Author | Harold Scott MacDonald Coxeter |
---|---|

Language | English |

Subject | Geometry |

Published | 1947, 1973, 1973 |

Publisher | Methuen, Pitman, Macmillan, Dover |

Pages | 321 |

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,

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] }

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] }

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

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.

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** "**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.

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.

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.

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 2 3 4 5 6 7 8 Goldberg, M., "Review of
*Regular Polytopes*",*Mathematical Reviews*, MR 0027148 - 1 2 3 4 5 6 Allendoerfer, C.B. (1949), "Review of
*Regular Polytopes*",*Bulletin of the American Mathematical Society*,**55**(7): 721–722, doi: 10.1090/S0002-9904-1949-09258-3 - 1 2 3 4 5 6 7 8 9 10 Cundy, H. Martyn (February 1949), "Review of
*Regular Polytopes*",*The Mathematical Gazette*,**33**(303): 47–49, doi:10.2307/3608432, JSTOR 3608432 - 1 2 Miller, J. C. P. (July 1949), "Review of
*Regular Polytopes*",*Science Progress*,**37**(147): 563–564, JSTOR 43413146 - ↑ Walsh, J. L. (August 1949), "Review of
*Regular Polytopes*",*Scientific American*,**181**(2): 58–59, JSTOR 24967260 - 1 2 Frueh, Jr., A. J. (November 1950), "Review of
*Regular Polytopes*",*The Journal of Geology*,**58**(6): 672, JSTOR 30071213 - 1 2 3 Wolfe, H. E. (February 1951), "Review of
*Regular Polytopes*",*American Mathematical Monthly*,**58**(2): 119–120, doi:10.2307/2308393, JSTOR 2308393 - ↑ Tóth, L. Fejes, "Review of
*Regular Polytopes*",*zbMATH*(in German), Zbl 0031.06502 - 1 2 3 4 Robinson, G. de B., "Review of
*Regular Polytopes*",*Mathematical Reviews*, MR 0151873 - 1 2 3 Goldberg, Michael (January 1964), "Review of
*Regular Polytopes*",*Mathematics of Computation*,**18**(85): 166, doi:10.2307/2003446, JSTOR 2003446 - 1 2 Primrose, E.J.F (October 1964), "Review of
*Regular Polytopes*",*The Mathematical Gazette*,**48**(365): 344–344, doi:10.1017/s0025557200072995 - 1 2 3 4 Yff, P. (February 1965), "Review of
*Regular Polytopes*",*Canadian Mathematical Bulletin*,**8**(1): 124–124, doi: 10.1017/s0008439500024413 - 1 2 Peak, Philip (March 1975), "Review of
*Regular Polytopes*",*The Mathematics Teacher*,**68**(3): 230, JSTOR 27960095 - 1 2 Wenninger, Magnus J. (Winter 1976), "Review of
*Regular Polytopes*",*Leonardo*,**9**(1): 83, doi:10.2307/1573335, JSTOR 1573335 - 1 2 3 Brown, Tricia Muldoon (October 2016), "Review of
*Regular Polytopes*",*MAA Reviews*, Mathematical Association of America

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