Sphere Eversion

How to turn a sphere inside-out ? One of the most interesting puzzles in topology.

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: 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: 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: The Geometry Center has more sphere eversion history, many pictures and links at: Visit Mike McGuffin's pages for pictures, movies, programs, and links: If you have any remarks, more references or visualizations, just E-mail me: