# Donald Coxeter's Books - Reviews

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

For extracts from Prefaces and from Publisher's information 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), 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, 6th ed. 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), 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)

*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 Prefaces and from Publisher's information 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), 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, 6th ed. 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), 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)

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

**1.1. Review by: Jeffrey Charles Percy Miller.**

*The Mathematical Gazette*

**23**(253) (1939), 105-107.

This pamphlet describes the application of the process of stellation to the regular solids. By stellation is meant the derivation of one polyhedron from another by extension of the faces of the original until they meet other such extensions, to form new faces, edges, and vertices. The definition is modified so as to include cases in which a "face" of the derived solid consists of several isolated portions all, of course, in the same plane. This process evidently leaves the number of such faces unaltered. In the paper reviewed here, restrictive rules have been formulated which mean, in effect, that the final polyhedron must retain the full symmetry of the parent regular solid, except possibly for reflexive symmetry. Thus the rules allow a solid which is not identical with its reflection; such a solid is called, rather unhappily,

*unselfrefiexible*. If identical with its reflection, the solid is called

*selfreflexible*. ...

The reasoning used in the derivation and description is clear and fairly elementary. Quite a number of the solids provide excellent opportunities for making simple models, whilst others, for example those which are merely vertex-connected, require all the tricks which a model-maker can devise. Although no mention of this is made in the paper, I believe I am right in saying that models of all the icosahedra have been constructed by Flather.

**1.2. Review (of 3rd edition) by: Henry Martyn Cundy.**

*The Mathematical Gazette*

**86**(506) (2002), 360-361.

This is an historic paper in the field of polyhedral stellations. Popularly, any polyhedron with 'starry' pointed vertices tends to be called 'stellated', but strictly a stellation is produced by extending the planes of a convex polyhedron until non-adjacent planes intersect - not simply by adding pyramids of any angle to their faces. Of the five convex regular polyhedra, the tetrahedron and cube obviously have no stellations. The octahedron has one, Kepler's

*stella octangula*, a compound formed by two tetrahedra which are images in their circumcentre. The dodecahedron has three, all regular if we give the pentagram its deserved status as a regular polygon. These are the

*small*and

*great stellated dodecahedra*with pentagram faces, and the great dodecahedron with pentagonal faces and pentagrammatic vertices. So far, so good. But when we come to the icosahedron each face can be cut by 18 others and the situation becomes very complicated. The best-known stellation is the

*great icosahedron*with equilateral triangular faces and pentagrammatic vertices, which completes the list of nine regular polyhedra in 3D, but there are many others. This was a challenge to Coxeter whose 'youthful exuberance' (as he recently expressed it) called in the help of three school and college friends to produce models of 59 stellations and a paper to explain them, published in Toronto in 1938. Patrick Du Val brought out a second edition published by Springer in 1982 and now Tarquin have reprinted this as a third edition with text copied electronically, together with newly calculated plates, a biographical foreword and a fascinating preface by Coxeter himself.

**2. Non-Euclidean Geometry (1942, 2nd ed. 1947, 3rd ed. 1957, 4th ed. 1961, 5th ed. 1965, 1968, 1978, 6th ed. 1998), H S M Coxeter.**

**2.1. Review by: Daniel Pedoe.**

*The Mathematical Gazette*

**27**(273) (1943), 35-36.

This second volume of the series "Mathematical Expositions" is learned, readable and attractive. All who are interested in geometry should browse over it. It is one of the mathematical books I should like to have in my pocket if Fate ever consigns me to a prison-camp (the present-day equivalent of a desert island). ...

The introductory chapters on real projective geometry are a model of clear ordered and powerful reasoning which should make the connoisseur purr with pleasure. (Incidentally, a reform in the teaching of projective geometry in our schools and colleges is long overdue. The way is very clearly indicated in Coxeter's opening chapters. A textbook which has profoundly influenced geometrical teaching in the U.S.A., Veblen and Young's

*Projective geometry*, has never "arrived" over here. Perhaps a book written in Canada by an English admirer of Veblen may meet with a happier fate.)

The various non-euclidean geometries are introduced synthetically and sympathetically and the reader soon finds himself led by his expert guide into brave new worlds constructed, as the references well show, by a truly international corps of mathematicians. On reaching the end of the book one feels that with its ample text, numerous references and excellent bibliography, it should be the standard textbook on non-euclidean geometry for a long time to come.

It is to be hoped that copies of this and other volumes in the series of "Mathematical Expositions" will soon be available in our bookshops. At the moment textbooks from across the Atlantic are evidently not given a high priority as cargo. There should be a little room between the cheeses for some mental food as well. If not, one would willingly sacrifice some of the cheese ration for the privilege of being able to read, on the rare occasions when there is a little energy left over from the common task, a book as stimulating and as satisfying as the work under review.

**2.2. Review by: Leonard Mascot Blumenthal.**

*Bull. Amer. Math. Soc.*

**29**(9) (1943), 679-680.

There seems to be a well established pattern for books on the non-Euclidean geometries, according to which a more or less elaborate historical sketch is followed by a development of the foundations of the geometries. There is usually little space left available for developing the geometries much beyond the foundations. Thus it not infrequently happens that many interesting results not intimately connected with the beginnings of the subject are declared "beyond the scope of this book."

Though the plan of the book under review presents no radical departure from such a pattern it does offer somewhat more of the subject proper than is usual, while the manner in which it accomplishes its aims sets a new high standard for such texts.

...

The book is heartily recommended as furnishing a very well written account of the fundamental principles of hyperbolic and elliptic geometry from the classical point of view. This was doubtless the author's aim. The reviewer feels constrained to observe, however, that as a modern treatment of an old subject, the book might have recognised some of the contributions to its field made available during the last few years. Thus to the three traditional avenues of approach to the non-Euclidean geometries there has been added a fourth - the abstract metric approach - which injects into the rigour of the classical axiomatic methods the stimulus of a rapid development. Still more within the spirit of the book, it seems, would have been a notice of the foundations of hyperbolic geometry due to Menger, Jenks, and Abbott based upon the sole operations of joining and intersecting.

**2.3. Review by: Henry Thomas Herbert Piagio.**

*Nature*

**151**(23 January 143), 94.

Dr Coxeter's book starts with an excellent introductory chapter, mainly historical. Chapters 2 and 3 are on projective geometry, and instead of defining polarity with reference to a conic, follow von Staudt by defining polarity as "a correlation of period two" and then look for a corresponding conic. Chapter 4 is on homogeneous coordinates. After these somewhat prolonged preliminaries, elliptic geometry is obtained from projective geometry, since every axiom of the former is valid in the latter. This idea is developed in Chapters 5, 6 and 7. In Chapters 8 and 9, Euclidean and hyperbolic geometry are derived from a general 'descriptive' geometry. The remaining five chapters are simpler, dealing with hyperbolic geometry in two dimensions circles and triangles, a general triangle of reference, rea, and Euclidean models.

