''Regular Polytopes'' is a

geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

book on

regular polytope In mathematics, a regular polytope is a polytope whose symmetry group acts transitively on its flags, thus giving it the highest degree of symmetry. All its elements or -faces (for all , where is the dimension of the polytope) — cells, ...

s written by

Harold Scott MacDonald Coxeter Harold Scott MacDonald "Donald" Coxeter, (9 February 1907 – 31 March 2003) was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century. Biography Coxeter was born in Kensington t ...

. It was originally published by Methuen in 1947 and by Pitman Publishing in 1948, with a second edition published by Macmillan in 1963 and a third edition by Dover Publications in 1973. The Basic Library List Committee of the

Mathematical Association of America The Mathematical Association of America (MAA) is a professional society that focuses on mathematics accessible at the undergraduate level. Members include university, college, and high school teachers; graduate and undergraduate students; pure a ...

has recommended that it be included in undergraduate mathematics libraries.

Overview

The main topics of the book are the

Platonic solid In geometry, a Platonic solid is a convex, regular polyhedron in three-dimensional Euclidean space. Being a regular polyhedron means that the faces are congruent (identical in shape and size) regular polygons (all angles congruent and all edges c ...

s (regular convex polyhedra), related polyhedra, and their higher-dimensional generalizations. It has 14 chapters, along with multiple appendices, providing a more complete treatment of the subject than any earlier work, and incorporating material from 18 of Coxeter's own previous papers. It includes many figures (both photographs of models by Paul Donchian and drawings), tables of numerical values, and historical remarks on the subject. The first chapter discusses

regular polygon In Euclidean geometry, a regular polygon is a polygon that is Equiangular polygon, direct equiangular (all angles are equal in measure) and Equilateral polygon, equilateral (all sides have the same length). Regular polygons may be either convex p ...

s, regular polyhedra, basic concepts of

Graph theory In mathematics, graph theory is the study of ''graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of '' vertices'' (also called ''nodes'' or ''points'') which are conne ...

, and the

Euler characteristic In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space ...

. Using the Euler characteristic, Coxeter derives a Diophantine equation whose integer solutions describe and classify the regular polyhedra. The second chapter uses combinations of regular polyhedra and their

duals ''Duals'' is a compilation album by the Irish rock band U2. It was released in April 2011 to u2.com subscribers. Track listing :* "Where the Streets Have No Name" and "Amazing Grace" are studio mix of U2's performance at the Rose Bowl, P ...

to generate related polyhedra, including the

semiregular polyhedra In geometry, the term semiregular polyhedron (or semiregular polytope) is used variously by different authors. Definitions In its original definition, it is a polyhedron with regular polygonal faces, and a symmetry group which is transitive on ...

, and discusses

zonohedra In geometry, a zonohedron is a convex polyhedron that is centrally symmetric, every face of which is a polygon that is centrally symmetric (a zonogon). Any zonohedron may equivalently be described as the Minkowski sum of a set of line segments in ...

and

Petrie polygon In geometry, a Petrie polygon for a regular polytope of dimensions is a skew polygon in which every consecutive sides (but no ) belongs to one of the facets. The Petrie polygon of a regular polygon is the regular polygon itself; that of a reg ...

s. Here and throughout the book, the shapes it discusses are identified and classified by their

Schläfli symbol In geometry, the Schläfli symbol is a notation of the form \ that defines regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, who generalized Euclidean geometry to more ...

s. Chapters 3 through 5 describe the symmetries of polyhedra, first as

permutation group In mathematics, a permutation group is a group ''G'' whose elements are permutations of a given set ''M'' and whose group operation is the composition of permutations in ''G'' (which are thought of as bijective functions from the set ''M'' to it ...

s and later, in the most innovative part of the book, as the

Coxeter group In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean refl ...

s, groups generated by reflections and described by the angles between their reflection planes. This part of the book also describes the regular

tessellation A tessellation or tiling is the covering of a surface, often a plane (mathematics), plane, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to high-dimensional ...

s of the

Euclidean plane In mathematics, the Euclidean plane is a Euclidean space of dimension two. That is, a geometric setting in which two real quantities are required to determine the position of each point ( element of the plane), which includes affine notions of ...

and the sphere, and the regular honeycombs of

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...

