''Polyhedra'' is a book on

polyhedra In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on t ...

, by Peter T. Cromwell. It was published by in 1997 by the

Cambridge University Press Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press A university press is an academic publishing hou ...

, with an unrevised paperback edition in 1999.

Topics

The book covers both the mathematics of polyhedra and its historical development, limiting itself only to three-dimensional geometry. The notion of what it means to be a polyhedron has varied over the history of the subject, as have other related definitions, an issue that the book handles largely by keeping definitions informal and flexible, and by pointing out problematic examples for these intuitive definitions. Many digressions help make the material readable, and the book includes many illustrations, including historical reproductions, line diagrams, and photographs of models of polyhedra. ''Polyhedra'' has ten chapters, the first four of which are primarily historical, with the remaining six more technical. The first chapter outlines the history of polyhedra from the ancient world up to

Hilbert's third problem The third of Hilbert's list of mathematical problems, presented in 1900, was the first to be solved. The problem is related to the following question: given any two polyhedra of equal volume, is it always possible to cut the first into finitely m ...

on the possibility of cutting polyhedra into pieces and reassembling them into different polyhedra. The second chapter considers the symmetries of polyhedra, 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 and

Archimedean solid In geometry, an Archimedean solid is one of the 13 solids first enumerated by Archimedes. They are the convex uniform polyhedra composed of regular polygons meeting in identical vertices, excluding the five Platonic solids (which are composed ...

s, and the honeycombs formed by

space-filling polyhedra 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 dime ...

. Chapter 3 covers the history of geometry in

medieval Islam The Islamic Golden Age was a period of cultural, economic, and scientific flourishing in the history of Islam, traditionally dated from the 8th century to the 14th century. This period is traditionally understood to have begun during the reign ...

and early Europe, including connections to astronomy and the study of visual perspective, and Chapter 4 concerns the contributions of

Johannes Kepler Johannes Kepler (; ; 27 December 1571 – 15 November 1630) was a German astronomer, mathematician, astrologer, natural philosopher and writer on music. He is a key figure in the 17th-century Scientific Revolution, best known for his laws ...

to polyhedra and his attempts to use polyhedra to model the structure of the universe. Among the remaining chapters, Chapter 5 concerns

angle In Euclidean geometry, an angle is the figure formed by two Ray (geometry), rays, called the ''Side (plane geometry), sides'' of the angle, sharing a common endpoint, called the ''vertex (geometry), vertex'' of the angle. Angles formed by two ...

s and

trigonometry Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. T ...

, 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 ...

, and the

Gauss–Bonnet theorem In the mathematical field of differential geometry, the Gauss–Bonnet theorem (or Gauss–Bonnet formula) is a fundamental formula which links the curvature of a surface to its underlying topology. In the simplest application, the case of a t ...

(including also some speculation on whether

René Descartes René Descartes ( or ; ; Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and science. Mathem ...

knew about the Euler characteristic prior to Euler). Chapter 6 covers Cauchy's rigidity theorem and

flexible polyhedra In geometry, a flexible polyhedron is a polyhedral surface without any boundary edges, whose shape can be continuously changed while keeping the shapes of all of its faces unchanged. The Cauchy rigidity theorem shows that in dimension 3 such ...

, and chapter 7 covers self-intersecting

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 ...

. Chapter 8 returns to the symmetries of polyhedra and the classification of possible symmetries, and chapter 9 concerns problems in

graph coloring In graph theory, graph coloring is a special case of graph labeling; it is an assignment of labels traditionally called "colors" to elements of a graph subject to certain constraints. In its simplest form, it is a way of coloring the vertices o ...

related to polyhedra such as 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 sh ...

. The final chapter includes material on

polyhedral compounds Polyhedral may refer to: *Dihedral (disambiguation), various meanings *Polyhedral compound * Polyhedral combinatorics * Polyhedral cone * Polyhedral cylinder * Polyhedral convex function * Polyhedral dice * Polyhedral dual * Polyhedral formula *Po ...

and metamorphoses of polyhedra.

Audience and reception

Most of the book requires little in the way of mathematical background, and can be read by interested amateurs; however, some of the material on symmetry towards the end of the book requires some background in

group theory In abstract algebra, group theory studies the algebraic structures known as group (mathematics), groups. The concept of a group is central to abstract algebra: other well-known algebraic structures, such as ring (mathematics), rings, field ...

. Reviewer

Bill Casselman William Allen Casselman (born November 27, 1941) is an American Canadian mathematician who works in representation theory and automorphic forms. He is a Professor Emeritus at the University of British Columbia. He is closely connected to the La ...

writes that it would probably not be appropriate to use as a textbook in this area, but could be valuable as additional reference material for an undergraduate geometry class. Reviewer Thomas Bending writes that "The writing is clear and entertaining", and reviewer Ed Sandifer writes that ''Polyhedra'' is "solid and fascinating ... likely to become the classic book on the topic ... worthy of many readings". Despite complaints about vague referencing of its sources and credits for its historical images, missed connections to modern work in group theory, difficult-to-follow proofs, and occasionally-clumsy illustrations, and typographical errors, Casselman also reviews the book positively, calling it "valuable and a labor of love". However, two experts on the topics of the book who also reviewed it,

polyhedral combinatorics Polyhedral combinatorics is a branch of mathematics, within combinatorics and discrete geometry, that studies the problems of counting and describing the faces of convex polyhedra and higher-dimensional convex polytopes. Research in polyhedral comb ...

specialist

Peter McMullen Peter McMullen (born 11 May 1942) is a British mathematician, a professor emeritus of mathematics at University College London. Education and career McMullen earned bachelor's and master's degrees from Trinity College, Cambridge, and studied at ...

and

historian of mathematics The history of mathematics deals with the origin of discoveries in mathematics and the mathematical methods and notation of the past. Before the modern age and the worldwide spread of knowledge, written examples of new mathematical developments ...

