In 1905, physician Paul Möbius included a photo of his long-dead grandfather’s (the great mathematician August Ferdinand Möbius) skull juxtaposed with the skull of Ludwig von Beethoven in his new book, Ueber den Schädel eines Mathematikers. The German title translates to Over the Skull of a Mathematician. It’s not completely clear how Paul obtained the photo of Grandpa Möbius’ skull, much less that of Beethoven’s, but there they are, captured in black and white.

This odd photo was made in the name of weird science. Paul Möbius was a neurologist of some distinction who dabbled in the bunk science of phrenology. He is thought to have compared the two skulls in an effort to prove that certain skull shapes were associated with mathematical and musical aptitude.

Although Möbius the Younger’s head bump research never panned out, he was right about one thing: August Möbius, professor of astronomy and observatory director at the University of Leipzig, was unquestionably a man of incredible mathematical ability. August Möbius’ scientific contributions are enormous: he invented a new type of calculus, he advanced the fields of celestial mechanics and astronomy, and he provided numerous important insights into the study of geometry.

But he is most famous now for his work in the realm of topology, the mathematical study of shape and form. Unfortunately, appreciating much of Möbius’ highly complex and arcane nature work requires a familiarity with concepts such as geometrical mechanics, polyhedral boundary theory, projective transformations, and other eye-glazing, albeit no doubt important, mathematical concepts.

But there is one Möbian concept that everyone can appreciate and that is the wonderful Möbius strip.

The Arrival of the Möbius Strip

In 1858, while working on some incredibly complex topological concepts involving geometrical solids, Möbius took a band of thin, flat material, gave it a single twist, and fastened the ends together. The Möbius strip was born. (To be fair, it was also independently discovered by another mathematician named Johann Listing at nearly the same time, but nobody calls it a Listing strip.)

In topological terms, a Möbius strip is a three-dimensional surface with only one side, and it has some amazing physical properties. For example, if you draw a line starting from the seam down the middle of the strip, the pencil line will meet back at the seam but on the “other side.” Also, if you cut a Möbius strip along the center line, you get not two separate strips, but rather one long strip with two complete axial twists. If you attach a Möbius strip to an object, say a bowling pin, and swing it around your head, the twisted strip resists kinking and curling, proving itself to be a superior attachment to a non-twisted one.

In the following activity, we exploit the properties of Dr. Möbius’ strip to make the Möbius strip coffee cup carrier, a Möbius strip–equipped cup holder that’s nearly spill-proof.

The Möbius Strip Coffee Cup Carrier

Every time you drive through a highway entrance cloverleaf, you and the passengers in your car experience centrifugal force, which pushes you toward one side of the car. Another common example of this is the way the clothes in your washing machine press against the rotating drum during the spin cycle.

For such a common occurrence, centrifugal force is a bit hard to explain. Physics books define it as “the apparent force, equal and opposite to the centripetal force, drawing a rotating body away from the center of rotation, caused by the inertia of the body.” Maybe that helps (probably not, though), but at least we can say with certainty that when something gets spun around in a big circle, centrifugal force causes it to press outward, opposite the central point of rotation. We’ll use this phenomenon and the anti-kinking abilities of the Möbius strip to design an anti-spill coffee cup holder.

Making the Cup Holder

Follow these steps to make the cup holder:

1. Cut a 3½-inch-wide strip from the acrylic plastic square with the jigsaw (or table saw if you’ve got one) as shown in Figure 5-1.

   

2. Measure your favorite coffee cup and then make appropriate bend lines on the 3½×12-inch piece of acrylic plastic as shown in Figure 5-2.

   

3. Place the marked plastic piece in the vise and direct the hot air from the air gun at the bend line.
4. When the plastic softens, bend the piece 90 degrees at the first bend line.
5. Make additional bends so the plastic takes the shape shown in Figure 5-3.

   

6. Using the bent plastic piece as a form, trace out and mark side pieces that are nominally 6 inches by 1½ inches (see Figure 5-4).

   

   Make sure to match the side contour of the main 12-inch piece of plastic as closely as possible when you are marking the plastic.

7. Cut out the marked pieces with the jigsaw (see Figure 5-5).
8. Now clamp the sides to the bent main piece (see Figure 5-6). Using acrylic adhesive, glue the side pieces to the main piece (see Figure 5-7).

   

   

   

9. After the adhesive/solvent has set and the parts are securely held together, drill a ¹/₈-inch-diameter hole in the top of the cup carrier assembly as shown in Figure 5-8.

   

10. Now glue the 3½-inch leather square to the bottom of the Coffee Cup Carrier assembly (see Figure 5-9).

    

Making the Möbius Strip Coffee Cup Carrier Handle

Follow these steps to make the coffee cup carrier handle:

1. Insert the 8-inch soft leather strip through one of the split rings.
2. Turn the leather into a Möbius strip by matching up the ends of the strip and making a half twist so that the top of one end matches up with the bottom of the other end.
3. Overlap the ends by about ¾-inch and fasten them securely either with needle and thread (see Figure 5-10; use a thimble to push the needle through the leather) or with a couple of pop rivets.

   

4. Now insert the split ring with the Möbius strip into one end of the barrel swivel.
5. Gently pry apart the other split ring and insert the other end of the barrel swivel.
6. Now insert the split ring into the ¹/₈-inch hole on the cup holder assembly.

   Your Möbius strip coffee cup carrier (see Figure 5-11) is ready to use.

   

Using Your Möbius Strip Coffee Cup Carrier

Now that you’ve assembled your coffee cup carrier, it’s time to put it to good use:

1. Place a paper or plastic cup filled with water on the leather pad of the Möbius strip coffee cup carrier.
2. Place one or two fingers through the leather handle. Begin gently swinging the cup holder in small to medium circles. It doesn’t really matter if the circles are large or small—you can even change directions—the water should stay in the cup.

   Keep in mind that the Möbius strip coffee cup carrier will splash or spill its contents if you jerk or whip the handle suddenly, but for most types of smoothly executed motion, the water will remain in the cup (see Figure 5-12).

   

How Does the Möbius Strip Coffee Cup Carrier Work?

To understand how the coffee cup carrier you made works, you must first consider why liquids spill. The physics behind the coffee stains on your carpet is a bit complex. Not only is an intricate interplay of accelerations, torques, and forces at work, but the biomechanics of bipedal human motion also come into play.

Researchers have found that the wave-like motion of coffee in a mug possesses a unique natural frequency that is related to the size of the mug. Typical coffee mugs produce oscillations that closely match the motion a person makes when walking. Now, if you walk, coffee cup in hand, at a steady and even pace, there is no spillage. But even small irregularities in your gait cause breathtakingly complex accelerations of liquid molecules that even Dr. Möbius would find a challenge to mathematically model. These irregularities amplify the liquid oscillations, which leads to sloshing, and ultimately, to stains on your carpet.

But if you add in the coffee cup carrier with its flexible handle, the math becomes much simpler. Because the handle flexes easily, all the forces acting on the coffee must act in line with the handle. As long as the flexible handle stays taut, there are no lateral (that is, side-to-side) accelerations, and the forces acting on the coffee merely push it toward the bottom of the cup holder. But if the handle bends, say due to a quick change in direction, the whipping action will cause the coffee to spill.

But you may be wondering, what does the Möbius strip have to do with all this? The handle, because it is twisted and tied into a Möbius strip (see Figure 5-13), resists kinking and snarling and makes the carrier even more spill resistant. Another great use for a Möbius strip!

By the way, here’s a joke that you’ll get, but will your friends? Maybe not!

Why did the chicken cross the Möbius strip? To get to the same side.