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## Further reading about turning the sphere inside out

*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.

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.

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.

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.

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.

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

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

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**Prev:** *Making Waves*

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