# Donald Coxeter's Books - Prefaces

We look below at the various books, with their later editions, which were written by H S M Coxeter. We give some information such as extracts from Prefaces, extracts from Publisher's information. We have put these books in order of publication of the first edition except for

For extracts from reviews of these books, see THIS LINK.

Click on a link below to go to the information about that book

The Fifty-Nine Icosahedra (1938, 2nd ed. 1982, 3rd ed. 1999, 2011, 2013), with P du Val, H T Flather and J F Petrie

Non-Euclidean Geometry (1942, 2nd ed. 1947, 3rd ed. 1957, 4th ed. 1961, 5th ed. 1965, 1968, 1978, 1998)

Regular Polytopes (1948, 2nd ed. 1963, 3rd ed. 1973)

The Real Projective Plane (1949, 2nd ed. 1955, 3rd ed. 1993)

Generators and Relations for Discrete groups (1957, 2nd ed. 1965, 3rd ed. 1972, 4th ed. 1980, 1984), with W O J Moser

Introduction to geometry (1961, 2nd ed. 1968)

Projective Geometry (1964, 2nd ed. 1973, 1987, 1994)

Geometry Revisited (1967), with S L Greitzer

Twelve Geometric Essays (1968, 1999)

Mathematical Recreations and Essays (12th ed. 1974, 13th ed. 1987), with W W Rouse Ball

Regular Complex Polytopes (1974, 2nd ed. 1991)

Zero-symmetric Graphs (1981), with R Frucht and S L Powers

*Mathematical Recreations and Essays*which was first published fifteen years before Coxeter was born but he became an editor and eventually a co-author. We note that dates given, as for example (1964, 2nd ed. 1973, 1987, 1994), mean that the work was first published in 1964, the second edition was published in 1973 and then two further reprints of this edition were published in 1987 and 1994.For extracts from reviews of these books, see THIS LINK.

Click on a link below to go to the information about that book

The Fifty-Nine Icosahedra (1938, 2nd ed. 1982, 3rd ed. 1999, 2011, 2013), with P du Val, H T Flather and J F Petrie

Non-Euclidean Geometry (1942, 2nd ed. 1947, 3rd ed. 1957, 4th ed. 1961, 5th ed. 1965, 1968, 1978, 1998)

Regular Polytopes (1948, 2nd ed. 1963, 3rd ed. 1973)

The Real Projective Plane (1949, 2nd ed. 1955, 3rd ed. 1993)

Generators and Relations for Discrete groups (1957, 2nd ed. 1965, 3rd ed. 1972, 4th ed. 1980, 1984), with W O J Moser

Introduction to geometry (1961, 2nd ed. 1968)

Projective Geometry (1964, 2nd ed. 1973, 1987, 1994)

Geometry Revisited (1967), with S L Greitzer

Twelve Geometric Essays (1968, 1999)

Mathematical Recreations and Essays (12th ed. 1974, 13th ed. 1987), with W W Rouse Ball

Regular Complex Polytopes (1974, 2nd ed. 1991)

Zero-symmetric Graphs (1981), with R Frucht and S L Powers

**1. The Fifty-Nine Icosahedra (1938, 2nd ed. 1982, 3rd ed. 1999, 2011, 2013), H S M Coxeter, P du Val, H T Flather and J F Petrie.**

**1.1. From the Preface to the 2nd edition.**

The Fifty-Nine Icosahedra was originally published in 1938 as No. 6 of "University of Toronto Studies (Mathematical Series)". Of the four authors, only Coxeter and myself are still alive, and we two are the authors of the whole text of the book, in which any signs of immaturity may perhaps be regarded leniently on noting that both of us were still in our twenties when it was written. Neither of the others was a professional mathematician. Flather died about 1950, and Petrie, tragically, in a road accident in 1972. Petrie's part in the book consisted in the extremely difficult drawings which constitute the left half of each of the plates (the much simpler ones on the right being mine). A brief biographical note on Petrie will be found on p. 32 of Coxeter's Regular Polytopes (3rd. ed. , Dover, New York, 1973); and it may be added that he was still a schoolboy when he discovered the regular skew polygons that are named after him, and are the occasion for the note on him in Coxeter's book. (Coxeter also was a schoolboy when some of the results for which he will be most remembered were obtained; he and Petrie were schoolboy friends and used to work together on polyhedron and polytope theory.)

