary thumbing reader easy. The mathematics is based almost entirely on elementary geometry and should be understood by anyone with a knowledge of school mathematics, provided that they have a mature outlook on the subject. The problems considered are quite fascinating and are much nearer those of the ordinary world than is common in pure mathematics. The field is one in which there is scope for the amateur mathematician to make important contributions.

C A Rogers.

2.1. Publisher's Description.

Regular Figures concerns the systematology and genetics of regular figures. The first part of the book deals with the classical theory of the regular figures. This topic includes description of plane ornaments, spherical arrangements, hyperbolic tessellations, polyhedral, and regular polytopes. The problem of geometry of the sphere and the two-dimensional hyperbolic space are considered. Classical theory is explained as describing all possible symmetrical groupings in different spaces of constant curvature. The second part deals with the genetics of the regular figures and the inequalities found in polygons; also presented as examples are the packing and covering problems of a given circle using the most or least number of discs. The problem of distributing $n$ points on the sphere for these points to be placed as far as possible from each other is also discussed. The theories and problems discussed are then applied to pollen-grains, which are transported by animals or the wind. A closer look into the exterior composition of the grain shows many characteristics of uniform distribution of orifices, as well as irregular distribution. A formula that calculates such packing density is then explained. More advanced problems such as the genetics of the protean regular figures of higher spaces are also discussed. The book is ideal for physicists, mathematicians, architects, and students and professors in geometry.

2.2. Preface.

On buildings, machines and other products of our civilisation, regularly arranged objects are often observed. The parquet blocks on the floor, the teeth on a cog-wheel, and the figures on a fancy cloth are all regularly arranged. Nature also produces a great variety of regular distributions, in the kingdom of the living and non-living as well. We recall the petals of flowers with various kinds of rotational symmetry, or the arrangement of atoms in crystals.

A discrete set of equal figures is said to be regularly arranged if each figure can be carried into any other one by a congruent transformation or isometry leaving the whole configuration unchanged. All such transformations together form a group, the symmetry group of the arrangement. The main aim of the classical theory of regular figures is to enumerate all possible symmetry groups in different spaces of constant curvature. This general problem is connected with a range of further problems: what kind of regular figures exists under certain restrictions? This question involves the theory of the regular polytopes, tessellations and lattices. Besides the enumeration of the various kinds of regular figures classical theory attempts to determine their metrical and topological properties. Thus it may be considered as the systematology and morphology of the regular figures. This theory is one of the oldest branches of science, the foundations of which were laid by Greek and Egyptian artists.

In the seventeenth century Kepler made essential contributions to the theory, but its golden age begins with the nineteenth century. This renascence of the regular figures of antiquity was due partly to the investigation of the inner structure of crystals, and partly to the discovery of the deep connection of regular figures with other branches of mathematics, especially with algebra, group theory, number theory and the theory of functions. The geometry of numbers is, today, one of the chief driving forces of the evolution of the theory.

Besides this classical theory, regular figures may be approached in another way, starting from the observation that extremum postulates often involve regularity. Classical theory starts with a more or less arbitrary definition of regularity. Here, in turn, regular arrangements are generated from unarranged, chaotic sets by the ordering effect of an economy principle, in the widest sense of the word. This theory may be called the genetics of regular figures.

Systematology plays a central part in directing the researches of genetics. On the other hand, the different extremum properties of the regular figures may be considered as precious contributions to the classical theory. This organic connection of the two approaches makes it reasonable to expound them in one book.

This book is divided into two parts: systematology and genetics of the regular figures. (Aspects of morphology are incorporated partly in the first, partly in the second part, where metrical properties of regular figures appear in various inequalities as the cases of equality.) In both parts we shall be content to present some typical, simple and interesting results and methods. For a more detailed discussion of the classical theory we refer to the excellent modern monographs of Coxeter (1948), Burckhardt (1947) and Coxeter and Moser (1957). Concerning recent theory the reader may consult the author's book (1953a) and the great number of original works on the subject quoted in the bibliography.

"C'est la dissymétrie qui crée le phénomène" writes Pierre Curie, expressing by these words the frequently observed tendency towards symmetry in fundamental physical structures. It is always an extremum postulate which lies at the bottom of this tendency. Thus we seem to be on the right track towards the wider aim of throwing some light on the causes of their origin, besides describing and systematizing the regular figures occurring in nature.

But, in writing this book, we have a much narrower aim in mind which may be best expressed by echoing the words of Clebsch wrote in his memoir on Plücker: "It is the joy in form, in a higher sense, that defines the geometer." I would like to awaken this noble joy in the reader, showing that we are all, in Clebsch's sense, geometers.

