Mathematics, Impossible

Geometric objects that cannot be constructed in three-dimensional space are considered "impossible." What makes them intriguing is that despite this difficulty, some representation of them is possible. Even though they can never be built or held, it is possible to imagine what they would look like and to learn about how these impossible objects behave.

Local versus Global

In the study of mathematics, distinctions are routinely made between the local properties of an object—what small pieces of the object look like—and the properties of the object as a whole. Impossible objects are all "wellbehaved" locally; it is only when we consider them globally that contradictions arise.

Consider two lithographs by Dutch artist M. C. Escher. (Although not shown in this entry, these artworks are easily viewed on various Internet web sites and on books of Escher's art.) The first is "Belvedere," created in 1958. At first glance, this seems to be a straightforward depiction of an open-air building with pillars and archways, with people scattered around at various points. But it does not take long to notice that this picture is filled with incompatibilities; many of the pillars do not start or end where they should, causing the building to twist around itself in unrealistic ways.

Similar trickery occurs in another Escher work, the 1961 lithograph entitled "Waterfall." This work depicts the peculiar phenomenon of water falling off a ledge and then proceeding to flow around a right-angled path until it reaches the top again. What makes this picture difficult to reconcile with reality is not that the water is fighting gravity but rather that the water is not fighting gravity. And once again, a close look at the picture uncovers all sorts of physical impossibilities.

In both examples, any small portion of the scene is a perfectly reasonable depiction of something that could exist in our three-dimensional world. It is only the way in which these small pieces are glued together that causes problems. This difference between local and global behavior leads us to the subject of topology , in which (among many other things) surfaces with similar behavior are studied.

Mathematical Surfaces

In mathematical terminology, a surface is an object that is "locally two dimensional." An example is a hollow sphere. Although the entire sphere is three-dimensional and cannot be squashed into a plane without radically altering it, any small patch on the sphere looks like a slightly curved piece of infinitesimally thin paper and consequently is considered to be two dimensional. The formal way to say this is that any small patch of the sphere is "topologically equivalent" to a small patch of a plane. Two objects are said to be topologically equivalent if one can be stretched or compressed to form the other. (Imagine that the objects are made of infinitely stretchable and infinitely thin rubber.)

Moreover, the sphere is an orientable surface. When considering the surface of the sphere itself, it has no boundary. If you were traveling "in" the surface rather than on it, you would never fall off the edge but instead would return to your starting point.

Building a Sphere. A number of surfaces can be built by gluing together regions in the plane. The sphere is an example of such a surface, since each hemisphere is (topologically) planar. Another way to view the sphere as a planar object with some gluing is as follows.

First, draw a circle in the plane. Draw two dots on it, one at either end of the "equator"—that is, the horizontal diameter. The sphere will be formed by taking the disc (the interior of the circle) and gluing its boundary (the circle) together in the following way: each point on the top half of the circle (above the two dots) is glued to the point directly below on the lower half of the circle. It may take a little practice to be able to visualize this without actually doing the cutting and pasting, but the result is a shape that fully encloses a hollow center—in other words, a shape that, topologically speaking, is a sphere.

Building Nonorientable Surfaces. Using the technique of creating surfaces from two-dimensional regions, one can build objects more complicated than a sphere and whose final shapes are not as easy (or not even possible) to visualize. Two of these objects—a torus and a Klein bottle—can be formed by gluing together the edges of a square.

When the left and right edges of a square (a plane) are glued to each other, every point on an edge is glued to the point directly across from it. Once glued in this way, the square has become rolled into a cylinder, and looks like the cardboard tube in a roll of paper towels. The top and bottom edges of the square, which previously were line segments, have now become the circles at the top and bottom of the cylinder. These two circles are then glued to each other by bending the cylinder into a circular shape. The end product is a hollow doughnut-shaped figure, called a torus. Like the sphere, the torus has no boundary, and it is an orientable surface. When a "surface traveler" goes off an edge, he returns on the opposite side.