**2.4. Review (of 6th edition) by: Tony Gardiner.**

*The Mathematical Gazette*

**86**(506) (2002), 364-365.

'Geometry is dead. Long live geometry!' For much of the 20th century, the first of these contrasting claims dominated. And for much of that time, H S M Coxeter remained one of the central figures who repeatedly showed - by example - that the geometrical viewpoint remains one of our most powerful ways of comprehending the mental universe of mathematics. Fortunately his very long life has allowed him to see his quiet stubbornness vindicated, as one mathematical domain after another has found fresh reasons to rediscover the geometrical viewpoint.

This new edition of

*Non-Euclidean geometry*(the original version was first published in 1942) is largely a straight reprint. However, a small number of corrections have been made; the introductory pages, a number of sections in the heart of the book, and the final bibliography have been mildly revised; and a completely new final section has been added. All of these changes have been effected using a well chosen modern font which enhances the overall effect.

...

Any book which lasts into a sixth edition must be a 'standard'. However, the potential reader should not be misled into believing that this is a transparent beginner's introduction. As with so many of Professor Coxeter's books, the reader is not only expected to work to fill in details, but is also assumed to have a high level of maturity in distinguishing between apparently isomorphic statements of 'fact' - some of which are indeed evident from the preceding text, while others can only be clarified with a considerable amount of additional work (and some even appear to be true only modulo some unstated result which may be clear to experts, but which can leave the beginner floundering).

...

Thus, this book would seem to presume a reader who is either ready to work fairly hard, or who has already made a moderately serious study of non-Euclidean geometry.

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

**3.1. Review by: Joseph Leonard Walsh.**

*Scientific American*

**181**(2) (1949), 58-59.

The early history of the polyhedra-geometrical figures bounded by portions of planes is "lost in the shadows of antiquity." The famous five regular solids, three of which, together with other polyhedra, occur in nature as crystals, were studied by the Pythagoreans, by Plato, and, more exhaustively, by Euclid. Excavations on Monte Loffa, near Padua, have uncovered an Etruscan dodecahedron, showing that this figure "was enjoyed as a toy at least 2,500 years ago." A law of symmetry related to Haüy's crystallographic "Law of Relativity" asserts the impossibility of the "inanimate occurrence of any pentagonal figure, such as the regular dodecahedron"; Thus while the tetrahedron, cube and octahedron are found as crystals, the dodecahedron and icosahedron cannot form crystals but need "the spark of life to occur naturally." Professor Coxeter of Toronto University, known for his revision of Ball's

*Mathematical Recreations and Essays*and for his papers on geometry, has assembled a treasure house of information on the two-, three- and four-dimensional polytopes. In addition to examining thoroughly, with the use of both elementary and advanced methods, the properties of ordinary and multidimensional figures, Coxeter has enriched this volume by interesting innovations of his own, numerous cuts and plates, historical summaries, tables and a useful bibliography. Of the last-named Coxeter remarks that the listing of the names of 30 German mathematicians, 27 British, 12 American, 11 French, 7 Dutch, 8 Swiss, 4 Italian, 2 Austrian, 2 Hungarian, 2 Polish, 2 Russian, 1 Norwegian, 1 Danish and 1 Belgian, "provides an instance of the essential unity of our Western civilisation, and the consequent absurdity of international strife."

**3.2. Review by: Jeffrey Charles Percy Miller.**

*Science Progress (1933-)*

**37**(147) (1949), 563-564.

This is a fascinating and stimulating book. The convex regular polyhedra, the five Platonic solids, have been known from time immemorial, but further developments as described in this book did not occur until much later. Kepler discovered two of the four regular star polyhedra - the two which have pentagrams (the simplest star polygon) as faces ; the two other star polyhedra, with triangles and pentagons as faces, but with star-shaped vertices, were only discovered about two centuries later, by Poinsot. Extension of the ideas connected with polygons and polyhedra to polytopes in four and more dimensions seems first to have originated with Schläfli, before 1853, but many other workers have studied such polytopes since then - mainly in isolation until the researches of Coxeter, into polytopes, and into the literature of polytopes, brought some unification to all these efforts. He has been indefatigable in searching the literature, in tracking down theses and unpublished work, and in making personal contact with workers in this field - British, Continental or American. The book now under review summarises the work of Coxeter and others to an extent not previously accomplished; he is meticulous in his tributes to earlier workers, although he rediscovered a very large part of the field himself before earlier work had been unearthed. The field is vast, so that this work is concentrated on regular polytopes, though other kinds are dealt with freely where they throw light on the regulars, and in such a way that exciting side tracks for further exploration are glimpsed, and in some cases partially explored

**3.3. Review by: Henry Martyn Cundy.**

*The Mathematical Gazette*

**33**(303) (1949), 47-49.

Those who have read the chapter on Polyhedra in the revised edition of Rouse Ball's

*Mathematical Recreations and Essays*will need no introduction to the author or his subject. Indeed, the book has been long awaited, and those who have ordered it, undaunted by the price, will have found that it has been worth the waiting. The mark of a great musical composition is the continual renewing and heightening of the brilliance and inspiration of its opening bars. It is also one of the marks of this book, though fewer may appreciate it in the more esoteric mathematical medium. The author obligingly introduces us to a new notation for his "music of the spheres", and the reader is invited to sample its potentialities on at the climax of a long and fascinating development.

**3.4. Review by: Alfred J Frueh Jr.**

*The Journal of Geology*

**58**(6) (1950), 672.

That a review of this text on a purely geometric subject should find its way into the

*Journal of Geology*might at first seem strange. Perhaps the title itself will attract the mineralogist and morphological crystallographer, as a few common crystal forms (tetrahedron, cube, and octahedron) can be classed as regular polyhedra. However, it is the chapters on rotation groups and on tessellations and honeycombs that should command the greatest attention of the structural crystallographers.

Professor Coxeter is not presenting a monograph on any original work but, rather, is giving us a textbook. The first two chapters are concerned mainly with the properties of regular and quasi-regular solids and the construction of graphs and maps. The third chapter is entitled "Rotation Groups" and presents a development of transformations and symmetry operations paralleled by a brief but adequate development of elementary group theory. The remaining portion of the chapter, as well as subsequent chapters on tessellations, honeycombs, and the kaleidoscope, show the applications of group theory to symmetry operations; and the limitations imposed upon discrete groups of displacements in a plane and in space are discussed. The presentation of this material, as well as the order in which it is discussed, is considerably different from the more rigorous derivations of the crystal classes and space groups more familiar to the average mineralogist. This in itself is refreshing and could give one a broader insight into the geometry of lattices. The last eight chapters of the book are devoted to the development and properties of figures in n-dimensional space.

Since the text is not designed especially the crystallographer, it would be unfair to criticise the fact that a derivation of the crystal classes or space groups is not presented beyond a development of group theory and a few applications to symmetry operations. However, much of the writing lacks clarity.