I wish to express my sincere gratitude to Professor H S M Coxeter for having encouraged me to write this book, for reading the whole manuscript and making many valuable suggestions. I offer also my friendly thanks to A Heppes for conscientiously criticising the manuscript, to J Molnár for the numerous expressive drawings and to I Pál for the beautiful anaglyphs. Last but not least I remember with grateful affection the enjoy-able winter semester (1960/61) at the University of Freiburg in Breisgau where I completed this book amid an inspiring circle of colleagues.

2.3. Abstracts of the Chapters.

CHAPTER I - PLANE ORNAMENTS

This chapter discusses the mathematical theory of plane ornaments. It presents a vivid introduction to one of the most fundamental notions of modern mathematics, the concept of a group. An isometry that leaves a figure invariant is called a symmetry operation. To classify the ornaments according to their symmetry operations, one has to investigate various isometries of the plane. The totality of the symmetry operations of a figure constitutes a group, in regard to the composition of its transformations as group operation. This group is called the symmetry group of the figure. It may happen that the symmetry group of a figure consists of the identity only. Then the figure is said to be asymmetrical. In all other cases, the figure is called symmetrical.

CHAPTER II - SPHERICAL ARRANGEMENTS

This chapter discusses some partial results from the theory of discrete groups of isometries in ordinary space, including the enumeration of the 32 crystallographic classes and of the finite symmetry groups of space. A rotatory-translation, or screw, is a rotation combined with a translation along the axis of the rotation; a glide-reflection is a reflection combined with a translation in a direction parallel to the reflecting plane; a rotatory-reflection is a rotation combined with a reflection in a plane perpendicular to the axis of the rotation. Multiplying each permutation by a certain fixed transposition we obtain again the totality of the permutations in some order. In other words, no rotation of the cube different from the identity can transform each diagonal into itself. If a diagonal is carried over into itself, it either coincides with the axis of rotation or else the operation is a half-turn about an axis orthogonal to two diagonals.

CHAPTER III - HYPERBOLIC TESSELLATIONS

This chapter focuses on the hyperbolic plane showing the missing tessellations. Among the five postulates underlying Euclid's elements there is one, the famous postulate of the unique parallel. Inversion, or reflection in a circle, is one of the most remarkable mappings of elementary geometry. Using the terminology that the image of zero is the point in infinity, the inversion sets up a bi-unique transformation of the plane onto itself. Inversion in a circle orthogonal to a given circle $C$ transforms the inside of $C$ into itself, and the circles orthogonal to $C$ into circles having the same property. A reflection of the hyperbolic plane in a line is the inversion in the circle representing the line. The isometries of the hyperbolic plane are defined as the products of reflections. As the isometries of the hyperbolic plane preserve the Euclidean angles in the model, the hyperbolic angles must be defined as the Euclidean angles of the model.

CHAPTER IV - POLYHEDRA

This chapter presents a brief account of some famous polyhedra including the star-polyhedra of Kepler and Poinsot, the Archimedean solids, and the space-fillers of Fedorov. A polygon means a finite set of segments arranged in such a way that exactly two segments meet at every segment-extremity, and no subset has the same property. The segments are called sides and their extremities vertices. A polyhedron is a finite set of polygons arranged in space in such a way that every side of each polygon belongs to just one further polygon, with the restriction that no subset has the same property. This definition is analogous to that of the polygon. Unfortunately, the definition of regularity has to be changed when passing from two to three dimensions. However, this difference disappears in higher dimensions, where both the definition of a polytope and that of its regularity run inductively.

CHAPTER V - REGULAR POLYTOPES

This chapter discusses regular polytopes. The arithmetisation of geometry, which started with Fermat and Descartes, gradually brought about a certain geometrisation of arithmetic. The geometrical interpretation became a guiding principle for solving analytical problems. It is convenient and useful that, sometimes, the relations of these geometries may be visualised and tested on objects of the world. However, that does not alter the fact that, in principle, geometry does not appeal to intuition. The main point is that abstraction should not lead to a bloodless, pathological aberration. The geometry of $n$-dimensional space lies in the line of healthy development: it represents a general, closed theory of intrinsic beauty; it permanently influences other branches of mathematics and takes impetus from them. It is an inseparable, organic part of one's mathematical knowledge and its purpose is the same as that of the whole of mathematics.

