Scroll beyond the pictures for a detailed explanation.
EVERTING THE SPHERE - A VISUALIZATION (c) 1996,98 by Erik de Neve
TOPOLOGY
The branch of mathematics called topology studies special properties of
shapes and surfaces. In topology, the size and to a certain extent the
shape of an object are irrelevant. In particular, two objects or
structures are considered topologically equivalent if one can be
stretched and deformed in a continuous way, to become exactly like the
other. Objects like the sphere and the torus (doughnut) are called
two-dimensional manifolds. Surfaces on such manifolds may also pass
through themselves, as long as this does not give rise to intermediate
stages with sharp points or creases.
EXAMPLE
The accompanying image sphere.gif shows some steps of my own approach
to everting the sphere (turning it inside-out). If you know enough
about topology you might want to try it yourself first, without
looking at the pictures and explanation.
First, try the rough approach of sequence A : push the bottom up, and
the top down, through each other. This results in an everted sphere (in
this case, imagine the inside was painted with a darker color) but also
leaves a ring-shaped fold, which you cannot get rid of just by making
it infinitely small, because that would create a sharp crease. Instead,
we pull the ring down and make it smaller.
Sequence B gives a recipe for doubling such a ring. Squeeze the
ring so its two sides touch and merge (first two pictures).
To see what happens when they merge, look at sequences C, D and E,
which provide cross sections of the two folds, in the same way as the
third and fourth picture in seq. A represent cross sections of the
whole sphere.
When one loop is made smaller, as in C, it is clear how it can traverse
the other, and in that case the two folds will cross each other. But
when pictured differently, the loops can actually lose their identity,
so two folds crossing each other (third picture in B) are topologically
equivalent to parallel folds touching each other (second picture in B).
To make this plausible, imagine how sequence C would change if the
two loops were gradually changed into completely overlapping circles of
the same size, as in the last picture of D.
If this does not convince you, follow C up to the middle picture
and then deform into two separate loops using the alternative in E.
Now, it is impossible to decide whether the loops have switched places
or not.
So, when two merging folds turn into two crossing folds, we can create
the figure 8 shape in the third picture of B. Using the same trick to
turn crossing folds into touching folds again, this time vertically, we
can end up with two rings (last picture).
Performing A twice will give the original sphere (outside out) with two
rings, one on the inside and one on the outside. Now, multiply the
outside ring as explained above. Then we can let two opposite rings
eliminate each other. For this, imagine the rings as concentric, one
around the other, then let them come closer and merge as indicated in
F. Now we are left with just one ring on the outside, and all that
remains is to execute seq. A in reverse.
Note that the procedures above can also be combined in a much simpler
way: perform F backwards, to generate two opposite ring folds out of
nothing; let the ring that's on the outside of the sphere spawn another
using B, then have them eliminate each other, leaving the one ring we
need for eversion. You don't even need to make two complete circles to
produce the extra circle - any stretch of fold will do.
HISTORY
Sphere eversion is a relatively recent mathematical discovery.
In 1958, mathematician Stephen Smale devised an abstract formula that
proved sphere eversion was possible. It was not until the 1970s that
the (blind !) mathematician Bernard Morin came up with a visualization,
based on work by Arnold Shapiro. Morin also developed an approach in
which the sphere is represented by a polyhedron (built from flat
triangular and square faces) and then everted.
Rob Kusner with others has developed a 'minimax' eversion, which
is the 'simplest' way to evert a sphere, simple meaning minimization
of an 'elastic bending energy'.
REFERENCES
I still don't know of any pictures of Morin's eversion. There is a 1966
Scientific American article on sphere eversion by Anthony Phillips,
and a computer-generated eversion by Thurston on the cover of Scientific
American, August 1993, which shows a symmetrical, 8-fold flower petal
pattern, with a balloon being blown up in the center, and the lower half
the original, shrinking sphere. However, no explanation is given in the
article itself, which is about computer proofs in mathematics. There have
been earlier articles in Scientific American which deal with everting a
torus, but only after cutting a hole in it first.
A well illustrated, *very* good introductory text on topology is:
The Shape of Space - How to Visualize Surfaces and
Three-Dimensional Manifolds. by Jeffrey R. Weeks
(1985) Marcel Dekker, Inc / New York - Basel ISBN: 0-8247-7437-X
Weeks is also the author of an article on the mathematics of
three-dimensional manifolds in Scientific American, July 1984.
The history of sphere eversion is discussed in:
Islands of Truth - a mathematical mystery cruise p. 46-
by Ivar S. Peterson (1990) W.H. Freeman & Company
Stephen Smale has received the Fields Medal (the 'Nobel prize' for
mathematics) in 1966 for his work in topology, and went on to become
one of the pioneers of chaotic dynamical systems. A description of his
career can be found in:
Chaos: making a new science. by James Gleick
(1987) Viking Penguin Inc. New York ISBN: 0 14 00.9250 1
The Geometry Center has more sphere eversion history, many
pictures and links at:
http://freeabel.geom.umn.edu/docs/outreach/oi/history.html
Visit Mike McGuffin's pages for pictures, movies, programs, and links:
http://www.csclub.uwaterloo.ca/~mjmcguff/eversion/
If you have any remarks, more references or visualizations,
just E-mail me :
Erik de Neve Erik AT_usefuldreams DOT org