Flather's part in the book consisted in making a very beautiful set of miniature models of all the fifty-nine figures. These are still in existence, and in excellent preservation. ...

Apart from partial precursors referred to in Section 1 of the present book, Coxeter was the first to devote serious study to stellations of the icosahedron. He was certainly the first to enumerate them completely ...

**1.2. From the Publisher of the 3rd edition.**

H S M Coxeter, P Du Val, H Flather Kate and David Crennell produced a third edition for Tarquin some years ago. The plans and illustrations of all 59 of the stellations of the icosahedron were redrawn and there was a new introduction by Professor Coxeter. This edition has now been revised and updated and now includes colour plates for the first time.

Many readers will wish to use the plates and descriptions included in The Fifty-nine Icosahedra to draw and make paper models of some or all of the stellations of the icosahedron and in so doing, gain a greater understanding of the stellation process. For a thorough understanding of the process of stellation and for splendid examples of beautiful polyhedra, this book will be a valuable addition to any mathematics library.

As Richard Guy put it in the Preface of the Third Edition: "We mathematicians know how beautiful our subject is, but are often frustrated when we try to explain this to the man-in-the-street. But here at least is one place where we can simply say, 'Look!'"

**2. Non-Euclidean Geometry (1942, 2nd ed. 1947, 3rd ed. 1957, 4th ed. 1961, 5th ed. 1965, 1968, 1978, 1998), H S M Coxeter.**

**2.1. From the Preface.**

The name non-Euclidean was used by Gauss to describe a system of geometry which differs from Euclid's in its properties of parallelism. Such a system was developed independently by Bolyai in Hungary and Lobachevsky in Russia, about 120 years ago. Another system, differing more radically from Euclid's, was suggested later by Riemann in Germany and Schläfli in Switzerland. The subject was unified in 1871 by Klein, who gave the names of parabolic, hyperbolic, and elliptic to the respective systems of Euclid-Bolyai- Lobachevsky, and Riemann-Schläfli. Since then, a vast literature has accumulated, and it is with some diffidence that I venture to add a fresh exposition.

After an historical introductory chapter (which can be omitted without impairing the main development), I devote three chapters to a survey of real projective geometry. Although many textbooks on that subject have appeared, most of those in English stress the connection with Euclidean geometry. Moreover, it is customary to define a conic and then derive the relation of pole and polar, whereas the application to non-Euclidean geometry makes it more desirable to define the polarity first and then look for a conic (which may or may not exist)!. This treatment of projective geometry, due to von Staudt, has been found satisfactory in a course of lectures for undergraduates.

In Chapters VIII and IX, the Euclidean and hyperbolic geometries are built up axiomatically as special cases of a more general "descriptive geometry." Following Veblen, I develop the properties of parallel lines (§8.9) before introducing congruence. For the introduction of ideal elements, such as points at infinity, I employ the method of Pasch and F Schur. In this manner, hyperbolic geometry is eventually identified with the geometry of Klein's projective metric as applied to a real conic or quadric (Cayley's Absolute, 8.1, 9.7). This elaborate process of identification is unnecessary in the case of elliptic geometry. For, the axioms of real projective geometry (2.1) can be taken over as they stand. Any axioms of congruence that might be proposed would quickly lead to the absolute polarity, and so are conveniently replaced by the simple statement that one uniform polarity is singled out as a means for defining congruence.

Von Staudt's extension of real space to complex space is logically similar to Pasch's extension of descriptive space to projective space, but is far harder for students to grasp; so I prefer to deal with real space alone, expressing distance and angle in terms of real cross ratios. I hope this restriction to real space will remove some of the mystery that is apt to surround such concepts as Clifford parallels (7.2, 7.5). But Klein's complex treatment is given as an alternative (at the end of Chapters IV-VII).

In order to emphasise purely geometrical ideas, I introduce the various geometries synthetically. But coordinates are used for the derivation of trigonometrical formulae in Chapter XII.

**2.2. From the Publisher of the 5th edition.**

This is a reissue of Professor Coxeter's classic text on non-Euclidean geometry. It begins with a historical introductory chapter, and then devotes three chapters to surveying real projective geometry, and three to elliptic geometry. After this the Euclidean and hyperbolic geometries are built up axiomatically as special cases of a more general 'descriptive geometry'. This is essential reading for anybody with an interest in geometry.

