|Click on the Chapter or Subchapter you wish
Here's one teacher's idea of how to knit a Moebius band. Since a Moebius band is just a twisted cylinder, why do you think the instructions are so complicated? How does this Moebius band differ from the one you'd get if you just knitted a traditional scarf and twisted one end before connecting the two ends?
"A mathematician named Klein/ Thought a Moebius band was divine..." Some math poetry. Do you think you could write a little ditty about some non-orientable surfaces? Try writing a limerick about a real projective plane. What about a poem about living on 3-torus?
Here's an example we wrote:
Thomas Tordu, the famous math professor from Universite de Marron, in Provence, France, recently retired to the town of Nouvel Havre. He left the Hôtel Gee one morning for his routine morning walk, and having gone off his normal route, he was surprised to come across what seemed to be an abandoned mansion. He considered entering, but looked at his watch and noticed that he was due to go to the real estate agency on rue Dunham, N°195a. He raced back to his hotel for breakfast, and filled up on bagels and doughnuts. Although perfect only in the mind of God, these particular doughnuts still had the topology of tori.
He dashed off to rue Dunham, and arrived breathless for his appointment with Monsieur Homme Loup, who had been helping Monsieur Tordu find a suitable house in Nouvel Havre.
“Bonjour, Monsieur Homme Loup, I think I have found the perfect house for me, down on rue de l’Homme Plié. There is an old abandoned mansion which seems just right.”
“Ah, non, Monsieur Tordu, you must never go into the old Lein estate! It is très dangereux. Some of our foremost citizens have disappeared into that house. August Ferdinand Moebius, Andreas Xenachis, Jimmy Hoffa are just a few names. Surely you do not want to add the illustrious Thomas Tordu to the list!”
“You have piqued my curiosity. Surely there is a mathematical explanation to the disappearances. I want to explore.”
“Do as you wish, you twisted man, but I will take no responsibility for your fate.”
That afternoon, Thomas Tordu packed a bag with some vittles, and made his way to the mysterious mansion. Easily unlatching the gate, he walked up the gravel path, and came to the big oak door which led into the Maison Lein. He noticed nothing unusual at first as he walked through the different rooms. As he reached the second floor and got to the last room of the hall, he started feeling an odd mathematical tingling, the same he had gotten when, as a boy, he first learned of the mysterious properties of Moebius bands. With an odd mixture of trepidation and excitement, he slowly opened the door, and the first thing that caught his eye was a pile of bones in the corner. He absent-mindedly let the door swing shut behind him as he walked towards what he assumed were the remains of his childhood hero, August Moebius.
In a sudden spell of vertigo, caused no doubt by the emotions of seeing such an august topologist reduced to so little, he leaned against the wall. “That’s odd,” he exclaimed, keeping his calm and rationality, “where is my arm?” He looked across the room, only to see a disembodied arm waving to him from the opposite wall. He realized that he was waving his missing arm and stopped, at which point the arm across the room stopped. This led him to deduce that the mysterious extra arm was, in fact, his own. To confirm his suspicion, he followed his arm, passing through the wall, reappearing where the arm had been -- and this taking only one step. It seemed to him that he had simply walked straight, although an exterior spectator would have said that he had walked all the way to the opposite side of the room. He suddenly saw the danger of which Monsieur Homme Loup had warned him: it was possible to get stuck going around in circles without ever escaping. He dashed to the door and flung it open. But instead of finding himself in the hallway that he took to enter the room, he found himself at the opposite diagonal corner of the room. “Just like in Clue,” he thought to himself, “it’s just as though I went through the secret passageway!” After dashing back and forth “through the door” twice more unsuccessfully (so he had gone through a total of three times), he sat down to think. He picked up a book that was lying on a nearby bookshelf, thinking that reading would let him relax a little before he tackled the problem. He opened the book, only to notice that it was written in an unfamiliar language. “Mais ce n’est pas du francais!” he exclaimed. “And it isn’t English either.” “Wait a minute. This is written backwards!” Suddenly it dawned on him: “I must be in some sort of a non-orientable three-manifold.” He stepped out again, this time only extending one leg beyond the wall. As he moved around his left leg, he realized that he was looking at his right leg moving around on the door side of the room. Professor Tordu spent some time playing around with his discovery, and then making his left side his left side again, until he realized it was time for him to leave the room. He did not find the prospect of spending the rest of his life in this particular three-manifold especially appealing. He also started to worry about getting turned inside out if he were to go out another wall in the room.