**3.5. Review by: Carl Barnett Allendoerfer.**

*Bull. Amer. Math. Soc.*

**55**(1949), 721-722.

Coxeter has spent a major portion of his mathematical career digging out the obscure references in early works, in making personal contact with contemporary gifted amateurs, and in developing his own outstanding contributions to the field. In this book he has poured forth all his devotion and scholarship and has produced a work which will be the standard treatise in this field for many years. It is beautifully illustrated with photographs of Donchian's models and with numerous drawings. Its value as a reference book is greatly enhanced by historical material at the end of each chapter, by tables giving the essential combinatorial and metric properties of polytopes of many varieties, by an exhaustive bibliography, and by a carefully constructed index. It is a particular pleasure to record this last feature; for its omission in so many mathematical books published in England greatly detracts from their value.

In his preface Coxeter follows the lead of Birkhoff-MacLane and says: "Anyone familiar with elementary algebra, geometry, and trigonometry will be able to appreciate this book." In a literal sense this is true, but let no one be deceived - this is a serious book, full of advanced ideas, and worthy of careful study by professional mathematicians. In the elementary section Coxeter gives a brief résumé of the Platonic solids, and then discusses other solids related to these which, though not regular, have many regular features. Examples are the cuboctahedron, the rhombic dodecahedron, and the zonohedra.

The serious mathematics begins with the third chapter in which Coxeter introduces the symmetry groups of the Platonic solids. After a full discussion of this important topic, he turns to degenerate polyhedra such as tessellations and honeycombs and their groups. These lead to results of crystallographic importance. Under the heading "The Kaleidoscope" he then describes the discrete groups generated by reflections. The exposition is greatly illuminated by his own "graphical notation" which makes complicated relations self-evident. The treatment of three-dimensional solids closes with a chapter on regular solids which are not quite polyhedra in the strict sense. These are obtained from the Platonic solids by "stellating" (adding pointed solid pieces) or "faceting" (removing solid pieces). This process raises the number of regular polyhedra from five to nine.

The remaining two-thirds of the book is devoted to polytopes of higher dimensions. The general program is similar to that carried out for ordinary polyhedra. It is shown that in four-space there are six regular polytopes and that in n-space (n > 4) there are only three regular polytopes. Explicit constructions are given for these and metrical properties are derived. There are even photographs of models of three-dimensional projections of some of these hypersolids. The methods used are increasingly analytic, but the underlying geometry is never lost among the formulae.

**3.6. Review by Harold E Wolfe.**

*The American Mathematical Monthly*58 (2) (1951), 119-120.

Professor Coxeter has collected and arranged the scattered material, enriched it with the results of his own research done over a period of twenty-five years, and has presented us with what is, and will probably be for many years, the only organised treatment of the subject. For this reason we should be glad that the work has been done by such a competent scholar and by one who is doubtless more thoroughly acquainted with this field than anyone else.

...

This book belongs in every university, college and high school library, and on the bookshelf of every mathematician. It is a book to read, to refer to and to recommend to students.

**3.7. Review (of 2nd edition) by: Michael Goldberg.**

*Mathematics of Computation*

**18**(85) (1964), 166.

This second edition is essentially the same as the first edition of 1948, but in paperback by another publisher, with slightly larger pages and appreciably larger plates. Corrections and minor additions have been made, and six pages have been revised.

It still remains the most extensive and authoritative summary of the derivations and enumeration of the n-space generalisations of the regular and quasi-regular polyhedra. It includes their metric, topological, and group properties, and the history of their development. Although the subject of polyhedra is quite ancient, new discoveries concerning these polytopes have been made since the first edition, many by the author. Some of the new work is mentioned in the text and in the extensive bibliography.

**3.8. Review (of 2nd edition) by: Peter Yff.**

*Canadian Mathematical Bulletin*

**8**(1) (1965), 124.

This excellent book has now been issued in paperback form, with a few changes since the first edition.

On page 74 the number h of sides of the Petrie polygon of {p, q} is expressed rationally in terms of p and q. On pages 228-232 there is a direct proof that the number of reflections in a symmetry group generated by four reflections is not less than 2h. These improvements result from recent work by R Steinberg.

Several figures have been re-drawn, and the plates have been enlarged in accordance with a somewhat larger page size. The bibliography has been brought up to date.

**3.9. Review (of 3rd edition) by: Magnus J Wenninger.**

*Leonardo*

**9**(1) (1976), 83.

Here is a book that is richly rewarding for those who wish to pursue the fascinating mathematical study of regular polytopes. Within its pages lies buried a mine of information, but as this metaphor implies, it calls for patient, painstaking penetration of the mind to arrive at the nuggets hidden within it. The interesting thing about this particular topic in the vast world of mathematics generally is that these nuggets do in fact have an appeal that attracts the eye as much as the mind. Or to put this in another way, this book deals with geometrical figures, lines, planes, solids and hypersolids, which have always called for drawings or models as aids for the mind to grasp the multiple interrelationships and symmetries that elude imagination left to itself. This introduces an aesthetic appeal that brings mathematics and art into a conjunction, enriching both. ...

So come, feast your mind on a book that undoubtedly will remain a classic in its field, authored by a man who has spent a lifetime of study and research, making him the foremost geometer of our times. This book will continue to bring truth and beauty to many of the present generation and to peoples yet unborn.

**3.10. Review (of 3rd edition) by: Tricia Muldoon Brown.**

*Mathematical Association of America*(30 October 2016).

https://www.maa.org/press/maa-reviews/regular-polytopes

Regular Polytopes is densely packed, with definitions coming rapid-fire and results following quickly .... Years of results are elegantly summarised with just enough details for clarity, but not so many as to increase the length to a burdensome amount. Most of the chapters are definition-heavy, but still very readable. The key vocabulary is italicised with definitions given more casually within the narrative rather than set apart in a formal style. Similarly, theorems, propositions, and proof also occur naturally in the text, with section headings giving reference to the theorem or proof being addressed.

The readability is further enhanced by the consistent use of concrete examples with each new topic. For instance, reflection groups are illustrated with mirrors (Chapter 5) and illustrations or diagrams of the polytopes are given throughout the text when possible. These illustrations and figures are not flashy, but are good, clear, and effective.

The book was last revised in 1973, so it is occasionally out-of-date, although not frustratingly so. For example, in Chapter 5, Coxeter discusses the "novelty" of the use of Dynkin diagrams. In Chapter 1, we are reminded that the four-colour problem is "unanswered." Vocabulary has also experienced some changes over time, i.e. Coxeter's use of "reciprocal" polytopes, which in more recent times are usually referred to as "dual" polytopes. Rather than detracting from the text, I found that these occasional differences give insight into the mathematical progress that has been made in the last half-century.

One of my favourite parts of this book are the historical remarks found at the end of each chapter. Coxeter carefully associates the results from the chapter with the major contributing mathematicians, but also adds a few interesting details setting the context for the mathematics. Personal details are also included, both about the mathematicians and occasionally about Coxeter himself; I particularly enjoyed reading about his acquaintance with Alicia Boole Stott in Chapter 13.

Overall, like the illustrations and diagrams, the book provides a well-written and comprehensive coverage of regular polytopes that is clear and effective without being elaborate or excessively detailed. Further, the historical perspective, found at the end of each chapter as well as in the treatment of the topics, gives this book a distinctly more entertaining style than a standard mathematical textbook.

