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A Topological Picturebook

Springer Verlag
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A Topological Picturebook, first published in 1987, is now available in a softcover edition. Having never seen the first edition, I was very pleasantly surprised when I opened the book. I had been expecting a large collection of computer-generated images of a variety of surfaces and geometrical constructions. While there are a few images like that, this is really a much richer book. Indeed, the approach it offers to drawing can have a significant impact on how we teach and think about mathematics.

The author’s begins modestly. His first line is: “This is a book about how to draw mathematical pictures.” He admits that his first inclination was to call his effort “A Graphical Calculus” with formalized procedures and rules for “calculating” how to produce good mathematical pictures. Fortunately, he settled on a less didactic approach and the result is a gem.

Historical connections for this work go back at least as far as Felix Klein with his approach to studying and explaining complicated Riemann surfaces using pictures and models. The author has provided drawings for Bill Thurston’s The Geometry and Topology of Three-Manifolds and he has much in common with both Thurston’s hands-on approach and his way of communicating mathematics. A first glance at the hand-drawn pictures in this book made me think immediately of Hilbert and Cohn-Vossen’s Geometry and the Imagination. But Francis’ book has much more the flavor of “how to” and “how to think about”.

Each chapter is a story about an area of mathematics — indeed, as the author says, a kind of “picture story”. The first chapter is a good example. It deals with descriptive topology and the relationship between the theory of general position and stable singularities of low-dimensional mappings. The author used the drawings here in lectures on Thom and Zeeman’s catastrophe theory where accurately drawn surfaces are critical for getting the ideas across. This chapter is particularly recommended for people who draw pictures during their lectures.

The second chapter is a loose collection of stories about “media and methods”, describing (among other things) a variety of ways that topologists can prepare illustrations for their papers. However, the advice is not just for publishing topologists, as the section about drawing on a blackboard with chalk amply demonstrates. There are a couple of color plates with photographs of gorgeous blackboard drawings of the dunce cap and a diapered (!) trefoil knot.

My favorite among the remaining chapters is the one on sphere eversion. As the author notes, so many graphical techniques have been applied to the associated visualization problem that it has become a paradigm of descriptive topology. The challenge is to show the motion of a spherical surface as it passes through itself in three-space without tearing or creasing until it is turned inside-out. Perhaps the most visually striking eversion in the one due to Bernard Morin; it is shown in Figures 8 and 13 of Chapter 6.

If you’re good at visualization and illustration, this book can help you become better yet. If you struggle to make comprehensible sketches during your lectures, this will give you concrete and specific suggestions for developing your skills. If you just appreciate skillful drawing and illustration, this book deserves a look.

Bill Satzer ( is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.


Date Received: 
Friday, November 3, 2006
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George K. Francis
Publication Date: 
William J. Satzer
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 Preface.- Descriptive Topology.- Methods and Media.- Pictures in Perspective.- The Impossible Tribar.- Shadows from Higher Dimension.- Sphere Eversions.- Group Pictures.- The Figure Eight Knot.- Postscript.- Bibliography.- Index.

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Thursday, July 23, 2009