''Regular Polytopes'' is a

geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

book on

regular polytope In mathematics, a regular polytope is a polytope whose symmetry group acts transitive group action, transitively on its flag (geometry), flags, thus giving it the highest degree of symmetry. In particular, all its elements or -faces (for all , w ...

s written by

Harold Scott MacDonald Coxeter Harold Scott MacDonald "Donald" Coxeter (9 February 1907 – 31 March 2003) was a British-Canadian geometer and mathematician. He is regarded as one of the greatest geometers of the 20th century. Coxeter was born in England and educated ...

. 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 A university () is an educational institution, institution of tertiary edu ...

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 polytope, convex, regular polyhedron in three-dimensional space, three-dimensional Euclidean space. Being a regular polyhedron means that the face (geometry), faces are congruence (geometry), congruent (id ...

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 ...

s, regular polyhedra, basic concepts of

graph theory In mathematics and computer science, graph theory is the study of ''graph (discrete mathematics), graphs'', which are mathematical structures used to model pairwise relations between objects. A graph in this context is made up of ''Vertex (graph ...

, 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's ...

. Using the Euler characteristic, Coxeter derives a

Diophantine equation ''Diophantine'' means pertaining to the ancient Greek mathematician Diophantus. A number of concepts bear this name: *Diophantine approximation In number theory, the study of Diophantine approximation deals with the approximation of real n ...

whose

integer An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative in ...

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, ...

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 point symmetry, centrally symmetric, every face of which is a polygon that is centrally symmetric (a zonogon). Any zonohedron may equivalently be described as the Minkowski addition, Minkows ...

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 List of regular polytopes and compounds, regular polytopes and tessellations. The Schläfli symbol is named after the 19th-century Swiss mathematician Ludwig Schläfli, wh ...

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 ...

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 ref ...

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, using one or more geometric shapes, called ''tiles'', with no overlaps and no gaps. In mathematics, tessellation can be generalized to higher dimensions and a variety ...

s of the

Euclidean plane In mathematics, a Euclidean plane is a Euclidean space of Two-dimensional space, dimension two, denoted \textbf^2 or \mathbb^2. It is a geometric space in which two real numbers are required to determine the position (geometry), position of eac ...

and the sphere, and the regular honeycombs of

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

. Chapter 6 discusses the star polyhedra 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 t ...

s, two chapters on higher-dimensional

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

Later editions

The second edition was published in

paperback A paperback (softcover, softback) book is one with a thick paper or paperboard cover, also known as wrappers, and often held together with adhesive, glue rather than stitch (textile arts), stitches or Staple (fastener), staples. In contrast, ...

; it adds some more recent research of

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

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 electrons as a source of illumination. It uses electron optics that are analogous to the glass lenses of an optical light microscope to control the electron beam, for instance focusing it ...

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 generally a long-form work on one (usually scholarly) subject, or one aspect of a subject, typically created by a single author or artist (or, sometimes, by two or more authors). Traditionally it is in written form and published a ...

, suggests that the parts of the book on the symmetries of space would likely be of great interest to

crystallographers A crystallographer is a type of scientist who practices crystallography, in other words, who studies crystals. Career paths The work of crystallographers spans several academic disciplines, including the life sciences, chemistry, physics, and m ...

; 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 In mathematics, the four color theorem, or the four color map theorem, states that no more than four colors are required to color the regions of any map so that no two adjacent regions have the same color. ''Adjacent'' means that two regions shar ...

, proved after its last update. However, she still evaluated it as "well-written and comprehensive".

References

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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. ...

, 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 scientists, including Albert Einstein and Nikola Tesla, have contributed articles to it, with more than 150 Nobel Pri ...

, 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 f ...

, 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 peer-reviewed scientific journal of mathematics. It was established by Benjamin Finkel in 1894 and is published by Taylor & Francis on behalf of the Mathematical Association of America. It is an exposi ...

, 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 a triannual peer-reviewed academic journal published by Cambridge University Press on behalf of the Mathematical Association. It covers mathematics education with a focus on the 15–20 years age range. The journ ...

, 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 , 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 ev ...

, jstor = 30071213 , page = 672 , title = Review of ''Regular Polytopes'' , volume = 58, doi = 10.1086/625793 {{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 Infrastru ...

, 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 , 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 The ''Canadian Mathematical Bulletin'' () is a mathematics journal, established in 1958 and published quarterly by the Canadian Mathematical Society. The current editors-in-chief of the journal are Antonio Lei and Javad Mashreghi. The journal p ...

, pages = 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://old.maa.org/press/maa-reviews/regular-polytopes Mathematics books Polytopes

Overview

Later editions

Reception

See also

References