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A topological picturebook. (English) Zbl 0612.57001
New York, NY.: Springer (ISBN 0-387-96426-6; 978-0-387-34542-0/ebook). xv, 194 p. (1987).
This book is about how to draw mathematical pictures. The author’s major purpose is to encourage mathematicians to illustrate their work. Thus he describes his own procedures and various techniques of this art which he appropriately calls ”graphical calculus”. He begins by describing the relationship between Whitney’s and Thom’s theory of general position, of stable singularities of low-dimensional mappings, and descriptive topology. He then shows how a topologist may produce his own pictures suitable for publication, and reviews the principles of perspective drawing from the viewpoint of projective geometry. He also mentions the non-perspective (”engineering”) drawing and the techniques of shading.
Next he studies optical illusions which serve as an introduction to the differential geometry of spaces of constant curvature. After an analysis of what a picture of an algebraic operation should look like, there follows a study of the visualization problems in differential topology. The last two chapters are on illustrating the topology behind the presentation of the mapping class groups and on geometry of the complement of fibred knots. The book contains excellent illustrations of several classical topological concepts, e.g. Whitney umbrella, dunce hat, Morin sphere eversion, Zeeman catastrophe machine, King Solomon seal, Boy surface, etc.
At the end there is a short discussion of using computer graphics in descriptive topology. The author is already well known for, among other things, his illustrations of two important papers and the celebrated lecture notes by Thurston, which were very helpful in understanding the material presented therein. This book is a most welcome addition to the mathematical literature.
Reviewer: D.Repovš

57-02 Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes
51N05 Descriptive geometry
51H05 General theory of topological geometry
57M99 General low-dimensional topology
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