Regular Polytopes (book)

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Regular Polytopes cover Dover.jpg
Cover of the Dover edition, 1973
Author Harold Scott MacDonald Coxeter
Published1947, 1973, 1973
PublisherMethuen, Pitman, Macmillan, Dover
ISBN 0-486-61480-8
OCLC 798003

Regular Polytopes is a geometry book on regular polytopes written by Harold Scott MacDonald Coxeter. It was originally published by Methuen in 1947 and by Pitman Publishing in 1948, [1] [2] [3] [4] [5] [6] [7] [8] with a second edition published by Macmillan in 1963 [9] [10] [11] [12] and a third edition by Dover Publications in 1973. [13] [14] [15] The Basic Library List Committee of the Mathematical Association of America has recommended that it be included in undergraduate mathematics libraries. [15]



The main topics of the book are the Platonic solids (regular convex polyhedra), related polyhedra, and their higher-dimensional generalizations. [1] [2] It has 14 chapters, along with multiple appendices, [3] providing a more complete treatment of the subject than any earlier work, and incorporating material from 18 of Coxeter's own previous papers. [1] It includes many figures (both photographs of models by Paul Donchian and drawings), tables of numerical values, and historical remarks on the subject. [1] [2]

The first chapter discusses regular polygons, regular polyhedra, basic concepts of Graph theory, and the Euler characteristic. [3] 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 to generate related polyhedra, [1] including the semiregular polyhedra, and discusses zonohedra and Petrie polygons. [3] Here and throughout the book, the shapes it discusses are identified and classified by their Schläfli symbols. [1]

Chapters 3 through 5 describe the symmetries of polyhedra, first as permutation groups [3] and later, in the most innovative part of the book, [1] as the Coxeter groups, groups generated by reflections and described by the angles between their reflection planes. This part of the book also describes the regular tessellations of the Euclidean plane and the sphere, and the regular honeycombs of Euclidean space. Chapter 6 discusses the star polyhedra including the Kepler–Poinsot polyhedra. [3]

The remaining chapters cover higher-dimensional generalizations of these topics, including two chapters on the enumeration and construction of the regular polytopes, two chapters on higher-dimensional Euler characteristics and background on quadratic forms, two chapters on higher-dimensional Coxeter groups, a chapter on cross-sections and projections of polytopes, and a chapter on star polytopes and polytope compounds. [3]

Later editions

The second edition was published in paperback; [9] [11] it adds some more recent research of Robert Steinberg on Petrie polygons and the order of Coxeter groups, [9] [12] appends a new definition of polytopes at the end of the book, and makes minor corrections throughout. [9] The photographic plates were also enlarged for this printing, [10] [12] and some figures were redrawn. [12] The nomenclature of these editions was occasionally cumbersome, [2] 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. [13] [14]


The book only assumes a high-school understanding of algebra, geometry, and trigonometry, [2] [3] but it is primarily aimed at professionals in this area, [2] and some steps in the book's reasoning which a professional could take for granted might be too much for less-advanced readers. [3] Nevertheless, reviewer J. C. P. Miller recommends it to "anyone interested in the subject, whether from recreational, educational, or other aspects", [4] and (despite complaining about the omission of regular skew polyhedra) reviewer H. E. Wolfe suggests more strongly that every mathematician should own a copy. [7] Geologist A. J. Frueh Jr., describing the book as a textbook rather than a 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. [6]

Already in its first edition the book was described as "long awaited", [3] and "what is, and what will probably be for many years, the only organized treatment of the subject". [7] In a review of the second edition, Michael Goldberg (who also reviewed the first edition) [1] called it "the most extensive and authoritative summary" of its area of mathematics. [10] 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". [15]

See also

Related Research Articles

Kepler–Poinsot polyhedron

In geometry, a Kepler–Poinsot polyhedron is any of four regular star polyhedra.

Polyhedron Three-dimensional shape with flat polygonal faces, straight edges and sharp corners

In geometry, a polyhedron is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. The word polyhedron comes from the Classical Greek πολύεδρον, as poly- + -hedron.

In elementary geometry, a polytope is a geometric object with flat sides (faces). It is a generalization in any number of dimensions of the three-dimensional polyhedron. Polytopes may exist in any general number of dimensions n as an n-dimensional polytope or n-polytope. In this context, "flat sides" means that the sides of a (k + 1)-polytope consist of k-polytopes that may have (k – 1)-polytopes in common. For example, a two-dimensional polygon is a 2-polytope and a three-dimensional polyhedron is a 3-polytope.