**3. Regular Polytopes (1948, 2nd ed. 1963, 3rd ed. 1973), H S M Coxeter.**

**3.1. From the Preface.**

A polytope is a geometrical figure bounded by portions of lines, planes, or hyperplanes; e.g., in two dimensions it is a polygon, in three a polyhedron. The word polytope seems to have been coined by Hoppe in 1882, and introduced into English by Mrs. Stott about twenty years later. But the concept, under the name polyscheme, goes back to Schläfli, who completed his great monograph in 1852.

The foundations for our subject were laid by the Greeks over two thousand years ago. In fact, this book might have been subtitled "A sequel to Euclid's Elements" But all the more elaborate developments (roughly, from Chapter V on) are less than a century old. This revival of interest was partly due to the discovery that many polyhedra ( including three of the regular ones) occur in nature as crystals. However, there is a law of symmetry (4.32) which prohibits the inanimate occurrence of any pentagonal figure, such as the regular dodecahedron. Thus the chief reason for studying regular polyhedra is still the same as in the time of the Pythagoreans, namely, that their symmetrical shapes appeal to one's artistic sense. (To be sure there is a little more to it than that: Klein's Lectures on the Icosahedron cast fresh light on the general quintic equation. But if Klein had not been an artist he might have expressed his results in purely algebraic terms.)

As for the analogous figures in four or more dimensions, we can never fully comprehend them by direct observation. In attempting to do so, however, we seem to peep through a chink in the wall of our physical limitations into a new world of dazzling beauty. Such an escape from the turbulence of ordinary life will perhaps help to keep us sane. On the other hand, a reader whose standpoint is more severely practical may take comfort in Lobachevsky's assertion that "there is no branch of mathematics, however abstract, which may not some day be applied to phenomena of the real world."

I have tried to make this book as self-contained as possible. Anyone familiar with elementary algebra, geometry, and trigonometry will be able to appreciate it, and may find in it some fresh applications of these subjects; e.g., Chapter III provides an introduction to the theory of Groups. All the geometry of the first six chapters is ordinary solid geometry; but the topic treated have been carefully selected as forming a useful background for the subsequent developments. If the reader is at all distressed by the multi-dimensional character of the rest of the book, he will do well to consult Manning's Geometry of four dimensions or Sommerville's Geometry of n dimensions.

It will be seen that most of our chapters end with historical summaries, showing which parts of the subject are already known. The history of polytope-theory provides an instance of the essential unity of our western civilisation, and the subsequent absurdity of international strife. ...

This book grew out of an essay on "Dimensional Analogy", begun in February 1923. It is thus the fulfilment of 24 years' work, which included the rediscovery of Schläfli's regular polytopes, Hess's star-polytopes and Gosset's semi-regular polytopes. Probably my own best contribution is the invention of the "graphical" notation, which facilitates the enumeration of groups generated by reflections, of polytopes derived from these groups by Wythoff's construction, of the elements of any such polytope, and of "Gosset's tetrahedra". This last instance, which looks like some bizarre notation for the Music of the Spheres, is essentially a device for computing the volumes of certain spherical tetrahedra without having recourse to the calculus. The same notation can be applied very effectively to the theory of regular honeycombs in hyperbolic space, but I have resisted the temptation to add a fifteenth chapter on that subject.

**3.2. From the Publisher of 3rd edition.**

Polytopes are geometrical figures bounded by portions of lines, planes, or hyperplanes. In plane (two dimensional) geometry, they are known as polygons and comprise such figures as triangles, squares, pentagons, etc. In solid (three dimensional) geometry they are known as polyhedra and include such figures as tetrahedra (a type of pyramid), cubes, icosahedra, and many more; the possibilities, in fact, are infinite! H S M Coxeter's book is the foremost book available on regular polyhedra, incorporating not only the ancient Greek work on the subject, but also the vast amount of information that has been accumulated on them since, especially in the last hundred years. The author, professor of Mathematics, University of Toronto, has contributed much valuable work himself on polytopes and is a well-known authority on them.

Professor Coxeter begins with the fundamental concepts of plane and solid geometry and then moves on to multi-dimensionality. Among the many subjects covered are Euler's formula, rotation groups, star-polyhedra, truncation, forms, vectors, coordinates, kaleidoscopes, Petrie polygons, sections and projections, and star-polytopes. Each chapter ends with a historical summary showing when and how the information contained therein was discovered. Numerous figures and examples and the author's lucid explanations also help to make the text readily comprehensible. Although the study of polytopes does have some practical applications to mineralogy, architecture, linear programming, and other areas, most people enjoy contemplating these figures simply because their symmetrical shapes have an aesthetic appeal. But whatever the reasons, anyone with an elementary knowledge of geometry and trigonometry will find this one of the best source books available on this fascinating study.