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

**4.1. Review by: Ralph G Sanger.**

*Mathematics Magazine*

**23**(2) (1949), 100-101.

At the present time the subject of Synthetic Projective Geometry is an area in which few mathematicians in this country are vitally interested. As a result, it is a pleasure to see a new work in this field which presents the material in a clear, rigorous, interesting manner. The book is clearly written, the printing is excellent, and the numerous diagrams are both clearly drawn and distinctly labelled.

The first seven chapters of this work deal with material covered in a first course in Synthetic Projective Geometry. Enough of a postulational basis is given to show the need for and use of, such a treatment. Conics are introduced via self conjugate points of a hyperbolic polarity. Emphasis is placed on the concept of correspondence and its relation to the theory of transformations.

The latter chapters deal with the specialisation of a projective geometry into an affine and a metric geometry, the concept of continuity, and an introduction into the study of coordinate systems, with especial emphasis on homogeneous projective coordinates.

**4.2. Review by: Daniel Pedoe.**

*The Mathematical Gazette*

**34**(308) (1950), 142-143.

This excellent book fills a gap in the ranks of geometrical textbooks. There are many on complex projective geometry, and some devote space to the axioms of real projective geometry, but few textbooks make the investigation of real projective geometry their main purpose. Although one of Professor Coxeter's aims is to show that all the properties we expect to find in the real projective plane can be derived from suitable axioms, he is careful not to bore his readers at the outset by insisting too much on the purity of his intentions. ....

... this is an admirable book, and a delight to read. It has, in the reviewer's opinion, only one flaw: the author uses "wo" for "with respect to", and says, in a footnote, "The preposition

*wo*(pronounced like 'woe') has been coined by some English mathematicians as a convenient abbreviation for 'with respect to' or 'with regard to'." It is difficult to see why

*wo*is more convenient as an abbreviation than "w.r.t."; if writing or saying "with respect to" makes too great a demand on time and energy, surely it would be better to adopt a symbol. ... Perhaps Professor Coxeter will lighten our gloom by getting rid of

*wo*from the new edition of his book which will surely soon be demanded.

**4.3. Review by: C N S.**

*Current Science*

**18**(10) (1949), 384-385.

This book presents the subject-matter of synthetic projective geometry in a very lucid and simple manner, developing the subject by a carefully chosen set of axioms of incidence and order. The development of the subject is primarily based on Von Staudt's definitions of projectivity and the conic. A projectivity between two ranges is a correspondence that transforms a harmonic set into a harmonic set. An involutary correlation or point-line correspondence is called a polarity and a conic is the locus of points that lie on their polars, or the envelope of lines that pass through their poles. The equivalence of these definitions with Poncelet's definition of projectivity and Steiner's definition of a conic is worked out in a very elegant manner. There is a chapter dealing with "generalised projectivity" on a conic, and with the theorem that any projectivity on a conic determines a collineation of the whole plane. There are two brief chapters giving the fundamental ideas of affine geometry, which is projective geometry minus the line at infinity and Euclidean geometry which is affine geometry possessing an orthogonal involution. A list of properties of conics which can lie considered as affine properties is worked out and properties of circles and some properties of conics are developed by the above conception of Euclidean geometry. The latter chapter fittingly ends with the proof of the focus-directrix property of a conic. In the last two chapters, "Analytical Geometry" is introduced in terms of the axioms and concepts of synthetic geometry, and is comparable to similar treatment in Veblen and Young's

*Projective Geometry*.

The handy size of the book makes it a convenient text-book for explaining the fundamental concepts of synthetic geometry, after which the student can proceed to the vast sea of geometrical properties of the conic sections in other books wherein the methods of projective and metrical geometries are freely mixed up.

**4.4. Review by Patrick Du Val.**

*Bull. Amer. Math. Soc.*

**56**(4) (1950), 376-378.

This book is an admirable introduction to the subject for students who know a fair amount of ordinary plane geometry, including at least something about conies, but have no idea at all of projective geometry. ...

...

... the work is severely and carefully argued from beginning to end, and ... within its limitations to the real field and to two dimensions it covers just about everything that one could think of including. The shelving of the serious discussion of continuity to a late stage, by assuming one of its chief results as a temporary axiom, probably makes greatly for the intelligibility of the book to beginners. A great number of admirably clear diagrams (probably more than one to every page on an average) illustrate the ideas. The proofs are lucid, and in nearly every case lay bare the fundamental ideas that are being used rather than obscuring these in a mass of detail. The whole book, indeed, is most readable; there are interesting historical notes on the genesis of the ideas presented and a very good bibliography. An appendix of only a couple of pages briefly indicates the nature of the step from real to complex geometry.

**4.5. Review (of 2nd edition) by: Daniel Pedoe.**

*The Mathematical Gazette*

**40**(332) (1956), 153.

This book was heartily recommended when it first appeared in an American imprint, and there is no need to add anything except to say that the author has removed certain minor blemishes of terminology which disturbed at least one reviewer.

It was suggested in the former review that this book should be studied before any systematic study of complex projective geometry is undertaken. Cambridge University could help by setting scholarship questions on real projective geometry, and lecturing on the subject. The geometric interpretation of algebraic equations is not the whole of geometry, and a change of emphasis is long overdue. If it should come, here is the book for the student to read and enjoy.

**4.5. Review (of 2nd edition) by: V.**

*Current Science*

**30**(8) (1961), 314-315.

This is an introductory university text-book on projective geometry, including a thorough treatment of conics and a rigorous presentation of the synthetic approach to co-ordinates. The restriction to real geometry of two dimensions makes it possible for every theorem to be adequately represented by a diagram. The subject is used to illustrate the development of a logical system from primitive concepts and simple axioms. Accordingly the treatment is mainly synthetic: analytic geometry is confined to the last two of the twelve chapters. The eighth and ninth chapters show how projective ideas can be used as a basis for metrical geometry.

In this second edition, several errors contained in the first edition have been corrected. There is an improved treatment of degenerate polarities, of the inside and outside of a conic, of the condition under which a quadrangle may be convex with respect to a line, and of Klein's classification of geometries according to the groups of transformations under which their properties are invariant.

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

**5.1. Review by: Jakob Nielsen.**

*Mathematica Scandinavica*

**5**(2) (1958), 291-293.

As is well known, the mathematical literature contains a rich material of abstract characterisations of given groups through generating and defining relations. It is to be welcomed that H S M Coxeter, whose previous work has contributed so much to this material, has now, in cooperation with W O J Moser, expanded the available knowledge in this area, especially regarding the groups of finite order, systematically arranged and collected it. The 133 pages of text contain an abundance of material; the presentation is brief, and some readers will use the abundant literature to help elucidate more difficult parts by reading the more detailed original papers. An essential aid for the reader is the tabular compilations given at the end, which also facilitate the use of the literature by specifying the names used by different authors for the same groups. If you want to spare your memory, you will see the table 12 summarising the names used by the authors for the groups used throughout. The book ends with a very detailed bibliography and an index.

