Jacob Siehler suggested this topic. I had to check several times that I hadn’t written an essay about the Möbius strip already. While I have talked about it some, mostly in comic strip essays, this is a chance to specialize on the shape in a way I haven’t before.
Möbius Strip.
I have ridden at least 252 different roller coasters. These represent nearly every type of roller coaster made today, and most of the types that were ever made. One type, common in the 1920s and again since the 70s, is the racing coaster. This is two roller coasters, dispatched at the same time, following tracks that are as symmetric as the terrain allows. Want to win the race? Be in the train with the heavier passenger load. The difference in the time each train takes amounts to losses from friction, and the lighter train will lose a bit more of its speed.
There are three special wooden racing coasters. These are Racer at Kennywood Amusement Park (Pittsburgh), Grand National at Blackpool Pleasure Beach (Blackpool, England), and Montaña Rusa at La Feria Chapultepec Magico (Mexico City). I’ve been able to ride them all. When you get into the train going up, say, the left lift hill, you return to the station in the train that will go up the right lift hill. These racing roller coasters have only one track. The track twists around itself and becomes a Möbius strip.
This is a fun use of the Möbius strip. The shape is one of the few bits of advanced mathematics to escape into pop culture. Maybe dominates it, in a way nothing but the blackboard full of calculus equations does. In 1958 the public intellectual and game show host Clifton Fadiman published the anthology Fantasia Mathematica. It’s all essays and stories and poems with some mathematical element. I no longer remember how many of the pieces were about the Möbius strip one way or another. The collection does include A J Deutschs’s classic A Subway Named Möbius. In this story the Boston subway system achieves hyperdimensional complexity. It does not become a Möbius strip, though, in that story. It might be one in reality anyway.
The Möbius strip we name for August Ferdinand Möbius, who in 1858 was the second person known to have noticed the shape’s curious properties. The first — to notice, in 1858, and to publish, in 1862 — was Johann Benedict Listing. Listing seems to have coined the term “topology” for the field that the Möbius strip would be emblem for. He wrote one of the first texts on the field. He also seems to have coined terms like “entrophic phenomena” and “nodal points” and “geoid” and “micron”, for a millionth of a meter. It’s hard to say why we don’t talk about Listing strips instead. Mathematical fame is a strange, unpredictable creature. There is a topological invariant, the Listing Number, named for him. And he’s known to ophthalmologists for Listing’s Law, which describes how human eyes orient themselves.
The Möbius strip is an easy thing to construct. Loop a ribbon back to itself, with an odd number of half-twist before you fasten the ends together. Anyone could do it. So it seems curious that for all recorded history nobody thought to try. Not until 1858 when Lister and then Möbius hit on the same idea.
An irresistible thing, while riding these roller coasters, is to try to find the spot where you “switch”, where you go from being on the left track to the right. You can’t. The track is — well, the track is a series of metal straps bolted to a base of wood. (The base the straps are bolted to is what makes it a wooden roller coaster. The great lattice holding the tracks above ground have nothing to do with it.) But the path of the tracks is a continuous whole. To split it requires the same arbitrariness with which mapmakers pick a prime meridian. It’s obvious that the “longitude” of a cylinder or a rubber ball is arbitrary. It’s not obvious that roller coaster tracks should have the same property. Until you draw the shape in that ∞-loop figure we always see. Then you can get lost imagining a walk along the surface.
And it’s not true that nobody thought to try this shape before 1858. Julyan H E Cartwright and Diego L González wrote a paper searching for pre-Möbius strips. They find some examples. To my eye not enough examples to support their abstract’s claim of “lots of them”, but I trust they did not list every example. One example is a Roman mosaic showing Aion, the God of Time, Eternity, and the Zodiac. He holds a zodiac ring that is either a Möbius strip or cylinder with artistic errors. Cartwright and González are convinced. I’m reminded of a Looks Good On Paper comic strip that forgot to include the needed half-twist.
Islamic science gives us a more compelling example. We have a book by Ismail al-Jazari dated 1206, The Book of Knowledge of Ingenious Mechanical Devices. Some manuscripts of it illustrate a chain pump, with the chain arranged as a Möbius strip. Cartwright and González also note discussions in Scientific American, and other engineering publications in the United States, about drive and conveyor belts with the Möbius strip topology. None of those predate Lister or Möbius, or apparently credit either. And they do come quite soon after. It’s surprising something might leap from abstract mathematics to Yankee ingenuity that fast.