CHAPTER VI - FIGURES IN THE EUCLIDEAN PLANE

This chapter discusses some covering and packing problems that lead to arrangements displaying the symmetry of some wall patterns. A greatest circle (closed disc) contained in a closed point set is called the incircle and the smallest circle containing the set is called the circumcircle of the set. The respective radii are known as the inradius and circumradius. Sometimes, it is convenient to attribute to the area of a triangle a positive or negative sign according as the vertices are named in the positive or negative sense. Analogous problems can be raised by considering a parallel strip or the whole plane instead of a circle. However, the number of the discs must be replaced by the packing-density and covering-density of the discs, defined by convenient limiting values. These problems, which may be of practical interest, offer a natural approach to all frieze groups and wall-pattern groups.

CHAPTER VII - SPHERICAL FIGURES

This chapter discusses an extremum property of the Platonic tessellations; showing a simple way in which the quasi-regular tessellations may be characterised by the means of an extremum hypothesis. Many extremum properties of the regular Euclidean polygons can be extended immediately to spherical polygons. The vertex angles of this triangle being all minimal angles, each vertex must belong to a further triangle. Thus, the whole tessellation can be built up step by step to form a tessellation of type. Consider a set of equal circles of given number and size, which can neither be placed onto the sphere without mutual overlapping, nor entirely cover the sphere. In spring milliards of pollen-grains are transported by wind, water, or insects, to transmit the life of their species. On the exterior hull, there are little orifices. Sticking on the stigma, through one orifice lying near to the point of adhesion, the pollen-grain sends out a tube that gives the spermatozoon access to the female nucleus.

CHAPTER VIII - PROBLEMS IN THE HYPERBOLIC PLANE

This chapter discusses the way certain completeness in regard to problems concerning the inner geometry of the sphere can be attained only by taking into consideration the hyperbolic plane. Without loss of generality, one may suppose that the packing is saturated, as otherwise one could saturate it by joining further circles. By this operation, the packing density does not decrease according to any sensible density definition. On the other hand, one may assume that in the covering the circles nowhere accumulate, as otherwise one could ascribe to the covering an infinite density. More precisely, defining the circle density on the whole sphere by a mean value of the circle densities in regard to the triangles of the tessellation, the theorem holds independently of the kind of the underlying average. Decomposing the supporting polygons of more than three sides by nonintersecting diagonals into triangles, one has constructed the desired tessellation.

CHAPTER IX - PROBLEMS IN 3-SPACE

This chapter discusses some fundamental extremum properties of the regular polyhedra. The first extremum problem concerning polyhedra that arrested the attention of mathematicians seems to have been the isoperimetric problem. Translating the faces of a convex polyhedron not circumscribed about a sphere through the same distance outwards, produces a better polyhedron. The chapter also discusses sphere-clouds. A set of nonoverlapping spheres lying between two parallel planes is said to form a cloud if each line orthogonal to the planes intersects at least one of the spheres. The lower bound of the distances of two such planes is called the width of the cloud. The solution of the analogous 2-dimensional problem is trivial: in the cloud of unit circles of minimal width the circles constitute a straight string. However, one can consider $k$-fold clouds in which each line orthogonal to the cloud intersects at least $k$ circles (spheres) and the corresponding problem becomes interesting even in the plane.

CHAPTER X - PROBLEMS IN HIGHER SPACES

This chapter presents some initial steps toward the genetics of the protean regular figures of higher spaces. There are several further extremum properties of the regular triangle that may be generalised without difficulty on the regular simplex. In the case of a simplex and cross polytope, Theorem 1 may be proved more directly by applying the symmetrisation immediately to the polytope, instead of to the bounding simplexes. Hence, by a symmetrisation the quotient of the circumradius and inradius does not increase. However, in the case of an irregular simplex and cross polytope, this quotient definitely decreases by at least one of the symmetrisations described above. This yields for a simplex a new proof of the inequality and for polytopes isomorphic to the cross polytope or measure polytope the inequality. The best way to understand the deeper causes of the difficulties inherent in the problems of closest sphere-packing and thinnest sphere-covering in Euclidean 3-space is to consider these problems from the standpoint of non-Euclidean geometry.

2.4. Review by: F A Sherk.
Mathematical Reviews MR0165423 (29 #2705).

The appeal of symmetry, displayed in a regular polyhedron or a wall frieze pattern, is universal. The mathematics behind the symmetry, while having its roots in antiquity, is by no means exhausted. These are the two main impressions that one is left with after reading this work.