**3.3. From the Preface to the 3rd edition.**

It is perhaps worthwhile to mention that the electron microscope has revealed icosahedral symmetry in the shape of many virus macromolecules. For instance, the virus that causes measles looks much like the icosahedron itself. The Preface to the First edition refers to a passage concerning the impossibility of any inorganic occurrence of this polyhedron. That statement must now be taken with a grain of borax, for the element boron forms a molecule B_12 whose twelve atoms are arranged like the vertices of an icosahedron.

**4. The Real Projective Plane (1949, 2nd ed. 1955, 3rd ed. 1993), H S M Coxeter.**

**4.1. From the Preface.**

This introduction to projective geometry can be understood by anyone familiar with high-school geometry and algebra. The restriction to real geometry of two dimensions makes it possible for every theorem to be illustrated by a diagram. The early books of Euclid were concerned with constructions by means of ruler and compasses; this is even simpler, being the geometry of the ruler alone. The subject is used, as metrical geometry was by Euclid, to reveal the development of a logical system from primitive concepts and axioms. Accordingly, the treatment is mainly synthetic; analytic geometry is confined to the last two of the twelve chapters.

The strict axiomatic treatment is followed far enough to show the reader how it is done, but is then relaxed to avoid becoming tedious. Continuity is introduced in Chapter 3 by means of an unusual but intuitively acceptable axiom. A more thorough treatment is reserved for Chapter 10, at which stage the reader may be expected to have acquired the necessary maturity for appreciating the subtleties involved.

The spirit of the book owes much to the great Projective Geometry of Veblen and Young. That dealt with geometries of various kinds in any number of dimensions; but the present book may be found easier because one particular geometry has been extracted for detailed consideration. Chapters 5 and 6 constitute what is perhaps the first systematic account in English of von Staudt's synthetic approach to polarities and conics as amplified by Enriques: A polarity is defined as an involutory point-to-line correspondence preserving incidence, and a conic as the locus of points that lie on their polars, or the envelope of lines that pass through their poles. This definition for a conic gives the whole figure at once and makes it immediately self-dual, a locus and an envelope, whereas Steiner's definition assigns a special role to two points on the conic, obscuring its essential symmetry. Moreover, the restriction to real geometry makes it desirable to consider not only the hyperbolic polarities which determine conics but also the elliptic polarities which do not. The latter are important because of their application to elliptic geometry. (In complex geometry this distinction is unnecessary, for an elliptic polarity determines an imaginary conic.) The linear construction for the polar of a given point (5·64) was adapted from a question in the Cambridge Mathematical Tripos, 1934, Part II, Schedule A.

The treatment of conics is followed in Chapter 8 by a description of affine geometry, where one line of the projective plane is singled out as a line at infinity, enabling us to define parallel lines. It is interesting to see how much of the familiar content of metrical geometry depends only on incidence and parallelism and not on perpendicularity. This includes the theory of area; the distinction between the ellipse, parabola and hyperbola; and the theory of diameters, asymptotes, etc. The further specialisation to Euclidean geometry is made in Chapter 9 by singling out an absolute involution on the line at infinity.

Chapter 10 introduces a revised axiom of continuity for the projective line, so simple that only eight words are needed for its enunciation. (This has not been published elsewhere save as an abstract in the Bulletin of the American Mathematical Society.) Chapter 11 develops the formal addition and multiplication of points on a conic and the synthetic derivation of coordinates. Finally, Chapter 12 contains a verification that the plane of real homogeneous coordinates has all the properties of our synthetic geometry. This proves that the chosen axioms are as consistent as the axioms of arithmetic.

Almost every section of the book ends with a group of problems involving the latest ideas that have been presented. All the difficult problems are followed by hints for solving them. The teacher can render them more difficult by taking them out of their context or by omitting the hints.

