Editor’s Note: Published in 1957, this article comes from Martin Gardner’s legendary Scientific American column Mathematical Games. Read more in our special digital issue, Fun and Games.
As many readers of this magazine are aware, a Möbius band is a geometrical curiosity which has only one surface and one edge. Such figures are the concern of the branch of mathematics called topology. People who have a casual interest in mathematics may get the idea that a topologist is a mathematical playboy who spends his time making Möbius bands and other diverting topological models. If they were to open any recent textbook of topology, they would be surprised. They would find page after page of symbols, seldom relieved by a picture or diagram. It is true that topology grew out of the consideration of geometrical puzzles, but today it is a jungle of abstract theory. Topologists are suspicious of theorems that must be visualized in order to be understood.
Serious topological studies nonetheless produce a constant flow of weird and amusing models. Consider, for example, the double Möbius band. This is formed by placing two strips of paper together, giving them a single half-twist as if they were one strip, and joining their ends as shown in the illustration below.
On supporting science journalism
If you're enjoying this article, consider supporting our award-winning journalism by subscribing. By purchasing a subscription you are helping to ensure the future of impactful stories about the discoveries and ideas shaping our world today.
[Play science-inspired games, puzzles and quizzes in our new Games section]
Double Möbius band is made by placing two strips of paper together (left), giving both of them a half twist and joining them as indicated by the straight lines at right.
James Egleson
We now have what appears to be two nested Möbius bands. Indeed, you can “prove” that there are two separate bands by putting your finger between the bands and running it all the way around them until you come back to the point at which you started. A bug crawling between the bands could circle them indefinitely, always walking on one strip with the other strip sliding along its back. At no point would he find the “floor” meeting the “ceiling.” An intelligent bug would conclude that he was walking between the surfaces of two separate bands.
Suppose, however, that the bug made a mark on the floor, and circled the bands until he reached the mark again. It would find the mark not on the floor but on the ceiling, and it would require a second trip around the bands to find it on the floor again! The bug would need considerable imagination to comprehend that both floor and ceiling were one side of a single strip. What appears to be two nested bands is actually one large band! When you open the model into the large band, you will find it difficult to restore it to its original form.
When the band is in its double form, two separate edges of it run parallel to each other; they circle the model twice. Imagine that these edges are joined and that the band is made of thin rubber. You would then have a tube which could be inflated to make a torus (the topologist’s term for the surface of a doughnut). The joined edges would form a closed curve that coiled twice around the torus. This means that a torus can be cut along such a curved line to form the double Möbius band.
The double band is identical, in fact, with a single band that is given four half-twists before its ends are joined. It is possible to cut a torus into a band with any desired even number of half-twists, but impossible to cut it so as to produce bands with an odd number of such twists. This is because the torus is a two-sided surface and only bands with an even number of half-twists are two-sided. Although two-sided surfaces can be made by cutting one-sided ones, the reverse is not possible. If we wish to obtain one-sided bands (bands with an odd number of half-twists) by cutting a surface without edges, we must resort to cutting a Klein bottle. This remarkable one-sided bottle is described in “Topology,” by Albert W. Tucker and Herbert S. Bailey, Jr. [Scientific American; January 1950].
The simple Möbius band is made by giving a strip one half-twist before joining the ends. Can the band somehow be stretched until this edge is a triangle? The answer is yes. The first man to devise such a model was Bryant Tuckerman, one of the four pioneers in the art of folding flexagons [“Mathematical Games”; December 1956]. The illustration below shows how a piece of paper can be cut, folded and pasted to create Tuckerman’s model.
Möbius band with triangular edge was devised by Bryant Tuckerman. If the flat figure is redrawn, preferably on a larger scale, the dimensional polyhedral model may be assembled as follows. First, cut out the figure. Second, fold it “down” along the solid lines. Third, fold it in the opposite direction along the broken lines. Fourth, by applying paste to the four tabs, join edges A and A, B and B, C and C, and D and D. The heavy lines in the finished polyhedron trace the triangular boundary of the Möbius surface.
James Egleson
Surfaces may not only have one or two sides; they may also differ topologically in the number and structure of their edges. Such traits cannot be altered by distorting the surface; hence they are called topological invariants. Let us consider surfaces with no more than two edges, and edges that are either simple closed curves or in the form of an ordinary single knot. If the surface has two edges, they may be independent of each other or linked. Within these limits we can list the following 16 kinds of surfaces (excluding edgeless surfaces such as the sphere, the torus and the Klein bottle):
ONE-SIDED, ONE-EDGED
1. Edge is a simple closed curve.
2. Edge is knotted.
TWO-SIDED, ONE-EDGED
3. Edge is a simple closed curve.
4. Edge is knotted.
ONE-SIDED, TWO-EDGED
5. Both edges are simple closed curves, unlinked.
6. Both edges are simple closed curves, linked.
7. Both edges are knotted, unlinked.
8. Both edges are knotted, linked.
9. One edge is simple; one knotted, unlinked.
10. One edge is simple; one knotted, linked.
TWO-SIDED, TWO-EDGED
11. Both edges are simple closed curves, unlinked.
12. Both edges are simple closed curves, linked.
13. Both edges are knotted, unlinked.
14. Both edges are knotted, linked.
15. One edge is simple; one knotted, unlinked.
16. One edge is simple; one knotted, linked.
Paper models are easily constructed to illustrate examples of each of these 16 surfaces. Models for surfaces 1 through 12 are depicted below. The reader may enjoy trying to construct models of the remaining four surfaces, drawings of which will appear in this department next month.
Topological models are 12 of the 16 that can be made from surfaces with no more than two edges, and edges which are a simple closed curve or a single knot. The simple Möbius band is 1.
James Egleson
When some of these models are cut with scissors in certain ways, the results are startling. As almost everyone who has played with a Möbius band knows, cutting the band in half lengthwise does not produce two separate bands, as one might expect, but one large band. (The large band has four half-twists; thus it can be made into the double Möbius band described earlier.) Not so well known is the fact that if you start the cut a third of the way between one edge and the other, and cut until you return to the starting point, the Möbius band opens into a large band linked with a smaller one.
Cutting surface 12 in half yields two interlocked bands of the same size, each exactly like the original one. Cutting surface 2 in half results in a large band that has a knot in it. This latter stunt was the subject of a booklet that enjoyed a wide sale in Vienna in the 1880s. The booklet revealed the secret of forming a knot in a cloth band without resorting to magical trickery.
In saying that two edges are “linked” we mean linked in the manner of two links in a chain. To separate the links it is necessary to open one link and pass the other through the opening. It is possible, however, to interlock two closed curves in such a manner that in order to separate them it is not necessary to pass one through an opening in the other. The simplest way to do this is shown by the upper curves in the illustration below. These curves can be separated by passing one band through itself at point A.
Interlocked curves may be separated without passing one through an opening in the other. The curves at the top may be separated by passing black curve through itself at A.
James Egleson
The three closed curves at the bottom of the illustration are also inseparable without being linked. If you remove any one curve, the other two are free; if you link any pair of curves, it frees the third one. This structure, by the way, is topologically identical with the familiar three-ring trademark of a well-known brand of beer. These are called Borromean rings because they formed the coat of arms for the Renaissance Italian family of Borromeo.