The book is divided into two parts. Part One, entitled "Systematology of the Regular Figures", outlines the classical theory. This is, of course, a group-theoretic approach. The author begins by considering the discrete symmetry groups of the Euclidean plane. All seventeen plane crystallographic groups are illustrated. The three regular and eight semiregular (Archimedean) tessellations are introduced. This leads naturally to circle-packings and circle-coverings, which occupy much of the author's attention later on. There follow chapters dealing with regular figures on the sphere and in the hyperbolic plane. The author then turns to Euclidean space of three dimensions, with a survey of regular and semiregular polyhedra, and the parallelohedra. Part One concludes with an introduction to regular polytopes of higher dimensions, especially those of dimension four.

Part Two, "Genetics of the Regular Figures", aims at exhibiting the regular figures as solutions of various extremum problems. To cite the best-known example, the arrangement of circles in the densest close-packing of equal circles in the Euclidean plane is such that the centres are the vertices of the regular tessellation {3, 6}. This part of the book fairly bristles with tantalising theorems, the statements of which are disarmingly simple, but whose proofs are often long and intricate. The author considers problems of this nature in the Euclidean plane, on the sphere, in the hyperbolic plane, and in Euclidean spaces of three and higher dimensions, thus paralleling the material in Part One. There are, of course, many gaps in this approach; it is not nearly as complete as the classical theory given in Part One. But enough is given to make the approach convincing and attractive. It will surely wet the appetites of geometers for many years to come.

Apart from the selection of the material itself, other noteworthy features of the book are the historical remarks and the many illustrations, which include twelve anaglyphs, complete with a viewer. There is a large bibliography and an adequate index.

2.5. Review by: J A Todd.
Proceedings of the Edinburgh Mathematical Society 14 (2) (1964), 174-175.

This interesting book falls into two distinct parts, of approximately equal length, which form a complete contrast to each other.

The first part, which the author calls "Systematology of the Regular Figures," is a formal development of the theory of regular and Archimedean polyhedra and of regular polytopes. The treatment, in which elementary group theory is well to the fore, is concise and complete. The chapter headings give a clear enough notion of the contents: I. Plane Ornaments (which contains a complete discussion of the two-dimensional crystallographic groups); II. Spherical arrangements (including an enumeration of the 32 crystal classes); III. Hyperbolic tessellations (essentially a discussion of the discrete groups generated by two operations whose product is involutary); IV. Polyhedra (including the enumeration of the regular solids, concave as well as convex, and of the convex Archimedean solids); V. Regular polytopes (which completes the enumeration of the regular figures in Euclidean space of higher dimension than three).

The author's exposition matches the elegance of his subject. This part of the book invites obvious comparison with the well-known work by Professor H S M Coxeter, which has clearly, in part, been its inspiration; however there are enough differences in approach and emphasis to make this account welcome in its own right. There are interesting historical notes at the end of each chapter; but surely, in the last line but two on p. 97, the reference should be to Legendre and not to Lagrange.

The second part of the book, which the author calls "Genetics of the Regular Figures," is of quite a different nature. Here, as it seems to the reviewer, we have no well-rounded theory, but rather a number of special problems, the solution of some of which is by no means complete. The subject matter consists of various extremal problems in which regular figures play a part. This is a subject to which the author has made notable contributions, and this seems to be the first time that current knowledge on the subject has been put together as a whole. Chapter VI deals with some relatively simple problems of packings and coverings of circles in a plane, and Chapter VII with tessellations on a sphere. We quote one theorem, out of many, as an example of the sort of thing the reader will find. Suppose that we have a tessellation of the unit sphere by convex faces of equal area, with $e$ edges, and suppose that $p$ and $q$ denote, respectively, the average number of edges in a face, and the average number of edges meeting in a vertex. Then the total length of the edges in the tessellation is not less than

          $2e \arccos (\cos \large\frac{\pi }{p}\normalsize \cosec \large\frac{\pi }{q}\normalsize )$

and this lower bound is attained if, and only if, the tessellation is regular. The remaining chapters deal with problems in the hyperbolic plane, and in Euclidean space of three or more dimensions. It is clear that this is a subject in which there is still much scope for research, and one which calls for considerable ingenuity in approaching its problems.

The book is copiously illustrated, with some anaglyphs in a folder at the end, and contains an extensive bibliography.

The excellence of the text is unfortunately not matched by a corresponding elegance of production, which falls a long way short of the standard one is now accustomed to expect in mathematical printing. There are numerous cases in which mathematical expressions or sentences are broken by the end of a line instead of being displayed and the number of obvious misprints in passages of plain text, not to mention weak letters, suggests that the press reader has not been sufficiently careful. The current standard of "higher mathematical printing" in this country is high, and it is a pity that it is not attained in this book which, both by its contents and its price, claims to be judged by the highest standards.