4-polytope Four-dimensional geometric object with flat sides

In geometry, a 4-polytope is a four-dimensional polytope. It is a connected and closed figure, composed of lower-dimensional polytopal elements: vertices, edges, faces (polygons), and cells (polyhedra). Each face is shared by exactly two cells. The 4-polytopes were discovered by the Swiss mathematician Ludwig Schläfli before 1853.

Harold Scott MacDonald Coxeter Canadian geometer

Harold Scott MacDonald "Donald" Coxeter, was a British and later also Canadian geometer. He is regarded as one of the greatest geometers of the 20th century.

A regular polyhedron is a polyhedron whose symmetry group acts transitively on its flags. A regular polyhedron is highly symmetrical, being all of edge-transitive, vertex-transitive and face-transitive. In classical contexts, many different equivalent definitions are used; a common one is that the faces are congruent regular polygons which are assembled in the same way around each vertex.

The term semiregular polyhedron is used variously by different authors.

Honeycomb (geometry) Tiling of 3-or-more dimensional euclidian or hyperbolic space

In geometry, a honeycomb is a space filling or close packing of polyhedral or higher-dimensional cells, so that there are no gaps. It is an example of the more general mathematical tiling or tessellation in any number of dimensions. Its dimension can be clarified as n-honeycomb for a honeycomb of n-dimensional space.


In five-dimensional geometry, a five-dimensional polytope or 5-polytope is a 5-dimensional polytope, bounded by (4-polytope) facets. Each polyhedral cell being shared by exactly two 4-polytope facets.

In geometry, a skew apeirohedron is an infinite skew polyhedron consisting of nonplanar faces or nonplanar vertex figures, allowing the figure to extend indefinitely without folding round to form a closed surface.

Petrie polygon

In geometry, a Petrie polygon for a regular polytope of n dimensions is a skew polygon in which every (n – 1) consecutive sides belongs to one of the facets. The Petrie polygon of a regular polygon is the regular polygon itself; that of a regular polyhedron is a skew polygon such that every two consecutive side belongs to one of the faces. Petrie polygons are named for mathematician John Flinders Petrie.

Regular 4-polytope Four-dimensional analogues of the regular polyhedra in three dimensions

In mathematics, a regular 4-polytope is a regular four-dimensional polytope. They are the four-dimensional analogues of the regular polyhedra in three dimensions and the regular polygons in two dimensions.

In geometry, the regular skew polyhedra are generalizations to the set of regular polyhedra which include the possibility of nonplanar faces or vertex figures. Coxeter looked at skew vertex figures which created new 4-dimensional regular polyhedra, and much later Branko Grünbaum looked at regular skew faces.


In six-dimensional geometry, a six-dimensional polytope or 6-polytope is a polytope, bounded by 5-polytope facets.

In geometry, a regular skew apeirohedron is an infinite regular skew polyhedron, with either skew regular faces or skew regular vertex figures.

Mathematical Models is a book on the construction of physical models of mathematical objects for educational purposes. It was written by Martyn Cundy and A. P. Rollett, and published by the Clarendon Press in 1951, with a second edition in 1961. Tarquin Publications published a third edition in 1981.

Regular Figures is a book on polyhedra and symmetric patterns, by Hungarian geometer László Fejes Tóth. It was published in 1964 by Pergamon in London and Macmillan in New York.


  1. 1 2 3 4 5 6 7 8 Goldberg, M., "Review of Regular Polytopes", Mathematical Reviews, MR   0027148
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  11. 1 2 Primrose, E.J.F (October 1964), "Review of Regular Polytopes", The Mathematical Gazette, 48 (365): 344–344, doi:10.1017/s0025557200072995
  12. 1 2 3 4 Yff, P. (February 1965), "Review of Regular Polytopes", Canadian Mathematical Bulletin , 8 (1): 124–124, doi: 10.1017/s0008439500024413
  13. 1 2 Peak, Philip (March 1975), "Review of Regular Polytopes", The Mathematics Teacher , 68 (3): 230, JSTOR   27960095
  14. 1 2 Wenninger, Magnus J. (Winter 1976), "Review of Regular Polytopes", Leonardo , 9 (1): 83, doi:10.2307/1573335, JSTOR   1573335
  15. 1 2 3 Brown, Tricia Muldoon (October 2016), "Review of Regular Polytopes", MAA Reviews, Mathematical Association of America