Mobius Strip (1 face, 1edge) and Cylindrical Loops (2 faces, 2 edges)

Building on our exploration of perimeter in one dimension and area in two dimensions, we entered the real world of three dimensions. Starting with a rectangular strip of paper about 1.5” x 11” (by cutting a 1.5” piece the long way from a piece of regular 8.5” X 11” paper), I taped the two short edges together to make a cylindrical loop. Asking the children what this was called, many said “circle.” Of course, the bases of a cylinder are circles, but this is a 3D object called a cylindrical loop. Then I asked them how many faces and edges it had. It took a little time to see that it is a two faced, two edge solid. Drawing a line down the middle of the inner and outer face proves this fact.

Then I took another thin rectangle and before connecting the short edges, gave one edge a 180 degree turn (1/2 twist) and then connected the edges. They still believed it to be a two faced, two edge solid. Drawing a line down the middle showed them that it covered all surfaces so there is only one face. Drawing a line along one of the edges also reveals that it is a continuous single edge. This discovery was made in 1858 by the German astronomer and mathematician August Ferdinand Möbius. Adding 0 twists, 2 twists, or more generally, an even number of twists, will always produce a two-faced, two-edged cylindrical loop. Similarly, an odd number of twists produce various one-faced surfaces. Interestingly, the boundary of a Mobius band or strip is a single closed curve with one face and one edge.

Our Mobius Strip Exploration included the following experiments:

1. faces and edges of a circular loop; 2. proving that a Mobius Strip has only one face and one edge; 3. creating one non-Mobius loop from a Mobius Strip by cutting halves; 4. creating one Mobius loop connected with a non-Mobius loop from a Mobius strip by cutting in thirds; 5. creating a square from two perpendicular connected cylindrical loops; 6. creating two interlocking hearts from two mirror image perpendicular connected Mobius strips. 

The pdf includes a work sheet to guide students through this exploration but it is much easier to have been in class with me rather than just following along from the instructions.

The children were amazed to find that a square was produced when linking two cylindrical loops together at a 90 degree or perpendicular angle and then cutting down the middle of each loop. Then the children immediately wanted to know what happened when we connected a cylindrical loop and a Mobius strip (same result); then of course, I had them do the same thing with two Mobius strips. IMPORTANT NOTE: HERE, THE MOBIUS STRIPS NEED TO BE MIRROR IMAGES OF EACH OTHER (TWIST THEM IN DIFFERENT DIRECTIONS). The result is two interlocking hearts. Makes for a great holiday ornament. 

Cutting single Mobius strips in half or in thirds revealed yet other amazing results. A half cut transformed the Mobius strip into a twisted cylindrical loop. A third cut transformed the Mobius strip into two interlocking loops (one a Mobius strip and the other a cylindrical loop). 

Several students tried their own experiments with extraordinary results. Try to connect the loops or strips at a 45 degree angle. 

Have fun manipulating Mobius’ discovery. His work led to the formation of a new area of mathematics called topology. Topology is used in many branches of mathematics, such as differentiable equations, dynamical systems, knot theory, and Riemann surfaces in complex analysis. It is also used in string theory in physics, and for describing the space-time structure of universe. A Mobius fan belt has been used in cars and will last twice as long as a standard cylindrical loop. 

There are many other versions of the Möbius puzzle, which can be achieved with strips of any shape and size, provided that when joining their ends we perform an odd number of turns. And that idea inspired another German mathematician, Felix Klein, to imagine in 1882 what we now know as “Klein bottles.” They are four-dimensional objects that we cannot build in our three-dimensional reality, but if we manage to visualise them, they will confuse us even more: they are theoretical containers that cannot contain a liquid because the inside and outside are mixed up.

Möbius strips and Klein bottles share a curious mathematical property within the field of topological study. They are not orientable, something that, to simplify, can be explained by imagining that if we draw an arrow on them, it’s impossible to conclude if that arrow points up or down. In a non-orientable world, our image and the one we see in the mirror would be indistinguishable.

But returning to our world and leaving aside theoretical mathematics, the great idea of ​​Möbius has been applied to conveyor belts that last longer (because all its surface is worn down at equal speed) and magnetic tapes for recording sound that don’t have to change sides: they can be used twice as long without interruption and they are used to play music in an infinite loop. Its application has also been patented in electronic components (such as a resistor that doesn’t produce magnetic interference) and its use is being investigated to achieve superconductors of high transition temperatures, molecular motors and graphene structures with new electronic characteristics.

Central part of a mosaic of a Roman villa in Sentinum (in Italy). Source: Glyptothek 

Such applications go far beyond what August Möbius imagined when he scientifically described this “impossible object” in 1858, although it should be acknowledged that this mathematician and theoretical astronomer was not the first to do so. Another German mathematician, Johann Benedict Listing, had come up with the same idea independently just a few months earlier. In fact, neither of them invented the single-sided strip: the concept is at least 1,600 years older, because a structure similar to the Möbius strip can be seen in Roman mosaics dating back to the third century.

However, the scientific weight of August Möbius—a student of the great mathematician Carl Friedrich Gauss and who went on to direct the astronomical observatory of the prestigious University of Göttingen—served to put his name to and to popularize this mathematical oddity whose greatest application, above all, has been to stimulate us to imagine beyond the space in which we live.