2.6. Review by: C A Rogers.
Journal of the London Mathematical Society 40 (1) (1965), 378.

As the author explains, this book divides naturally into two halves. The first half studies symmetry groups and the way in which regular polyhedra and tessellations arise as objects invariant under such groups. The second half studies extremal problems and the way in which regular polyhedra and tessellations arise as their solutions.

The first half of the book naturally has a certain amount in common with H S M Coxeter's Regular Polytopes, but it goes considerably further in certain directions. In particular it gives complete determinations for possible symmetry groups for plane ornaments and for the possible finite symmetry groups for configurations in 3-space. After a digression to study regular tessellations in hyperbolic 2-space, the results on symmetry groups are used to give an elegant account of the convex polyhedra and of the 3-dimensional uniform polyhedra, regular star-polyhedra, and parallelohedra.

The second half of the book is naturally similar in outlook to Fejes Tóth's own Lagerungen in der Ebene auf der Kugel und im Raum, but contains many results that have been since discovered or proved. The author gives an account of his well known results on the packing of convex domains into the plane, and on covering the plane by convex domains. He goes on to give his results on the perimeters of equi-areal convex domains packed into a hexagon. He gives a detailed account of his work with help from Florian, Heppes and Molńar, on the packing of unequal circles in the plane, and on the covering of the plane with unequal circles. He obtains similar, but not so extensive, results on the surface of a sphere and in 2-dimensional hyperbolic space. Perhaps the most striking results in the whole book are the author's results, some obtained with assistance from Florian, establishing very precise inequalities for the volumes and surface areas of convex 3-dimensional polyhedra in terms of their inradii, circumradii, and the numbers of their faces, edges and vertices. The author also states many related results and discusses many related problems, offering conjectures and possible lines of attack.

While, in the main, the book is written in a very clear style I occasionally came across passages where I could not accept the author's arguments as complete. I had particular difficulty on pages 96, 165 and 170-171. As evidence that I am not alone in having difficulty with these arguments, I remark that J W S Cassesls has based a proof of one of Fejes Tóth's assertions on a lemma which may be shown to be false by a simple-counter example.

The book is illustrated by many diagrams, three beautiful coloured plates and a set of striking "anaglyphs" drawn by I Pál. When viewed through the coloured spectacles provided, these anaglyphs stand out as striking 3-dimensional figures. I found that my eyes needed quite a bit of training before I could see the diagrams properly; it is I think best to leave the more complicated ones until one is seeing the simpler ones clearly.

2.7. Review by: Patrick du Val.
American Mathematical Monthly 73 (7) (1966), 799.

This is in many respects a most charming book. It is in two parts, which are likely to be of interest to rather different types of readers. Part I deals with the geometry of the regular figures and their groups of symmetry, and is extremely lucid and well illustrated with figures. The first of its five chapters deals with plane crystallography, describing and illustrating the seventeen groups, as well as the point groups, and those that leave a line invariant. Here there are three extremely fine coloured plates, of Egyptian, Arabic, and other ornamental designs. The next two chapters deal with similar questions on the sphere (including the 32 point groups of crystallography) and in the hyperbolic plane; and the other two with regular and semi-regular polyhedra, and with polytopes in higher space. The three dimensional figures are admirably illustrated in a series of anaglyphs (stereographic figures in two colours, with coloured eyepieces provided) which are separate in a pocket at the end of the book.

Part II deals with all kinds of extremal problems, such as close packing and covering by spheres and other figures, maximum and minimum areas and volumes, and so forth, whose solution involves regular figures in some way; the maximal density of the hexagonal packing of circles, and the minimal area of the honeycomb, are typical. It is in five chapters, corresponding in scope to those of Part I, namely the Euclidean plane, the sphere, the hyperbolic plane, three dimensional space, and higher space. This part of the book seems to be, for the author, its main raison d'être, and it includes many original results, as well as much up-to-date work of others. For this reviewer it is much harder to read than the first part, but it will undoubtedly be of great interest and value to specialists in this field.

2.8. Review by: W J Edge.
The Mathematical Gazette 49 (369) (1965), 343-345.

A quarter of a century has now passed since the author of the book before us began to publish his long series of original papers. These, together with his Lagerungen in der Ebene, auf der Kugel und im Raum which appeared as one of Springer's Grundlehren in 1953, have established the author in a position of high authority and ensure that any work from his pen will arouse lively expectation. This new book is in two parts; it is the second part which includes many of its author's discoveries and contains material cognate to, and indeed all but identical with, parts of the Lagerungen.

