# A Topological Picturebook

## George K. Francis

Language: English

Pages: 194

ISBN: 0387345426

Format: PDF / Kindle (mobi) / ePub

Praise for George Francis's ** A Topological Picturebook**:

Bravo to Springer for reissuing this unique and beautiful book! It not only reminds the older generation of the pleasures of doing mathematics by hand, but also shows the new generation what ``hands on'' really means.

- John Stillwell, University of San Francisco

* The Topological Picturebook* has taught a whole generation of mathematicians to draw, to see, and to think.

- Tony Robbin, artist and author of **Shadows of Reality: The Fourth Dimension in Relativity, Cubism, and Modern Thought**

*The classic reference for how to present topological information visually, full of amazing hand-drawn pictures of complicated surfaces.*

**- John Sullivan, Technische Universitat Berlin**

** A Topological Picturebook** lets students see topology as the original discoverers conceived it: concrete and visual, free of the formalism that burdens conventional textbooks.

- Jeffrey Weeks, author of **The Shape of Space**

** A Topological Picturebook** is a visual feast for anyone concerned with mathematical images. Francis provides exquisite examples to build one's "visualization muscles". At the same time, he explains the underlying principles and design techniques for readers to create their own lucid drawings.

- George W. Hart, Stony Brook University

In this collection of narrative gems and intriguing hand-drawn pictures, George Francis demonstrates the chicken-and-egg relationship, in mathematics, of image and text. Since the book was first published, the case for pictures in mathematics has been won, and now it is time to reflect on their meaning. ** A Topological Picturebook** remains indispensable.

- Marjorie Senechal, Smith College and co-editor of the Mathematical Intelligencer

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Rainich, who first introduced me to the wonders of projective geometry, held that the presence of rotations in the Euclidean group of congruences reflects the Greek habit of drawing figures on the ground as the students stood about in a circle. Be that as it may, the vertical blackboard is now the mathematician's prime medium of pedagogical expression. No amount of explanation accompanying a complete figure on a page can match the information transmitted while creating the same figure at the

of the chalk strengthens the appearance of curvature. A fair command of these techniques should be as common a virtue among topologists as their glib command of the language of sets and the notation of symbolic logic. The kind of picture to draw on the board depends, of course, on its purpose. In the first instance, the lecturer wants a pictograph to accent the exposition. Here speed and simplicity are essential, while artistry is not. The exposition could proceed quite well without the picture.

have a common "infinite line" has the following perspective interpretation. 46 A TOPOLOGICAL PrCTUREBOOK As the object point at the end of the sightline moves away from the observer along a straight line in space, its trace on the picture plane moves along a line segment until it stops at the vanishing point of the object line, which is now parallel to the sight line. For this reason, the vanishing point remains fixed as long as the object line moves parallel to itself in space. As long as

cube in space. Recall that the set of 3 by 3 matrices P whose transpose is equal to their inverse, p T = p-I, form the orthogonal group. Such a matrix is also called orthonormal because ppT = I says that the three column vectors are mutually perpendicular unit vectors. Generally, we prefer a right handed system, so that det(P) = + 1 and the third column vector is the cross product of the first two. These matrices constitute the group SO(3) of "special" or sense-preserving orthogonal

their maxima near motion north pole, as the surface is made to wobble a bit. A I-parameter family of functions is recorded by a path on the polar cap. As the function "crosses" one of the three solid radii, two of the saddles are at the same height. The same happens at the dotted radii, but being "inessential," no choice between competing cut circles is necessary. Note, in passing, that 3-saddled N has all three borders horizontal, hence they could be capped by three extrema of the height