Outside In: Further reading about turning the sphere inside out

Up: Introduction
Prev: Making Waves

Further reading about turning the sphere inside out

[Levy 1995]

Making Waves, a Guide to the Ideas behind Outside In, Silvio Levy, A K Peters, Wellesley, MA, 1995.

This 48-page full-color book introduces more precisely the mathematical ideas behind Outside In and develops them further. It requires very little mathematical background.

[Francis 1987]

George K. Francis, A Topological Picturebook, Springer, New York, 1987.

This richly illustrated book is a guide to ``descriptive topology''. Chapter 6 is entirely devoted to sphere eversions. Following the text requires a certain familiarity with topology, but even mathematically naïve readers will find the book worth looking at just for the figures.

[Max 1977]

Nelson Max, ``Turning a Sphere Inside Out'', International Film Bureau, Chicago, 1977 (video).

This early triumph of computer animation explains Morin's eversion, illustrating it with real-life models (made by Charles Pugh) as well as computer-animated sequences rendered by Jim Blinn, based on a digitization of Pugh's models. A frame from the video is included here.

[Francis and Morin 1979]

George K. Francis and Bernard Morin, ``Arnold Shapiro's Eversion of the Sphere'', Math. Intelligencer, 2 (1979), 200--203.

Although Shapiro was probably the first person who had a detailed idea of how an explicit eversion might be realized, his method only became well-known many years after his death, largely thanks to this article. The level of the article is intermediate: it requires some topology and a good spatial imagination, but is not very technical.

[Phillips 1966]

Anthony Phillips, ``Turning a surface inside out'', Scientific American, May 1966, 112--120.

In this clear and accessible article, a visual ``recipe'' for turning the sphere inside out was published for the time. One of the original drawings by Phillips appears here.

[Smale 1958]

Steve Smale, ``A classification of immersions of the two-sphere'', Trans. Amer. Math. Soc. 90 (1958), 281--290.

This paper started the whole subject of sphere eversions, because it contains a general theorem (which unfortunately requires very technical language to state), one of the consequences of which is that the sphere can be turned inside out by means of smooth motions and self-intersections. The paper is accessible only to mathematicians.

References in Making Waves

[Gould 1987]

Stephen Jay Gould, Time's Arrow, Time's Cycle: Myth and Metaphor in the Discovery of Geological Time, Harvard University Press, Cambridge, MA, 1987.

Up: Introduction
Prev: Making Waves

The Geometry Center Home Page