**5.2. Review by Marshall Hall.**

*Bull. Amer. Math. Soc.*

**64**(3.1) (1958), 106-108.

It is refreshing to find a book that not only studies groups but also deals with many particular and interesting groups. The major theorems of group theory have substance only insofar as they apply to actual groups. The Mathematician with any feeling for groups will welcome this monograph and its rich display of groups pf many kinds.

...

There are two main faces to the study of finitely presented groups. The obverse is the problem of studying the properties of a group defined by given relations. Among other things we wish to know if the group is finite and if so, what its order is. The reverse is the problem of finding a simple set of defining relations for a given group. Both these problems are studied in this monograph, and a variety of methods, mostly geometrical, are employed. Since the word problem for groups is unsolvable, we are relieved of the necessity of searching for an all embracing method and may enjoy the elegance of several diverse approaches.

**5.3. Review (of 2nd edition) by: Wilhelm Magnus.**

*Quarterly of Applied Mathematics*

**24**(3) (1966), 285.

This is the second edition of a monograph that appeared first in 1957. The new edition differs from the first one by minor corrections and additions and by a newly inserted brief account of the use of electronic computers for enumerating cosets in a finite abstract group.

The appearance of a second edition is a testimonial to the great achievement of the authors who managed to organise what, at first sight, looks like a complete and nearly infinite chaos. Their success is due in part to the use of geometrical ideas as a guiding principle. To the applied mathematician, the chapters on abstract crystallography and on groups generated by reflections are likely to be the most useful ones. The chapters on the symmetric, alternating, modular, and linear fractional groups provide an abundance of technical information on these important groups. Nine pages of tables and fourteen pages of references facilitate orientation and access to additional information.

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

**6.1. Review by: Eric John Fyfe Primrose.**

*The Mathematical Gazette*

**48**(365) (1964), 343.

There has been a tendency during this century for geometry to become more and more algebraic. The algebraic method is admittedly very useful, but frequently the geometrical ideas tend to get lost. Professor Coxeter's aim has been to write a book about geometry, though he does not hesitate to use coordinates where necessary.

As the author says in his preface, "the unifying thread that runs through the whole work is the idea of a group of transformations or, in a single word, symmetry." In accordance with this idea, he devotes three chapters of Part I to congruent and similar transformations in two and three dimensions. This is, as far as I know, the only complete account of this work in English and is, in my view, one of the most important parts of the book. It gives a dynamic, as opposed to a static, view of geometry. These transformations are applied to such topics as crystallography, tessellations, and repeating patterns in art.

Part II is concerned with various topics such as complex numbers and their representation on the Argand diagram, regular solids, and golden section, with its application to phyllotaxis.

In part III various geometries - ordered, affine, projective, absolute, hyperbolic - are discussed. Although much of this has appeared before (for example, in Veblen and Young's

*Projective Geometry*), there are some novel ways of treating some of the work.

Differential geometry of curves and surfaces is worked out in some detail in part IV, and an introduction to the tensor notation is given. Finally there are chapters on topology and four-dimensional geometry.

It will be seen that the book covers a wide range, in fact all of what is usually known as elementary geometry and much beyond. I consider it to be one o the most important books on geometry to be published for many years, and would urge all those who teach the subject to get a copy.

The book is beautifully written and well designed. Each section starts with a quotation: these are always relevant, and sometimes amusing. The list of references at the end would keep a conscientious reader busy for years.

**6.2. Review by: Hans Freudenthal.**

*Bull. Amer. Math. Soc.*

**68**(2) (1962), 55-59.

Coxeter presents classical style geometry in 22 chapters, which are reasonably self-contained, though tied together by a modern spirit of reinterpretation of classical matter. If geometry can be rewritten in a modern style without losing its classical character, is it fair to call it out of date? The answer of dogmatics to this rhetorical question will still be: yes, it is. They will emphasise this answer when they read the table of contents of the first chapter "Triangles": 1. Euclid, 2. Primitive concepts and axioms, 3.

*Pons asinorum*, 4. The medians and the centroid, 5. The incircle and the circumcircle, 6. The Euler line and the orthocentre, 7. The nine-point circle, 8. Two extremum problems, 9. Morley's theorem. Of course they will never read this chapter (or the others either). If they are endowed with a sense of mathematical beauty, this is to be regretted.

Fortunately there are people left, who like mathematical still-life. If they read this chapter they will admire not only the choice of subjects, but also the condensed style as opposed to the verbosity of many older geometry texts, and the compact lucid proofs in which every definition and conclusion is completely to the point. These are characteristics not only of the first chapter. They will make reading the book a pleasure to everybody who honestly tries to appreciate the subject itself in a positive sense. Another feature is the rich variety of subjects in the main text and in the exercises. Geometrical transformations and groups penetrate the interpretation as much as possible, axiomatics, and non-Euclidean geometry are not neglected, there is even a short course of differential geometry included, and topology is represented by the four-colour problem. The main feature, however, is didactical. Abstraction is approached not by blunt decree, but by a scale of seemingly tentative generalisations. Owing to this feature, Coxeter's book belongs to the very few from which textbook authors might learn how to write.

**6.3. Review by: Howard Levi.**

*The Journal of Philosophy*

**60**(1) (1963), 19-21.

The preface to this book states that "For the last thirty or forty years Americans have somehow lost interest in geometry. The present book constitutes an attempt to revitalise this sadly neglected subject." Presumably the revitalisation is to take place by confronting students of mathematics with enough of what has been done in geometry so that they will be pleased by the subject and stimulated to contribute to it. To this end a great variety of subjects have been assembled, including individual theorems of elementary geometry, groups of transformations, axiomatic development of several geometries, some differential geometry of curves and surfaces, and some topology of surfaces. Most of the elementary topics are treated from a contemporary point of view, but some of the more advanced material is presented in a much more traditional fashion.

The range of subjects is so great that anyone with any capacity for appreciating geometry is very likely to find something in this book to arrest and edify him ...

...

If this book does not revitalise the subject it will not be because of imperfections in the material or in its presentation, but will be attributable to a flaw in the strategy employed. There are many pressures the mathematical community exerts on its members to produce more mathematics, and these tend to focus research activities onto newly opened fields, which have not been so thoroughly mined as classical geometry. From the point of view of expediency it can be predicted that geometry will furnish imagery and analogies for investigations into new fields, rather than be itself the subject of further ex- tensive study. Introduction to Geometry is well suited for intensifying this role for geometry.

**6.4. Review (of 2nd edition) by: Donald C Foss.**

*The Mathematics Teacher*

**83**(1) (1990), 73.

Coxeter states that the first eleven chapters would constitute a course for students who have some knowledge of Euclidean and elementary analytic geometry but who have not mathematics or for enterprising high school teachers who wish to get a glimpse of topics just beyond the curriculum they usually teach. It would also be possible to develop a short course for teachers based on a few selected chapters, since they are fairly self-contained. Some nice features of this book are the many interesting quotes, the carefully cross-referenced topics, and answers for most of the exercises. This book could serve as the text for a geometry course for teachers and thereafter would make a great reference.