**4.2. From the Preface to the 2nd Edition.**

Why should one study the real plane? To this question, put by those who advocate the complex plane, or geometry over a general field, I would reply that the real plane is an easy first step. Most of the properties are closely analogous, and the real field has the advantage of intuitive accessibility. Moreover, real geometry is exactly what is needed for the projective approach to non· Euclidean geometry. Instead of introducing the affine and Euclidean metrics as in Chapters 8 and 9, we could just as well take the locus of 'points at infinity' to be a conic, or replace the absolute involution by an absolute polarity.

**4.3. From the Preface of the 3rd edition.**

Along with many small improvements, this revised edition contains van Yzeren's new proof of Pascal's theorem (§1.7) and, in Chapter 2, an improved treatment of order and sense. The Sylvester-Gallai theorem, instead of being introduced as a curiosity, is now used as an essential step in the theory of harmonic separation (§3.34). This makes the logical development self-contained: the footnotes involving the References (pp. 214-216) are for comparison with earlier treatments, and to give credit where it is due, not to fill gaps in the argument.

**5. Generators and Relations for Discrete groups (1957, 2nd ed. 1965, 3rd ed. 1972, 4th ed. 1980, 1984), H S M Coxeter and W O J Moser.**

For the Preface see THIS LINK.

**6. Introduction to geometry (1961, 2nd ed. 1968), H S M Coxeter.**

**6.1. From the Preface.**

For the last thirty or forty year, most Americans have somehow lost interest in geometry. The present book constitutes an attempt to revitalise this sadly neglected subject.

The four parts correspond roughly to the four years of college work. However, most of Part II can be read before Part I, and most of Part IV before Part III. The first eleven chapters (that is, Parts I and II) will provide a course for students who have some knowledge of Euclid and elementary analytic geometry but have not yet made up their minds to specialise in mathematics, or enterprising high school teachers who wish to see what is happening just beyond their usual curriculum. Part III deals with the foundations of geometry, including projective geometry and hyperbolic non-Euclidean geometry. Part IV introduces differential geometry, combinatorial topology, and four dimensional Euclidean geometry.

In spite of the large number of cross references, each of the twenty-two chapters is reasonably self-contained; many of them can be omitted on first reading without spoiling one's enjoyment of the rest.

...

The customary emphasis on analytic geometry is likely to give students the impression that geometry is merely a part of algebra or analysis. It is refreshing to observe that there are some important instances (such as the Argand diagram) in which geometrical ideas are needed as essential tools in the development of these other branches of mathematics. The scope of geometry was spectacularly broadened by Klein in his Erlanger Programm (Erlangen Programme) of 1872, which stressed the fact that, besides plane and solid Euclidean geometry, there are many other geometries equally worthy of attention. For instance, many of Euclid's own propositions belong to the wider field of affine geometry, which is valid not only in ordinary space but also in Minkowski's space-time, so successfully exploited by Einstein in his special theory of relativity.

Geometry is useful not only in algebra, analysis and cosmology, but also in kinematics and crystallography (where it is associated with the theory of groups), in statistics (where finite geometries help in the design of experiments), and even in botany. The subject if topology has been developed so widely that it now stands on its own feet instead of being regarded as part of geometry; but it fits into the Erlangen Programme, and its early stages have the added appeal of a famous unsolved problem: that of deciding whether every possible map can be coloured with four colours.

The material grew out of courses of lectures delivered at summer institutes for school teachers and others at Stillwater, Oklahoma; Luneburg, Nova Scotia; Ann Arbor, Michigan, Stanford, California; and Fredericton, New Brunswick, along with several public lectures given to friends of Scripta Mathematica in New York City by invitation of the late Professor Jekuthiel Ginsburg. The most popular of these separate lectures was the one on the golden section and phyllotaxis, which is embodied in Chapter II.

**6.2. From the Preface to the 2nd edition.**

I am grateful to the readers of the first edition who have made suggestions for improvement. Apart from some minor corrections, the principal changes are as follows.

The equation connecting the curvatures of four mutually tangent circles, now known as the Descartes Circle Theorem, is proved along the lines suggested by Mr Beecroft of "The Lady's and Gentleman's Diary for the year of our Lord 1842, being the second after Bissextile, designed principally for the amusement and instruction of students in Mathematics: comprising many useful and entertaining particulars, interesting to all persons engaged in that delightful pursuit."

For similarity in the plane, a new treatment was suggested by A L Steger when he was a sophomore at the University of Toronto. For similarity in space, a different treatment was suggested by Professor Maria Wonenburger. A new exercise introduces the useful concept of inversive distance. Another has been inserted to exhibit R Krasnodebski's drawings of symmetrical loxodromes.