. Chapter 6 discusses the

star polyhedra In geometry, a star polyhedron is a polyhedron which has some repetitive quality of nonconvexity giving it a star-like visual quality. There are two general kinds of star polyhedron: *Polyhedra which self-intersect in a repetitive way. *Concave p ...

including the Kepler–Poinsot polyhedra. The remaining chapters cover higher-dimensional generalizations of these topics, including two chapters on the enumeration and construction of the

s, two chapters on higher-dimensional

s and background on

quadratic form In mathematics, a quadratic form is a polynomial with terms all of degree two ("form" is another name for a homogeneous polynomial). For example, :4x^2 + 2xy - 3y^2 is a quadratic form in the variables and . The coefficients usually belong to a ...

s, two chapters on higher-dimensional

s, a chapter on cross-sections and projections of polytopes, and a chapter on star polytopes and

polytope compound In geometry, a polyhedral compound is a figure that is composed of several polyhedra sharing a common centre. They are the three-dimensional analogs of polygonal compounds such as the hexagram. The outer vertices of a compound can be connecte ...

Later editions

The second edition was published in paperback; it adds some more recent research of

Robert Steinberg Robert Steinberg (May 25, 1922, Soroca, Bessarabia, Romania (present-day Moldova) – May 25, 2014) was a mathematician at the University of California, Los Angeles. He introduced the Steinberg representation, the Lang–Steinberg theorem, t ...

s and the order of

s, appends a new definition of polytopes at the end of the book, and makes minor corrections throughout. The photographic plates were also enlarged for this printing, and some figures were redrawn. The nomenclature of these editions was occasionally cumbersome, and was modernized in the third edition. The third edition also included a new preface with added material on polyhedra in nature, found by the

electron microscope An electron microscope is a microscope that uses a beam of accelerated electrons as a source of illumination. As the wavelength of an electron can be up to 100,000 times shorter than that of visible light photons, electron microscopes have a hi ...

Reception

The book only assumes a high-school understanding of algebra, geometry, and trigonometry, but it is primarily aimed at professionals in this area, and some steps in the book's reasoning which a professional could take for granted might be too much for less-advanced readers. Nevertheless, reviewer J. C. P. Miller recommends it to "anyone interested in the subject, whether from recreational, educational, or other aspects", and (despite complaining about the omission of regular skew polyhedra) reviewer H. E. Wolfe suggests more strongly that every mathematician should own a copy. Geologist A. J. Frueh Jr., describing the book as a textbook rather than a

monograph A monograph is a specialist work of writing (in contrast to reference works) or exhibition on a single subject or an aspect of a subject, often by a single author or artist, and usually on a scholarly subject. In library cataloging, ''monograph ...

, suggests that the parts of the book on the symmetries of space would likely be of great interest to crystallographers; however, Frueh complains of the lack of rigor in its proofs and the lack of clarity in its descriptions. Already in its first edition the book was described as "long awaited", and "what is, and what will probably be for many years, the only organized treatment of the subject". In a review of the second edition, Michael Goldberg (who also reviewed the first edition) called it "the most extensive and authoritative summary" of its area of mathematics. By the time of Tricia Muldoon Brown's 2016 review, she described it as "occasionally out-of-date, although not frustratingly so", for instance in its discussion of the four color theorem, proved after its last update. However, she still evaluated it as "well-written and comprehensive".

References

{{reflist, refs= {{citation , last = Allendoerfer , first = C.B. , author-link = Carl B. Allendoerfer , doi = 10.1090/S0002-9904-1949-09258-3 , issue = 7 , journal =

Bulletin of the American Mathematical Society The ''Bulletin of the American Mathematical Society'' is a quarterly mathematical journal published by the American Mathematical Society. Scope It publishes surveys on contemporary research topics, written at a level accessible to non-experts. I ...

, pages = 721–722 , title = Review of ''Regular Polytopes'' , volume = 55 , year = 1949, doi-access = free {{citation , last = Goldberg , first = M. , journal = Mathematical Reviews , mr = 0027148 , title = Review of ''Regular Polytopes'' {{citation , last = Robinson , first = G. de B. , author-link = Gilbert de Beauregard Robinson , journal = Mathematical Reviews , mr = 0151873 , title = Review of ''Regular Polytopes'' {{citation , last = Walsh , first = J. L. , date = August 1949 , issue = 2 , journal =

Scientific American ''Scientific American'', informally abbreviated ''SciAm'' or sometimes ''SA'', is an American popular science magazine. Many famous scientists, including Albert Einstein and Nikola Tesla, have contributed articles to it. In print since 1845, it i ...