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

**7.1. Review by: Frank C Ogg.**

*The Mathematics Teacher*

**59**(3) (1966), 291.

Professor Coxeter has offered a small (176 pages) book on projective geometry, but one so packed with meat that many much larger books are less comprehensive. The style and method are indicated by a statement in an early chapter: "The best possible advice to the reader is to set aside all his previously acquired knowledge (such as trigonometry and analytic geometry) and use only the axioms and their consequences." The reviewer considers this good advice but admits that it is now and then rather hard to follow. A rigorously logical procedure with synthetic methods and a compact notation keep the book from being easy reading. However, who wants an easy book? An experienced geometrician can derive real pleasure and profit from the ingenious methods used. A novice will find the book an excellent introduction to the subject and one which will not foster bad habits in procedure.

**7.2. Review (of 2nd edition) by: Robert P Burn.**

*The Mathematical Gazette*

**58**(405) (1974), 236-237.

This is a beautiful book, with the classical theorems on conics in a Pappian projective plane of characteristic ≠ 2 at its heart. The treatment in Chapters 1 to 9 is a simplified version of Veblen and Young's

*Projective geometry*, Vol. 1 and an expanded version of the chapter on projective geometry in the author's well known

*Introduction to Geometry*. The exercises have the same air of economy and freshness that Maxwell caught in his

*Geometry for advanced pupils*, and in days gone by one would have happily recommended this book to the able sixth former already afire with Maxwell or Durell. Coxeter's aims are explicit: real projective space models them, but so do projective spaces over most finite fields. Coxeter's methods are synthetic: his basic tools are perspectivities, and thus he can write a self-contained account.

There is, however, no mention of groups (except in the preface to the second edition) or of vector spaces. There is no topology or continuity (apparently) here. So it is easy to see why those needing to find room for new areas in the undergraduate curriculum can tread on the fewest toes by pushing out projective geometry. Beauty is not a sufficient criterion when choosing a mathematics syllabus because there is too much beauty available. Beauty with relevance has a stronger case, and when beautiful and relevant mathematics bristles with unsolved problems it has the strongest case of all.

...

Projective geometry has a place in the undergraduate curriculum of the future (and not just in a vector space course) because of the way diagrams illuminate algebra and feed the mathematician's intuition; this book only hints in its later chapters at what that place will be.

**7.3. Review (of 2nd edition) by: Philip Peak.**

*The Mathematics Teacher*

**68**(3) (1975), 230.

This is a second edition of a book first published in 1963. It has been a successful text for more than ten years. The author's experiences as an excellent teacher are reflected in his texts. The second edition has some new material, some up-to-date symbolic representation, and some new exercises. The usual topics of projective geometry are covered. There are challenging exercises and well-drawn figures. Answers are provided in the appendix.

**7.4. Review (of 2nd edition) by: Des MacHale.**

*The Mathematical Gazette*

**74**(467) (1990), 82.

If geometry is currently the Cinderella of mathematics, then surely projective geometry is the Cinderella of geometry. I was lucky enough to have taken a course in projective geometry in my first year in college and the memory of its beauty and elegance has remained ever since.

Historically, projective geometry seems to have arisen from the 15th century interest of artists and architects in the theory of perspective, although as early as the third century AD Pappus of Alexandria had discovered several theorems of a truly projective nature. Projective geometry dispenses with the Euclidean notions of distance, angles, between-ness and parallelism and concentrates on incidence, collinearity and other properties of perspective. What remains is a beautiful and intricate system of theorems, simpler in many ways than Euclid's, and considered by many to be more fundamental and by a few, even more interesting.

*Projective geometry*is an elegant, carefully written and beautifully produced introduction to the fundamental concepts and development of projective geometry, building up from basic topics such as projectivities, quadrangles, harmonic sets, and duality, to polarities, conics, and finite projective planes, an exotic world undreamed of by those who know only of Euclidean geometry. As always, Coxeter has a great feeling for the intrinsic beauty of his subject, but the reader is never left behind, because the book is written to inform rather than impress. The diagrams, illustrations, examples, and charts are clear and helpful and there are over twenty pages of answers to exercises, a delightful luxury for a geometry book!

For me, the highlight of the book was chapter 10, where Coxeter examines in detail the finite projective plane PG(2, 5) which contains precisely 31 points and 31 lines, with exactly 6 points on each line and 6 lines through each point. For this system he counts the triangles (3875 in all), the conics (3100 of these) and other objects such as the projectivities, involutions, collineations and harmonic homologies. I can think of no better way of really coming to grips with such concepts than by counting their occurrences in a non-trivial system.

*Projective geometry*is strongly recommended, especially for its excellent sense of history. Certainly it should be in every school and college library and on the bookshelf of anyone who dares to claim the ancient and sacred title of "geometer".

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

**8.1. Review by: Edwin Arthur Maxwell.**

*The Mathematical Gazette*

**52**(381) (1968), 286.

The authors present two formidable names, from whom much is expected and by whom, indeed, much is given. Written with skill and enthusiasm, the book should give valuable help in restoring geometry to a more prominent place in a curriculum from which it has too often been banished. (But whether revisited will be a proper description for much of the subject matter is, sadly, doubtful.)

The chapter headings are: Points and lines connected with the triangle, some properties of circles, collinearity and concurrence, transformations, an introduction to inversive geometry, and an introduction to projective geometry.

It will be seen that the net is wide. The treatment, though brisk and pointed throughout, moves with that leisurely air of browsing that is essential to all good geometry.

...

Twenty-seven pages of Hints and Answers, twenty-nine texts for reference, and six pages of glossary bring to a triumphant conclusion a book which any school or college library neglects at its peril and which all interested in the teaching and reading of geometry will insist on obtaining.

**8.2. Review by: Peter N Ruane.**

*Mathematical Association of America*(9 November 2008).

In 1961 a book appeared with the widely embracing title Introduction to Geometry. Its author was H S M Coxeter who, in the preface, said that 'For the past thirty or forty years, most Americans have somehow lost interest in geometry. The present book constitutes an attempt to revitalise this sadly neglected subject'.

It's hard to say what effect that book has had in terms of the author's overall aim; but I know that it was highly valued by many mathematical educators of that era. Unfortunately, it seemed to exert (at most) transient influence over the development of geometry in school or university mathematics. This, I feel, was because it isn't a textbook as such, but rather, in the spirit of Birkhoff and Mac Lane, more akin to a 'Survey of modern geometry'. Moreover, it was regarded as too challenging for direct student use, and it is still hard to see how one could base an introductory course on it.

Six years later, however, there appeared this book, co-authored with S.L. Greitzer. Its purpose was the same as Coxeter's book of 1961, but the contents and structure are entirely different, and much more likely to achieve the stated aim.

Of course, it may seem strange to be reviewing a book that was published over forty years ago but, having been unable to locate any previous review of this excellent book, it is surely a case of 'better late than never'.