If it did. It’s not hard to explain why mechanical belts didn’t consider Möbius strip shapes before the late 19th century. Their advantage is that the wear of the belt distributes over twice the surface area, the “inside” and “outside”. A leather belt has a smooth and a rough side. Many other things you might make a belt from have a similar asymmetry. By the late 19th century you could make a belt of rubber. Its grip and flexibility and smoothness is uniform on all sides. “Balancing” the use suddenly could have a point.
I still find it curious almost no one drew or speculated about or played with these shapes until, practically, yesterday. The shape doesn’t seem far away from a trefoil knot. The recycling symbol, three folded-over arrows, suggests a Möbius strip. The strip evokes the ∞ symbol, although that symbol was not attached to the concept of “infinity” until John Wallis put it forth in 1655.
Even with the shape now familiar, and loved, there are curious gaps. Consider game design. If you play on a board that represents space you need to do something with the boundaries. The easiest is to make the boundaries the edges of playable space. The game designer has choices, though. If a piece moves off the board to the right, why not have it reappear on the left? (And, going off to the left, reappear on the right.) This is fine. It gives the game board, a finite rectangle, the topology of a cylinder. If this isn’t enough? Have pieces that go off the top edge reappear at the bottom, and vice-versa. Doing this, along with matching the left to the right boundaries, makes the game board a torus, a doughnut shape.
A Möbius strip is easy enough to code. Make the top and bottom impenetrable borders. And match the left to the right edges this way: a piece going off the board at the upper half of the right edge reappears at the lower half of the left edge. Going off the lower half of the right edge brings the piece to the upper half of the left edge. And so on. It isn’t hard, but I’m not aware of any game — board or computer — that uses this space. Maybe there’s a backgammon variant which does.
Still, the strip defies our intuition. It has one face and one edge. To reflect a shape across the width of the strip is the same as sliding a shape along its length. Cutting the strip down the center unfurls it into a cylinder. Cutting the strip down, one-third of the way from the edge, divides it into two pieces, a skinnier Möbius strip plus a cylinder. If we could extract the edge we could tug and stretch it until it was a circle.
And it primes our intuition. Once we understand there can be shapes lacking sides we can look for more. Anyone likely to read a pop mathematics blog about the Möbius strip has heard of the Klein bottle. This is a three-dimensional surface that folds back on itself in the fourth dimension of space. The shape is a jug with no inside, or with nothing but inside. Three-dimensional renditions of this get suggested as gifts to mathematicians. This for your mathematician friend who’s already got a Möbius scarf.
Though a Möbius strip looks — at any one spot — like a plane, the four-color map theorem doesn’t hold for it. Even the five-color theorem won’t do. You need six colors to cover maps on such a strip. A checkerboard drawn on a Möbius strip can be completely covered by T-shape pentominoes or Tetris pieces. You can’t do this for a checkerboard on the plane. In the mathematics of music theory the organization of dyads — two-tone “chords” — has the structure of a Möbius strip. I do not know music theory or the history of music theory. I’m curious whether Möbius strips might have been recognized by musicians before the mathematicians caught on.
And they inspire some practical inventions. Mechanical belts are obvious, although I don’t know how often they’re used. More clever are designs for resistors that have no self-inductance. They can resist electric flow without causing magnetic interference. I can look up the patents; I can’t swear to how often these are actually used. There exist — there are made — Möbius aromatic compounds. These are organic compounds with rings of carbon and hydrogen. I do not know a use for these. That they’ve only been synthesized this century, rather than found in nature, suggests they are more neat than practical.
Perhaps this shape is most useful as a path into a particular type of topology, and for its considerable artistry. And, with its “late” discovery, a reminder that we do not yet know all that is obvious. That is enough for anything.
There are three steel roller coasters with a Möbius strip track. That is, the metal rail on which the coaster runs is itself braced directly by metal. One of these is in France, one in Italy, and one in Iran. One in Liaoning, China has been under construction for five years. I can’t say when it might open. I have yet to ride any of them.
This and all the other 2020 A-to-Z essays should be at this link. Both the 2020 and all past A-to-Z essays should be at this link. I am hosting the Playful Math Education Blog Carnival at the end of September, so appreciate any educational or recreational or simply fun mathematics material you know about. And, goodness, I’m actually overdue to ask for topics for the latters P through R; I’ll have a post for that tomorrow, I hope. Thank you for your reading and your help.
. The area of a circle, represented with A, would be 
. I’m not sure what ‘Vol.C’ would mean, although ‘Volume of a cylinder’ would make sense … if the next line didn’t start “Vol.Cyl”. The volume of a circular cylinder is 
, where r is the radius and h the height. For a non-circular cylinder, it’s the area of a cross-section times the height. So that last line may be right, if it extends out of frame. 
nebusresearch 