A nonorientable surface and an "impossible" object known as a Klein bottle can be created by changing how the edges of the square are glued. For this construction, an intermediate object known as a Möbius strip must first be created. Bring together the opposite sides of a square as before, but first twist them 180 degrees. The resulting object is almost a cylinder, but it is a cylinder with a twist. A point once on the upper or lower edge will now be glued to the point diagonally opposite it. For example, points on the left half of the top edge are attached to points on the right half of the bottom edge.

The name for this object is a Möbius strip, and it has many unusual properties. The Möbius strip has a boundary like a cylinder, but it has only one edge instead of two, and it has only one side. If you travel along the edge, you will cover the entire boundary, eventually returning to the place you started, but mirror-reversed. (Remember that traveling in a surface means traveling inside the surface, not on top of it.) Hence, the Möbius strip is considered nonorientable.

The Klein bottle results from sewing together two Möbius strips along their single edge. Another way to make a Klein bottle is from a cylinder (as when constructing a torus), but by bringing together opposite edges of the boundary circles. But because of the twist, this is not as simple as before. The only way to picture this in three-dimensional space is to pass the surface through itself so that one circle can be glued to the other "from behind." However, this is not a realistic procedure because this self-intersection causes points to be attached to each other which were not supposed to be.

Thus the Klein bottle is a surface that cannot truly exist in three-dimensional space, even though a three-dimensional model is used to show its general shape. To get a general idea, stretch the neck of a bottle through its side and join its end to a hole in the base. But realize that a true Klein bottle requires four dimensions because the surface must pass through itself without the hole !

A Klein bottle therefore has one side and no boundary: it passes through itself; its inside is its outside; and it contains itself.

As with the Escher pictures, any small portion of the Klein bottle is perfectly easy to visualize in three-dimensional space; it is only the way in which these small pieces are attached to each other that yields a globally "impossible" object.

One more important "impossible" surface is called the projective plane. To construct it, start with a disc. Join points in the top half of the boundary (the circle) with points in the bottom half, but attach each point to the one diametrically opposite (that is, exactly halfway around the circle). This gives an "impossible" surface that cannot be constructed in three-dimensional space, even though it is locally two dimensional.

Compact Surfaces

It would seem as though there are an infinite number and variety of surfaces one could construct by gluing boundaries of planar regions. By drawing more elaborate shapes in the plane, one will come up with all sorts of combinations of edges to glue, with an occasional twist thrown in. But although there are an infinite number of surfaces that can be constructed, surprisingly enough their variety is limited and can be very precisely described.

If the entire boundary of the planar region is glued together in some way, the resulting object will be what is called a compact surface. A compact surface is a finite surface without a boundary—this rules out the plane itself and a surface such as a cylinder, which has a boundary. Every compact surface is topologically equivalent to one of the following: a sphere, a "sum" of finitely many tori (hollow doughnuts), or a "sum" of finitely many projective planes.

To "add" two surfaces means to cut a small hole in each one and glue the two surfaces together along the boundaries of these holes. For example, when three tori are added, the result is a three-holed torus. It is, of course, much harder to visualize the result of adding together a collection of projective planes, but interestingly enough, the sum of two projective planes is a Klein bottle. The remarkable fact that all conceivable surfaces can be described in this straightforward manner is called the Classification Theorem for Compact Surfaces.

see also Escher, M. C.; Möbius, August Ferdinand; Topology.

Naomi Klarreich

Bibliography

Escher, M. C. The Graphic Work of M. C. Escher. New York: Wings Books, 1996.

Weeks, Jeffrey R. The Shape of Space. New York: Marcel Dekker, Inc., 1985.

Internet Resources

"The Math of Non-Orientable Surfaces." <http://pantheon.yale.edu/~jar55/math/project/math.htm>.

VIDEO GAMES AND TOPOLOGY

The surface of a torus is topological space found in most video games, where a vehicle (for example) goes off the right-hand side of the screen only to reappear on the left, or off the top only to reappear on the bottom.