...

Among many beautiful and surprising theorems included in this book are those by Ceva, Menelaus, Morley, Brianchon, Steiner-Lehmus and Feuerbach. The proofs in themselves are always interesting, and often non-standard.

But, apart from triangle theorems, many fascinating aspects of circles, quadrilaterals and conics are revealed, and the book explains the importance of the transformational viewpoint, beginning with the isometries and leading nicely into inversion and projective transformations.

Overall, I found the chapter that introduces inversive geometry particularly enjoyable; it includes a metric for inversive distance that relates very nicely to Steiner's porism. As for the book's final chapter, the approach to projective geometry is synthetic and perhaps, to quote an English saying, 'not everyone's cup of tea'.

Lastly, the exercises in the book are of equal interest as the main text itself. They extend the ideas previously introduced and encourage investigational participation.

Having read this book from cover-to-cover and worked through every problem, I can declare it to totally error-free. It deserves never to be out of print and could readily form the basis of an introductory geometry course in any undergraduate programme.

**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*.

**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. 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 (1939) and eventually became a co-author of the book.

**10.2. Review (of 11th edition) by: W D R.**

*The Mathematics Teacher*

**33**(1) (1940), 46.

This revised edition is for the most part the tenth edition reprinted, and most teachers are familiar enough with the old edition to make further comment here unnecessary. However, the revised edition has omitted three chapters of the old edition, namely, "String Figures," "Bees and Their Cells," and "Mechanical Recreations." Chapter II has been radically changed as has also Chapter VIII on "Map Colouring Problems." A great deal of Chapter III on "Geometric Recreations" and Chapter VII on "Magic Squares" is new; Chapter XII has been broken up and distributed among the first, third, fourth, and eleventh chapters. Finally, Chapter V on "Polyhedra" is new, and Chapter XIV on "Cryptographs and Ciphers" has been completely rewritten. Dr Coxeter has been ably assisted in this revision by Professor D H Lehmer, Abraham Sinkov, a cryptanalyst in the United States War Department, by J M Andreas, and P S Donchian. This new book will be of great interest and help particularly to secondary teachers of mathematics.

**10.3. Review (of 11th edition) by: Harold T Davis.**

*National Mathematics Magazine*

**14**(6) (1940), 357-358.

For nearly half a century this delightful classic has furnished amusement to many who would explore mathematical realms for recreation, and it has stimulated others to attack the unsolved problems of earlier editions and to generalise the results of some of the known theorems. It is thus a real event to have at hand a new edition published 17 years after the tenth, which appeared in 1992.

The revision has been extensive, although there has been no essential change in the character of the material included in the volume. The fifth chapter of the old edition on mechanical recreations, the eighth on bees and their calls, and the fifteenth on string figures have been omitted. The twelfth, devoted to miscellaneous problems (Chinese rings, problems connected with packs of cards, etc.), has been broken up and distributed among other chapters. The fourteenth, on cryptographs and ciphers, has been completely rewritten by Abraham Sinkov, a cryptanalyst in the U. S. War Department. The new material is found in arithmetical recreations, Chapter 2; geometrical recreations, Chapter 3; polyhedra, Chapter 5, which is entirely new; magic squares, Chapter 7; and map-colouring problems, Chapter 8. Chapter 5 is handsomely illustrated by two half-tone plates and numerous figures, which greatly illuminate the text.

The reviewer does not regret the omissions which the reviser has found necessary to make, since these recreations are still available in the tenth edition, reprinted last in 1937, and the new material adds greatly to our ever growing fund of problems which belong to the realm of mathematical aesthetics. One has but to turn to earlier editions than the tenth to note this increasing supply of recreations. The reviewer still recalls the interest which he found in a chapter on the nature of the ether (if there is an ether), which has long since disappeared from new editions.

It would be impossible to attempt to describe the new material in detail, or even to mention the many new topics that have been introduced. One notes with interest a considerable increase in material derived from the theory of numbers, and the numerous problems in which techniques derived from the theory of continued fractions are employed. One notes with interest a competent account of the problem of the distribution of prime numbers, a good description of Lehmer's ingenious machine for factoring large numbers, a device which the author characterises as "this wonderful machine", and the theory and application of Fibonacci numbers. One also finds among the new sections, accounts of the minimal problems of Besicovitch and Kakeya, the Platonic and Archimedean solids, zonohedra, the theory of the kaleidoscope, the present status of the map-colouring problem, the problem of colouring the icosahedron, tessellation, or the covering of a plane area with mosaics, problems in probability, continued fractions and lattice points, magic domino squares, magic cubes, etc.

**10.4. Review (of 11th edition) by: Howard F Munch.**

*The High School Journal*

**26**(1/2) (1943), 44-45.

This is the last edition of an older book by Mr Ball by the same title. The original edition came out in 1892. The book has been so popular that eleven different editions were published. Besides these there were four different reprints of the tenth and two of the eleventh editions. This alone gives one some idea of the place this book has taken in our mathematics literature.

The book consists of fourteen chapters. Chapters I and II deal with "Arithmetical Recreations," Chapters III and IV with "Geometrical Recreations," Chapter V with "Polyhedra," Chapter VI with "Chess-Board Recreations," Chapter VII with "Magic Squares," Chapter VIII with "Map-Colouring Problems," Chapter IX with "Unusual Problems," Chapter X with "Kirkman's School-girl Problem," Chapter XI with "Miscellaneous Problems," Chapter XII with "Three Classical Geometrical Problems," Chapter XIII with "Calculating Prodigies," while Chapter XIV deals with cryptography and cryptanalysis.

In these chapters the reader finds excellent treatments of many of the interesting problems that have intrigued men's minds throughout the ages. It is amazing that such a variety of problems have presented themselves to mankind for solution and still more amazing that they have been successfully solved. The ingenious devices that have been invented to solve these problems are really more wonderful than the mechanical devices which have so richly blessed us.

There is no doubt that this is the outstanding book in this field. It is really a classic. Any person who has leisure time and the urge to follow some of the interesting by-paths which some of our greatest intellects have blazed for us should treasure this book. The reading of it and the study of the problems here presented would keep one's mind out of mischief for many hours and would challenge its alertness and acumen.

**10.5. Review (of 11th edition) by: F P White.**

*The Mathematical Gazette*

**23**(256) (1939), 422.

This is still Rouse Ball, but with a difference. A large part of the tenth edition is reprinted without change; this needs no description here. Three chapters have, however, been omitted entirely, the one on string figures, which Rouse Ball himself had very much cut down after the appearance of his separate book on the subject in 1920, the short chapter on bees and their cells, and the chapter on mechanical recreations. This last, which begins with Zeno's paradoxes and includes a mention of cut and spin, might well be expanded, as Rouse Ball himself suggested, into an independent work; it is about time that someone dealt authoritatively with the mystery of the "new ball" in cricket.

Recent work in the theory of numbers has made a good deal of change necessary in Chapter II; here Dr Coxeter has had the help of Professor D H Lehmer, whose work on the computational side is well known. Similarly, the chapter on map-colouring problems has undergone considerable revision in the light of modern developments in topology.