Pages 203-208 have been revised so as to clarify the treatment of affinities (which preserve collinearity) and equiaffinities (which preserve area). The new material includes some challenging exercises. For the discovery of finite geometries, credit has been given to von Staudt, who anticipated Fano by 36 years.

Page 395 records the completion, in 1968, by G Ringel and J W T Youngs, of a project begun by Heawood in 1890. The result is that we now know, for every kind of surface the minimal number of colours that will suffice for colouring every map on the surface, though for anyone dissatisfied with a computer-generated proof, there remains a modicum of doubt in the case of the sphere (or plane).

**7. Projective Geometry (1964, 2nd ed. 1973, 1987, 1994), H S M Coxeter.**

**7.1. From the Preface.**

In Euclidean geometry, constructions are made with a ruler and compass. Projective geometry is simpler: its constructions require only the ruler. We consider the straight line joining two points, and the point of intersection of two lines, with the further simplification that two lines never fail to meet!

In Euclidean geometry we compare figures by measuring them. In projective geometry we never measure anything; instead, we relate one set of points to another by a projectivity. Chapter 1 introduces the reader to this important idea. Chapter 2 provides a logical foundation for the subject. The third and fourth chapters describe the famous theorems of Desargues and Pappus. The fifth and sixth make use of projectivities on a line and in a plane, respectively. In the next three we develop a self-contained account of von Staudt's approach to the theory of conics, made more "modern" by allowing the field to be general (though not of characteristic 2) instead of real or complex. This freedom has been exploited in Chapter 10, which deals with the simplest finite geometry that is rich enough to illustrate all our theorems nontrivially (for instance, Pascal's theorem concerns six points on a conic, and in PG(2, 5) these are the only points on the conic). In Chapters 11 and 12 we return to more familiar ground, showing the connections between projective geometry, Euclidean geometry, and the popular subject of "analytic geometry."

The possibility of writing an easy book on projective geometry was foreseen as long ago as 1917 when D N Lehmer wrote:

The subject of synthetic projective geometry is ... destined shortly to force its way down into the secondary schools.More recently, A N Whitehead recommended a revised curriculum beginning with Congruence, Similarity, trigonometry, Analytic geometry, and then:

In this ideal course of Geometry, the fifth stage is occupied with the elements of projective geometry.This "fifth" stage has one notable advantage: its primitive concepts are so simple that a self-contained account can be reasonably entertaining, whereas the foundations of Euclidean geometry are inevitably tedious.

**7.2. From the Publisher.**

In Euclidean geometry, constructions are made with a ruler and compass. Projective geometry is simpler: its constructions require only a ruler. In projective geometry one never measures anything, instead, one relates one set of points to another by a projectivity.

This classic book introduces the important concepts of the subject and provides the logical foundations, including the famous theorems of Desargues and Pappus and a self-contained account of von Staudt's approach to the theory of conics. The modern approach used in this account is then utilised to deal with the simplest finite geometry that is rich enough to illustrate all the theorems nontrivially. The book concludes by demonstrating the connections among projective, Euclidean, and analytic geometry.

**8. Geometry Revisited (1967), H S M Coxeter and S L Greitzer.**

**8.1. From the Preface.**

The mathematics curriculum in the secondary school normally includes a single one-year course in plane geometry or, perhaps, a course in geometry and elementary analytic geometry called tenth-year mathematics. This course, presented early in the student's secondary school career, is usually his sole exposure to the subject. In contrast, the mathematically minded student has the opportunity of studying elementary algebra, intermediate algebra, and even advanced algebra. It is natural, therefore, to expect a bias in favour of algebra and against geometry. Moreover, misguided enthusiasts lead the student to believe that geometry is "outside the main stream of mathematics" and that analysis or set theory should supersede it.