Judith Grabiner Judith Victor Grabiner (born October 12, 1938) is an American mathematician and historian of mathematics, who is Flora Sanborn Pitzer Professor Emerita of Mathematics at Pitzer College, one of the Claremont Colleges. Her main interest is in mathema ...

, were much less positive. McMullen writes that "There appears to be some degree of carelessness in the preparation of the book", pointing to errors including calling the

Dehn invariant In geometry, the Dehn invariant is a value used to determine whether one polyhedron can be cut into pieces and reassembled ("dissection problem, dissected") into another, and whether a polyhedron or its dissections can Honeycomb (geometry), tile s ...

a number, mis-dating

Hilbert's problems Hilbert's problems are 23 problems in mathematics published by German mathematician David Hilbert in 1900. They were all unsolved at the time, and several proved to be very influential for 20th-century mathematics. Hilbert presented ten of the pro ...

, misspelling the name of artist

Wenzel Jamnitzer Wenzel Jamnitzer (sometimes Jamitzer, or Wenzel ''Gemniczer'') (1507/1508 – 19 December 1585) was a Northern Mannerist goldsmith, artist, and printmaker in etching, who worked in Nuremberg. He was the best known German goldsmith of his e ...

and misattributing to Jamnitzer an image by

M. C. Escher Maurits Cornelis Escher (; 17 June 1898 – 27 March 1972) was a Dutch graphic artist who made mathematically inspired woodcuts, lithographs, and mezzotints. Despite wide popular interest, Escher was for most of his life neglected in t ...

, and using idiosyncratic and occasionally incorrect names for polyhedra. McMullen writes of these errors that "every time I look at the book, I find more", casting into doubt the other less-familiar parts of the book's content. And Grabiner faults the book's history as naive or mistaken, citing as examples its claims that the discovery of

irrational number In mathematics, the irrational numbers (from in- prefix assimilated to ir- (negative prefix, privative) + rational) are all the real numbers that are not rational numbers. That is, irrational numbers cannot be expressed as the ratio of two integ ...

s ended

Pythagorean Pythagorean, meaning of or pertaining to the ancient Ionian mathematician, philosopher, and music theorist Pythagoras, may refer to: Philosophy * Pythagoreanism, the esoteric and metaphysical beliefs purported to have been held by Pythagoras * Ne ...

mysticism, and that pre-Keplerian astronomy consisted only of observation and record-keeping. She accuses Cromwell of basing his narrative on secondary sources rather than checking the original sources he cites, points to sloppy sourcing of historical quotations, and complains about the book's minimal coverage of Islamic and medieval geometry. She writes that the book can be enjoyed as "a treasury" of "beautiful models" and "examples of the impact of polyhedra on the imagination of artists" but should not be relied on for historical insights.

References

{{reflist, refs= {{citation , last = Bending , first = Thomas , date = March 1999 , doi = 10.2307/3618744 , issue = 496 , 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 = 3618744 , pages = 178–179 , title = Review of ''Polyhedra'' , volume = 83 {{citation , last = Böhm , first = J. , journal =

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

, title = Review of ''Polyhedra'' , zbl=0888.52012 {{citation , last = Casselman , first = Bill , author-link = Bill Casselman (mathematician) , date = September 1998 , issue = 8 , journal =

Notices of the American Mathematical Society ''Notices of the American Mathematical Society'' is the membership journal of the American Mathematical Society (AMS), published monthly except for the combined June/July issue. The first volume appeared in 1953. Each issue of the magazine since ...

, pages = 978–980 , title = Review of ''Polyhedra'' , url = https://www.ams.org/notices/199808/review-casselman.pdf , volume = 45 {{citation , last = Grabiner , first = Judith V. , authorlink = Judith Grabiner , date = December 1998 , doi = 10.1086/384173 , issue = 4 , journal =

Isis Isis (; ''Ēse''; ; Meroitic: ''Wos'' 'a''or ''Wusa''; Phoenician: 𐤀𐤎, romanized: ʾs) was a major goddess in ancient Egyptian religion whose worship spread throughout the Greco-Roman world. Isis was first mentioned in the Old Kingd ...

, jstor = 236751 , pages = 714–715 , title = Review of ''Polyhedra'' , volume = 89 {{citation , last = McMullen , first = Peter , author-link = Peter McMullen , journal =

Mathematical Reviews ''Mathematical Reviews'' is a journal published by the American Mathematical Society (AMS) that contains brief synopses, and in some cases evaluations, of many articles in mathematics, statistics, and theoretical computer science. The AMS also pu ...

, mr = 1458063 , title = Review of ''Polyhedra'' , year = 1998 {{citation , last = Sandifer , first = Ed , date = February 1999 , journal = MAA Reviews , publisher =

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 ...

, title = Review of ''Polyhedra'' , url = https://www.maa.org/press/maa-reviews/polyhedra Polyhedra Mathematics books 1997 non-fiction books

Topics

Audience and reception

See also

References