The first chapter of the book deals with plane ornaments; these are classified as rosettes, friezes and wall-patterns by means of their symmetry groups of congruent transformations. Diagrams are provided generously, indeed one might say lavishly, here and throughout the book. The three consecutive pages of coloured plates adorning this first chapter, culled from two mid-nineteenth century treatises on ornamentation, are very fine. All the finite groups of symmetries in Euclidean space of three dimensions are found in chapter II, and are represented by diagrams. Some of them, in consequence of a fundamental theorem here stated but not proved, cannot occur as crystallo-graphic groups; when they are excluded 32 "crystal classes" remain. In chapter III the tessellations of a hyperbolic plane $H$ are found. Poincaré's map of $H$ by the interior of a circle $C$ in the Euclidean plane is used: arcs inside $C$ of circles orthogonal to $C$, together with the diameters of $C$, map the lines of $H$ and reflection in a line of $H$ is mapped by inversion in the mapping circular arc (or by reflection in the mapping diameter). The model serves to establish formulae of hyperbolic trigonometry. This chapter, as well as other passages of the book (one may mention the section on permutations in chapter II and the first eight pages of chapter V) combine lightness of touch and conciseness of exposition in a quite delightful way. One must also commend the informative paragraphs, expounding the history of the subject, that close each chapter. The two chapters, IV and V, on regular polyhedra and regular polytopes, concern topics made familiar by the writing of Coxeter.

The second part of the book starts with chapter VI, which opens appropriately by proving Jensen's inequality for convex functions of one and of two variables; this opening is appropriate because the inequality is used some score of times in this and the next three chapters. About one third of the subject matter of the Lagerungen is, with variations, treated in this present book. The very first application of Jensen's inequality is to prove, for star polygons, inequalities given for convex polygons in the German book, though there arrived at by geometrical reasoning. These inequalities exemplify the author's thesis: that regular figures are, basically, solutions of extremum problems. Jensen's inequality also plays, on p. 221, a vital part in establishing an in-equality between the integrals of a monotonic function over a definite spherical triangle and over the polygonal faces of a tessellation of the sphere; it is interesting to compare this argument with pages 81-84 and 137-141 of the German text. This inequality between the integrals is used to obtain an upper bound for the density of a packing of equal circles on the sphere, as well as a lower bound for the covering of the sphere by equal circles. Other related passages of the two books discuss the lower bound of the volume of a convex polyhedron $P$ in relation to that of a sphere lying wholly inside $P$, and its upper bound in relation to the volume of a sphere wholly surrounding $P$. Others explain how to locate $n$ points on a sphere so that the least distance between any two of them is as large as possible; in the present book it is pointed out that this problem can occur naturally in connection with orifices on grains of pollen. Another topic common to both books is an isoperimetric property of convex polyhedra: the ratio of the cube of the surface area to the square of the volume can never be less than

          $54(f - 2) \tan \phi (4 \sin^{2}\phi - 1)$

where $f$ is the number of faces and $(f - 2) \phi = \large\frac{\pi f}{6}$. The author claims not to have discovered this inequality but to have given the first sound proof of it. Here, too, is an extremum problem. Equality can occur, but only for those three regular Platonic solids-tetrahedron, cube and dodecahedron - having three faces meeting at each vertex.

There is a discussion in chapter IX of the still unsolved problem of the upper bound for the density of a packing of equal spheres. This is connected with the question, a matter of contention as long ago as Newton's time, whether it is possible for 13 equal spheres all to touch one sphere $S$ of the same size. The vertices of any regular icosahedron inscribed in $S$ afford eligible contacts for 12 such spheres; room to manoeuvre is then available because these 12 spheres are not themselves in contact. But manoeuvring will not leave sufficient room to allow a further equal sphere inserted among the 12 to sink low enough to touch $S$ too. The author is constrained to say that the proof of the conjecture that the upper bound is $\large\frac{\pi}{√18}\normalsize$ is a remote aim. He remarks, as he did in the Lagerungen, that the problem is "reduced to the determination of the minimum of a function of a finite number of variables"; he now adds the aspiration that computers may be helpful.