Perhaps the inferior status of geometry in the school curriculum stems from a lack of familiarity on the part of educators with the nature of geometry and with advances that have taken place in its development. These advances include many beautiful results such as Brianchon's Theorem (Section 3.9), Feuerbach's Theorem (Section 5.6), the Petersen-Schoute Theorem (Section 4.8) and Morley's Theorem (Section 2.9). Historically, it must be remembered that Euclid wrote for mature persons preparing for the study of philosophy. Until our own century, one of the chief reasons for teaching geometry was that its axiomatic method was considered the best introduction to deductive reasoning. Naturally, the formal method was stressed for effective educational purposes. However, neither ancient nor modern geometers have hesitated to adopt less orthodox methods when it suited them. If trigonometry, analytic geometry, or vector methods will help, the geometer will use them. Moreover, he has invented modern techniques of his own that are elegant and powerful. One such technique is the use of transformations such as rotations, reflections, and dilatations, which provide shortcuts in proving certain theorems and also relate geometry to crystallography and art. This "dynamic" aspect of geometry is the subject of Chapter 4. Another "modern" technique is the method of inversive geometry, which deals with points and circles, treating a straight line as a circle that happens to pass through "the point at infinity". Some flavour of this will be found in Chapter 5. A third technique is the method of projective geometry, which disregards all considerations of distance and angle but stresses the analogy between points and lines (whole infinite lines, not mere segments). Here not only are any two pants joined by a line, but any two lines meet a t a point; parallel lines are treated as lines whose common point happens to lie on "the line at infinity". There will be some hint of the content of this subject in Chapter 6.

Geometry still possesses all those virtues that the educators ascribed to it a generation ago. There is still geometry in nature, waiting to be recognised and appreciated. Geometry (especially projective geometry) is still an excellent means of introducing the student to axiomatics. It still possesses the aesthetic appeal it always had, and the beauty of its results has not diminished. Moreover, it is even more useful and necessary to the scientist and practical mathematician than it has ever been. Consider, for instance, the shapes of the orbits of artificial satellites, and the four-dimensional geometry of the space-time continuum.

Through the centuries, geometry has been growing. New concepts and new methods of procedure have been developed: concepts that the student will find challenging and surprising. Using whatever means will best suit our purposes, let us revisit Euclid. Let us discover for ourselves a few of the newer results. Perhaps we may be able to recapture some of the wonder and awe that our first contact with geometry aroused.

**9. Twelve Geometric Essays (1968, 1999), H S M Coxeter.**

**9.1. Note.**

The 1999 republication was given the title

*The Beauty of Geometry: Twelve Essays*.

**9.2. From the Publisher.**

Written by a distinguished mathematician, the dozen absorbing essays in this versatile volume offer both supplementary classroom material and pleasurable reading for the mathematically inclined.

The essays promise to encourage readers in the further study of elementary geometry, not just for its own sake, but also for its broader applications, which receive a full and engaging treatment. Beginning with an analytic approach, the author reviews the functions of Schlafli and Lobatschefsky and discusses number theory in a dissertation on integral Cayley numbers. A detailed examination of group theory includes discussion of Wythoff's construction for uniform polytopes, as well as a chapter on regular skew polyhedra in three and four dimensions and their topological analogues. A profile of self-dual configurations and regular graphs introduces elements of graph theory, followed up with a chapter on twelve points in PG(5, 3) with 95040 self-transformations. Discussion of an upper bound for the number of equal non-overlapping spheres that can touch another same-sized sphere develops aspects of communication theory, while relativity theory is explored in a chapter on reflected light signals.

Additional topics include the classification of zonohedra by means of projective diagrams, arrangements of equal spheres in non-Euclidean spaces, and regular honeycombs in hyperbolic space. Stimulating and thought-provoking, this collection is sure to interest students, mathematicians, and any math buff with its lucid treatment of geometry and the crucial role geometry can play in a wide range of mathematical applications.

**10. Mathematical Recreations and Essays (12th ed. 1974, 13th ed. 1987), W W Rouse Ball and H S M Coxeter.**

**10.1. Note.**

The first edition of this book with author W W Rouse Ball, appeared in 1892 with title

*Mathematical recreations and Problems*. A second edition appeared, also in 1892 and further editions 3rd 1896, 4th 1905, 5th 1911, 6th 1914, 7th 1917, 8th 1919, 9th 1920, 10th, 1922, 11th 1939. Following Rouse Ball's death in 1925, H S M Coxeter edited the 11th edition and became a co-author of the book by the 12th.

**10.2. From the Preface to the 1987 Dover reprint.**

Rouse Ball's first edition (with the slightly different title

*Mathematical recreations and Problems*) appeared nearly a hundred years ago. I am grateful to Dover Publications for undertaking to keep the book alive by publishing this new edition. I wish to thank also the many friends who have helped me to bring the material up to date. I have, of course, retained the "new" chapters X and XIV, so kindly contributed by J J Seidel (for the twelfth edition) and Abraham Sinkov (for the eleventh).