He looked around the room again. He had gotten in through the door, but he already knew that it was no use trying that escape. Suddenly the ventilation system caught his eye. Perhaps the creator of this dimensionally-distorted room had forgotten to include the ventilating system in his spell…he unscrewed the ventilator and slithered down the pipe, only to find himself flying through the air for a few seconds before he hit the ground with a loud thump.
He looked around and found himself in front of the main door of the mansion, which he no longer had any interest in renting. As he stared at the old worn off letters which spelled out Maison Lein, he noticed the faint outline of a K preceding Lein.
Kekule went on to identify benzene’s shape: a hexagon composed of six carbon atoms and six hydrogen atoms at its vertices. Since this first inquiry into the geometry of molecules, chemists have discovered even more complex shapes, like DNA’s remarkable double helix. This hand-in-handedness (no pun intended) of science and mathematics continues even into the realm of non-orientability.
David Walba and co-workers at the University of Colorado at Boulder found a way in their laboratories to synthesize the marvelous Moebius Molecule. They began with a molecule shaped like a ladder with three rungs, each rung a carbon-carbon double bond. The ladder is “bent around,” and the ends are joined, so that “half the time the loop will simply be a circular band, but the other half of the time the loop will be a Mobius strip.”
What holds true for a paper Moebius strip holds true too for Walba’s microscopic biochemical one. Breaking the “ladder’s” bonds corresponds to cutting a Moebius strip up its middle, which produces one longer loop. When divided this way, the Moebius Molecule, too, becomes “a single band with twice the circumference of the original.”
The scenario is a good entrance point into the greater philosophical chicken-and-egg problem for science and mathematics. Which came first -- the molecule or the math? Another way, “...do ideas in the physical sciences inspire new ideas in mathematics, or is it the other way around?” Followers of Plato would insist “their discipline is divorced from physical reality,” that “numbers would exist even if there were no objects we could count.” This claim would be immediately disputed, however, by physicists who could cite Isaac Newton’s “invention” of calculus in order to study “exceedingly small intervals of space and time” -- here, it seems, the math was modeled after the world of physical phenomena.
And the debate rages on -- into the realm of a Klein bottle universe and beyond.
These paragraphs are abridged from Archimedes’ Revenge: The Joys and Perils of Mathematics by Paul Hoffman (New York: Ballantine Books, 1989). Find the book and write a paragraph on it. You can also use it as a springboard to answer these questions:
From a letter in The Independent (a British newspaper), 17 October 1990: “If the world condemns Saddam Hussein for his rape of Kuwait, it can avoid condemning Israel for its rape of Palestine only by resort to an ethical Mobius strip that starts off on one moral plane and ends on another.”
From a book review in The Observer (another British paper), 13 October 1991: “This is a Moebius strip of a novel: a thriller which, halfway through, contorts itself into a novel of ideas, and finally into a romance. In so doing, it snaps its backbone of psychological truth.”
Musicians seem to use the word Moebius a lot. Look at these examples in Chapter 5 and explain why they are or are not really Moebius bands.
What do you think is Escher's purpose in creating this piece of artwork? Does it lend any insights to your understanding of Moebius bands?
For homework, answer the following questions:
This section written by all.
The picture of Heinlein (when he was in his 30s) is from wegrokit.com, a Heinlein fansite. The image of the cover of Archimedes' Revenge is from Amazon. "Mobius Strip II" comes from The World of Escher.
For further reading, look at the following books:
Fauvel, John, et al., eds. Moebius and His Band. Oxford: Oxford University Press, 1993.
Heinlein, Robert. ". . . And He Built a Crooked House." In Clifton Fadiman, Fantasia Mathematica. New York: Simon and Schuster, 1958. (Republished in 1997 by Springer-Verlag). pp 70-90. [Also available online.]
Hoffman, Paul. Archimedes' Revenge: The Joys and Perils of Mathematics. New York: Ballantine, 1989.
On the web, you can look at the following sites:
For more information on sources and other ideas for further reading, see the bibliography.