Chapters VIII and X are mostly compounded of information that has appeared in the author's papers over the last decade, and these same papers have of course provided material for the other chapters. Chapter VIII, on problems in a hyperbolic plane $H$, naturally uses the facts set out in Chapter III. The packings and coverings of $H$ by equal circles are investigated by using "supporting" circles and polygons, these polygons composing a tessellation of $H$. There is also a study of the packings and coverings of $H$ by horocycles - the curves mapped in Poincaré's model by circles touching $C$ internally. Chapter X is concerned with problems in higher space; in it the author says

The best way to understand the deeper causes of the difficulties inherent in the problems of closest sphere-packing and thinnest sphere-covering in Euclidean 3-space is to consider these problems from the standpoint of non-Euclidean geometry.

Left pondering upon some tantalising conjectures we lay down the book wondering when and how they will be answered, and wishing the author well.

The reviewer has felt it incumbent upon himself to refurbish his earlier study of the Lagerungen, and if he deems the page of Springer-Verlag to be more comely, and the print more satisfying, than those of the Pergamon Press these are matters, after all, of taste. But proof-reading is not, and so erudite and precise a scholar as our author was entitled to better service. To print > instead of < is a slip made only too easily, but it is a slip to be especially wary of, and wariness did not prevail at the top of p. 273. The appearances of convexity, when there ought to be concavity, lower down on the same page, and again on p. 276, are equally unfortunate. At one place on p. 140 we are offered the impossible [3, 4, 4] instead of [3, 4, 3]. The numerator of the fraction towards the bottom of p. 296 ought to be 27, not 2; however, the decimal on the right is correct, and not equal to the misprinted fraction on the left.

One must not conclude without warmly thanking and congratulating those two colleagues of the author, one of whom has provided the anaglyphs pocketed at the end of the volume and the other the liberal supply of handsome diagrams. Every reader of the book shares the author's indebtedness to them.

2.9. Review by: H S M Coxeter.
Science, New Series 146 (3549) (1964), 1288.

This book seems to have everything that could be desired in a mathematical monograph - a pleasant style, careful explanation of all technical terms used, a great variety of topics with a single unifying idea, a good bibliography and index, and many beautiful illustrations, including four plates and a pocketful of stereoscopic anaglyphs. Although the work is mostly concerned with geometry, it has connections with art, crystallography, biology, city planning, and the standardisation of industrial products.

Chapter 1, "Plane ornaments," includes the first complete and readable proof in English that there are exactly 17 essentially different wallpaper pat-terns, a fact that was utilised unconsciously by the Moors in decorating the Alhambra. It was first proved by E S Fedorov in 1891 and has been rediscovered as a mathematical theorem at least three times since then. This chapter ends with an almost incredible tour de force due to H Voderberg - a systematic (but completely asymmetric) tessellation of the Euclidean plane with congruent tiles, each of which is an irregular enneagon (9-gon).

Using the notation of L Schläfli (1814-1895), Fejes Tóth defines the regular tessellation $\{p, q\}$ as consisting of equal regular $p$-gons, $q$ round each vertex. In chapter 2 he uses the same notation for a regular spherical tessellation, and proves that such a pattern exists for every pair of positive integers (except $p = q = 1$) satisfying the inequality

          $\large\frac{1}{p}\normalsize + \large\frac{1}{q}\normalsize > \large\frac{1}{2}\normalsize$

For instance, {2, 1} has one face (a digon), one edge (a semicircle such as a meridian), and two vertices (the north and south poles). In this connection, he should perhaps have mentioned the beautifully illustrated paper by H Emde, "Homogene Polytope" [Math. Revs. 21, 1105 (1960)].

In chapter 3, reversing the above inequality, Fejes Tóth obtains the infinite family of hyperbolic (non-Euclidean) tessellations $\{p, q\}$. As background for this discussion, he gives a clear but concise account of the theory of inversion, the invariance of cross ratio, and the two conformal models of the hyperbolic plane. On page 97 the drawings of {3, ∞} and {∞, 3} are particularly striking. (Some readers may be puzzled by the unusual notation $\sin^{-1}\beta$ for $\\co\sec \beta$ on page 93.) It is interesting to be reminded of the contrast between the successful career of Lobachevsky and the tragic life of Bolyai, "unhappy in his marriage, broken in health, hated and cast out by the philistines of a small town owing to his uncompromising straight-forwardness, separated from mathematical life, books and periodicals, but fully aware of the significance of his epoch-making discovery and in full possession of his sound judgment."

Chapter 4 deals with the most important kinds of polyhedra-regular, semiregular, and so on. Fedorov's five parallelohedra are constructed by the method of B N Delaunay, who "succeeded in giving a complete enumeration of their 4-dimensional analogues (whose number of types turned out to be 52)." This remark leads naturally to the treatment of regular polytopes in chapter 5, where the discovery of n-dimensional geometries (by Grassmann and Schläfli) is described as enabling us "to create an infinite set of new universes, the laws of which are within our reach, though we can never set foot in them."