During the 61 years since Rouse Ball died, mathematical knowledge has increased enormously, but most of the results that interested him still remain valid. Among arithmetical recreations, his lists of Mersenne primes and Fermat composites have been extended with the aid of the computer. ...

A remarkable convergence of pure and applied mathematics arose about 1980, when the geometric concept of a "quasi-lattice" was discovered just in time to provide a possible explanation for the physical observation of "quasicrystal."

**11. Regular Complex Polytopes (1974, 2nd ed. 1991), H S M Coxeter.**

**11.1. From the Preface.**

This book has occupied much of my time and attention for nearly twenty years. It was inspired by the dissertation of G C Shephard, which had the same title. I have made an attempt to construct it like a Bruckner symphony, with crescendos and climaxes, little foretastes of pleasure to come, and abundant cross-references. The geometric, algebraic and group-theoretic aspects of the subject are interwoven like different sections of the orchestra.

Its relation to my earlier

*Regular Polytopes*resembles that of

*Through the Looking-Glass*to

*Alice's Adventures in Wonderland*. The sequel is more profound; it is essentially self-contained, but some of the characters reappear with recognisable but slightly changed names, and there are many new characters of the same sort, but even more fantastic.

The term complex polytope was first used by D M Y Sommerville in 1928. His 'complex polygon' may have more than two vertices on an edge, or more than two edges at a vertex. He seems to have used the word 'complex' in its colloquial sense without noticing how natural the idea becomes when the coordinates are complex numbers and a Hermitian form is used to define a unitary matrix.

**11.2. From the Publisher of the 2nd edition.**

The properties of regular solids exercise a fascination which often appeals strongly to the mathematically inclined, whether they are professionals, students or amateurs. In this classic book Professor Coxeter explores these properties in easy stages, introducing the reader to complex polyhedra (a beautiful generalisation of regular solids derived from complex numbers) and unexpected relationships with concepts from various branches of mathematics: magic squares, frieze patterns, kaleidoscopes, Cayley diagrams, Clifford surfaces, crystallographic and non-crystallographic groups, kinematics, spherical trigonometry, and algebraic geometry. In the latter half of the book, these preliminary ideas are put together to describe a natural generalisation of the Five Platonic Solids. This updated second edition contains a new chapter on Almost Regular Polytopes, with beautiful 'abstract art' drawings. New exercises and discussions have been added throughout the book, including an introduction to Hopf fibration and real representations for two complex polyhedra.

**12. Zero-symmetric Graphs (1981), H S M Coxeter, R Frucht and S L Powers.**

**12.1. From the Preface.**

At the Conference on Graph Theory and Combinatorial Analysis held at the University of Waterloo in 1966, Ronald Foster presented a

*Census of trivalent symmetrical graphs*, a draft of which was distributed to a dozen colleagues. ("Symmetrical" means edge-transitive as well as vertex-transitive.) In the same year, in a letter to the first author of this book, Foster suggested the study of those finite trivalent graphs whose automorphism group acts regularly on the vertices, coining for them the term "0-symmetric." Loosely speaking, these are trivalent graphs that are just vertex-transitive, in the sense that they have no further symmetry.

In 1975, in

*Notes*distributed again only to a reduced number of friends, he began the study of these 0-symmetric graphs and also of the "t-symmetric" graphs, which represent an intermediate class between the 0-symmetric and the symmetrical. In particular, he studied the most numerous family of 0-symmetric graphs, those whose automorphism group is isomorphic to a dihedral group. In Table 22.1 we list, from Foster's work, the 350 graphs of this type having not more than 120 vertices (the upper limit we have fixed, somewhat arbitrarily, for this study). For these and other contributions, the authors dedicate this book to him.

We also wish to acknowledge the contributions of Mark Watkins, who found the first examples of the graphs studied in Sections 23 and 24.

From the preceding it should already be clear to the reader that the aim of this book is to describe all of the 0-symmetric graphs with not more than 120 vertices that we have found during several years of intensive search. In spite of our intentions, we very likely have overlooked some 0-symmetric graphs or erroneously included some that are not 0-symmetric because of a hidden symmetry. We will be most grateful if a reader finding any omission or error would communicate the facts to any or all of the authors.

Last Updated July 2020