, jstor = 24967260 , pages = 58–59 , title = Review of ''Regular Polytopes'' , volume = 181 {{citation , last = Goldberg , first = Michael , date = January 1964 , doi = 10.2307/2003446 , issue = 85 , journal =

Mathematics of Computation ''Mathematics of Computation'' is a bimonthly mathematics journal focused on computational mathematics. It was established in 1943 as ''Mathematical Tables and other Aids to Computation'', obtaining its current name in 1960. Articles older than fi ...

, jstor = 2003446 , page = 166 , title = Review of ''Regular Polytopes'' , volume = 18 {{citation , last = Miller , first = J. C. P. , author-link = J. C. P. Miller , date = July 1949 , issue = 147 , journal = Science Progress , jstor = 43413146 , pages = 563–564 , title = Review of ''Regular Polytopes'' , volume = 37 {{citation , last = Wolfe , first = H. E. , date = February 1951 , doi = 10.2307/2308393 , issue = 2 , journal =

American Mathematical Monthly ''The American Mathematical Monthly'' is a mathematical journal founded by Benjamin Finkel in 1894. It is published ten times each year by Taylor & Francis for the Mathematical Association of America. The ''American Mathematical Monthly'' is an e ...

, jstor = 2308393 , pages = 119–120 , title = Review of ''Regular Polytopes'' , volume = 58 {{citation , last = Cundy , first = H. Martyn , author-link = Martyn Cundy , date = February 1949 , doi = 10.2307/3608432 , issue = 303 , journal =

The Mathematical Gazette ''The Mathematical Gazette'' is an academic journal of mathematics education, published three times yearly, that publishes "articles about the teaching and learning of mathematics with a focus on the 15–20 age range and expositions of attractive ...

, jstor = 3608432 , pages = 47–49 , title = Review of ''Regular Polytopes'' , volume = 33 {{citation , last = Wenninger , first = Magnus J. , author-link = Magnus Wenninger , date = Winter 1976 , doi = 10.2307/1573335 , issue = 1 , journal =

Leonardo Leonardo is a masculine given name, the Italian, Spanish, and Portuguese equivalent of the English, German, and Dutch name, Leonard Leonard or ''Leo'' is a common English masculine given name and a surname. The given name and surname originate ...

, jstor = 1573335 , page = 83 , title = Review of ''Regular Polytopes'' , volume = 9 {{citation , last = Peak , first = Philip , date = March 1975 , issue = 3 , journal =

The Mathematics Teacher Founded in 1920, The National Council of Teachers of Mathematics (NCTM) is a professional organization for schoolteachers of mathematics in the United States. One of its goals is to improve the standards of mathematics in education. NCTM holds an ...

, jstor = 27960095 , page = 230 , title = Review of ''Regular Polytopes'' , volume = 68 {{citation , last = Frueh , first = Jr., A. J. , date = November 1950 , issue = 6 , journal =

The Journal of Geology ''The Journal of Geology'' publishes research on geology, geophysics, geochemistry, sedimentology, geomorphology, petrology, plate tectonics, volcanology, structural geology, mineralogy, and planetary sciences. Its content ranges from planetary evo ...

, jstor = 30071213 , page = 672 , title = Review of ''Regular Polytopes'' , volume = 58 {{citation , last = Tóth , first = L. Fejes , author-link = László Fejes Tóth , journal =

zbMATH zbMATH Open, formerly Zentralblatt MATH, is a major reviewing service providing reviews and abstracts for articles in pure and applied mathematics, produced by the Berlin office of FIZ Karlsruhe – Leibniz Institute for Information Infrastruct ...

, language = de , title = Review of ''Regular Polytopes'' , zbl = 0031.06502 {{citation , last = Primrose , first = E.J.F , date = October 1964 , doi = 10.1017/s0025557200072995 , issue = 365 , journal = The Mathematical Gazette , pages = 344–344 , title = Review of ''Regular Polytopes'' , volume = 48 {{citation , last = Yff , first = P. , date = February 1965 , doi = 10.1017/s0008439500024413 , issue = 1 , journal = Canadian Mathematical Bulletin , pages = 124–124 , title = Review of ''Regular Polytopes'' , volume = 8, doi-access = free {{citation , last = Brown , first = Tricia Muldoon , date = October 2016 , journal = MAA Reviews , publisher =

, title = Review of ''Regular Polytopes'' , url = https://www.maa.org/press/maa-reviews/regular-polytopes Mathematics books Polytopes

Overview

Later editions

Reception

See also

References