It is, however, to the entirely new chapter on polyhedra that one turns with the greatest expectations, which are not disappointed. Dr Coxeter is an unrivalled authority on the subject, and in these 32 pages he gives a masterly description of the five Platonic solids, the thirteen Archimedean solids, the Kepler-Poinsot polyhedra and so on. The diagrams are excellent, and there are two admirable plates from photographs by P S Donchian.

Rouse Ball's Recreations has been deservedly popular since its first appearance in 1892. It may be predicted with confidence that Dr. Coxeter's admirable edition will give it a new lease of life.

**10.6. Review (of 12th edition) by: Philip Peak.**

*The Mathematics Teacher*

**69**(1) (1976), 86.

It goes without saying this book is an established work, as the twelve editions of it since 1892 verify. This twelfth edition follows the eleventh, which appeared in 1939. The changes in this edition include new material on primes, recent work in factorisation, consideration of some new unsolved problems, and the latest in formation on the map colouring problems. Some maze problems have been added, and problems which mathematical ingenuity can solve that the computer cannot. Naturally, if you have the 1939 edition you would have much of what is in this 1974 edition, but 35 years of new material still makes this a must to add to your mathematics library.

**10.7. Review (of 13th edition) by: Donald Johnson.**

*The Mathematics Teacher*

**81**(4) (1988), 325.

Ball and Coxeter's

*Mathematical Recreations and Essays*"contains descriptions of various problems of the kind usually termed Mathematical Recreations, and a few Essays on some analogous questions."

Generally, the text, which has had "all matter which involved advanced mathematics" deleted, is clearly written, and information on both notation and problem origin is carefully referenced. Although not remarkable for the materials included because it is a republication of Rouse Ball's first edition (1892), the text is nonetheless a publication that in its revised form would be an excellent supplementary text for the high school student. Coxeter's revision contains fourteen chapters with a very useful table of contents. As a supplementary book, its sections on calculating prodigies, magic squares, and cryptographic systems alone present great motivational devices. I would highly recommend it as an addition to any educator's personal library. It is clearly written, informative, and very well referenced.

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

**11.1. Review by: Rolf L E Schwarzenberger.**

*The Mathematical Gazette*

**59**(409) (1975), 196-197.

This is an astonishing book - the ideal present for a mathematician who enjoys patterns, whether geometric or algebraic, and who is sometimes depressed by the arid axiomatic theories of so much published mathematics. If you have a friend in this category then there are two good reasons why you should buy this book on his behalf.

The first is that he will not buy it himself. Even if he is lucky enough to see it he will be deterred by the price and by the unfamiliarity of the notation. He will be attracted by the sumptuous full-page diagrams of beautiful patterns with names like the great grand stellated 120-cell but will be uncertain what they represent. He will resist the temptation to purchase by arguing that complex polytopes are hardly a central topic in mathematics, that he does not need the book for his teaching or for his research, and that the book appears to consist mainly of examples with no clear development of general theory.

The second reason is simply that on all these counts he will have been quite wrong. As to the price, for example, the dozen largest and most beautiful diagrams could easily be sold singly in art shops: there are indeed analogies between this book and much more expensive sets of art reproductions or of symphonic recordings. The author writes in an unusually compact style and has selected his material with such skill that he is able to compress into one book what could have been five distinct monographs: (1) a masterly 28-page survey of two- and three-dimensional symmetries; (2) a 25-page section which explains polytopes in four-dimensional space and introduces the author's technique of studying their spherical trigonometry by means of frieze patterns; (3) an 18-page exposition of quaternions including the classification and geometric significance of subgroups; (4) a 20-page study of n-dimensional unitary space with a complete enumeration of finite reflection groups when n = 2; (5) the promised detailed treatment of complex polygons, complex polytopes and complex honeycombs, culminating in a complete classification of the various symmetry groups, within 50 pages. As in the real case, the number drops sharply in higher dimensions and thus justifies the concentration on low-dimensional examples.

This conciseness of exposition has not been achieved by concentrating on general theory. On the contrary, the exposition proceeds primarily by examples, and each example is illustrated by one or more diagrams. Notations, definitions and general theory are justified by the examples and follow them unobtrusively. The examples point towards all kinds of unexplored relationships with other areas of mathematics (crystallography, algebraic groups, algebraic geometry, combinatorial theory and graph theory come im- mediately to mind) but the author is content to leave these for the reader to explore with the help of brief but illuminating historical remarks. Nor has the conciseness been achieved by omitting proofs or by assuming advanced results: the proofs are there in full detail, but are selected for their power and brevity; they are understandable to anyone who knows the definition of a group and a few facts about complex numbers. The conciseness is primarily the result of a very careful selection of material, of skilful use of diagrams, and of illuminating notation (much of it due to the author and his students). It belies the fact that every single page of this book contains the material for half a dozen pages of works written in more garrulous style and at least another half dozen pages worth of food for thought! The owner of this book will want to read it very slowly, to follow up references to other more easily accessible works (for example, the same author's book on real polytopes which is completely independent of, yet closely related to, the present book), to enjoy a chapter and to try out a notation in other contexts before moving on to the next, and to leave the book lying on the coffee table or at the bedside to pick up at odd moments. He will find it difficult to read at first, but more and more rewarding; whether he teaches symmetry at the most elementary level, or does research in geometry or group theory at the most advanced level, he will find it constantly stimulating

**11.2. Review (of 2nd edition) by: Lawrence O Cannon.**

*The Mathematics Teacher*

**85**(4) (1992), 316.

H S M Coxeter, Emeritus Professor of Mathematics at the University of Toronto, is one of the most influential geometers of our time. In addition to his research and teaching, through workshops and his several books he has shared his appreciation of the beauties of geometry with hundreds of teachers and has inspired thousands of students. This volume distils the essence of nearly forty years of work by Coxeter and other geometers, and explores the properties of regular polyhedra with all sorts of generalisations. The book is a work of art, as well as a work of very substantial mathematics, having both aesthetic and scholarly appeal.

The author compares his com position of the book to a Bruckner symphony, "with crescendos and climaxes, little foretastes of pleasure to come, and abundant cross-references." The analogy is apt in many respects. Just as Bruckner's music is a source of a great deal of pleasure to many listeners and can be enjoyed on many levels, Coxeter's geometry can appeal to readers of diverse back grounds. But the symphonies of Bruckner require substantial musical maturity for full appreciation. Likewise, Coxeter's book demands considerable mathematical maturity for any kind of thorough understanding.

*Regular Complex Polytopes*can be picked up and casually admired, but its careful analysis and thorough proofs cannot be followed without considerable effort. Those who have not pre pared themselves to follow "the geometric, algebraic and group theoretic aspects of the subject ... interwoven like ... sections of the orchestra" will find the reading dense and difficult. This is a beautiful book, but its presence on the library shelves of most secondary or undergraduate schools will be at best only partially appreciated.

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

**12.1. Note.**

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Last Updated July 2020