Despite the high quality of these first five chapters, it is in the remaining five that the author reveals the full scope of his mathematical ingenuity.

Here we see many examples of the extremal problems for which he and other Hungarian geometers are justly famous. The spaces considered are the same (and in the same order) as in chapters 1-5. For instance, chapter 7, on spherical figures, deals with such problems as distributing n points on a sphere so that the minimal distance between pairs is maximised. This "problem of Tammes" is introduced on page 226 with a charming description of its botanical origin. Its difficulty is made evident by the fact that, despite the best efforts of the world's geometers for 35 years, it has only been solved for $n$ ≤ 12 and for $n$ = 24 (the last case by R M Robinson in 1961).

We have another example on page 253-two particularly fine drawings to illustrate the theorem that, if the hyperbolic plane is packed or covered by equal circles (of finite or infinite radius), then the packing density ≤ $\large\frac{3}{\pi}\normalsize$ and the covering density ≥ $\large\frac{2√3}{\pi}\normalsize$.

2.10. Review by: Michael Goldberg.
Mathematics of Computation 19 (89) (1965), 166.

The discrete groups of isometries in the plane are the basis for ornamental patterns. The simplest geometrical representations for them are displayed, as well as ten beautifully coloured classic designs based on them.

Their extension to three-space is seen in spherical arrangements and the classical geometrical crystal classes. Since these do not exhaust the permutations of the group parameters, it is necessary to go to hyperbolic space to complete the utilisation of all the possible values of the parameters. Their groups are derived, and tessellation examples are displayed.

The regular and semi-regular polyhedra are derived and exhibited in photo-graphs and well-drawn figures. They include the Platonic and Archimedean solids, the Kepler-Poinsot star-polyhedra, and the regular honeycombs. The convex regular polytopes and Euclidean tessellations in all higher dimensions are derived from purely combinatorial considerations.

Problems concerning the most efficient packing of congruent figures in the plane are considered, as well as the most economical covering of the plane by congruent figures. These have technical applications as well as artistic applications. The problems are generalised to the use of sets of non-congruent figures. Also, multiple coverage and corresponding problems on a sphere are considered. The results are compared with biological patterns such as those occurring in pollen grains. The problems in this field are very difficult and many are still unsolved.

Problems in three-space, which involve the properties of polyhedra, include the isoperimetric problems, covering with clouds of spheres, sphere packing, and honey-combs. Extensions to higher spaces are made. Many beautiful results are derived, but there are many promising avenues to be explored.

A six-page bibliography and a good index make the book an excellent reference work.

The excellent three-dimensional anaglyphs in the book-pocket are not mentioned in the table of contents or the index. Their proper use is not explained in the text. For best viewing, these plates should be horizontal with the near edge about a foot from the eyes. They should be viewed through the coloured spectacles with the green lens before the right eye and the red lens before the left eye. The line of sight should be depressed about 45°.

3.1. Preface.

Since the first edition of this book, the theory of storages and coverings has been enriched by numerous results. Several of these results are discussed in the "Notes." In keeping with the spirit of this book, which is already expressed in the title, we have mainly limited ourselves to the most intuitive elementary geometric spaces. With regard to the $n$-dimensional theory, we refer to the fundamental monograph by Rogers.

Several problems are raised in the book that were still unsolved at the time of the first edition's publication, but which have since been solved, partly due to the influence of the book itself. These problems are marked in the margin with a black triangle (►). The numbers refer to the corresponding page numbers in the Notes.

I owe sincere thanks to my friend Professor A Florian (Salzburg), who assisted me in preparing the second edition.

Budapest, March 1971.

L Toth Fejes.

4.1. From the Publisher.

The publication of the first edition of Lagerungen in der Ebene, auf der Kugel und im Raum in 1953 marked the birth of discrete geometry. Since then, the book has had a profound and lasting influence on the development of the field. It included many open problems and conjectures, often accompanied by suggestions for their resolution. A good number of new results were surveyed by László Fejes Tóth in his Notes to the 2nd edition. The present version of Lagerungen makes this classic monograph available in English for the first time, with updated Notes, completed by extensive surveys of the state of the art. More precisely, this book consists of: a corrected English translation of the original Lagerungen, the revised and updated Notes on the original text, eight self-contained chapters surveying additional topics in detail. The English edition provides a comprehensive update to an enduring classic. Combining the lucid exposition of the original text with extensive new material, it will be a valuable resource for researchers in discrete geometry for decades to come.