Other Topics

Click on the Chapter or Subchapter you wish to read. 
    Preface for Teachers 
  1. Introduction to Orientability:  A Fable 
  2. The Math of Non-Orientable Surfaces 
    1. Surface and Manifold
    2. Non-Orientable Surface 
    3. Orientable Surfaces:  Sphere, Torus
    4. Möbius Band
    5. Klein Bottle 
    6. Real Projective Plane 
    7. And Beyond: 3-Manifolds 
    8. What Would it Be Like to Live on a...?
    9. Homework Exercises about Math
  3. The History and Philosophy of Non-Orientability 
    1. The Original Topological Tyrant
    2. Klein Bottles and Kant
    3. Homework Exercises about History and Philosophy
  4. Literature 
    1. "The No-Sided Professor"
    2. "A Subway Named Moebius"
    3. Extra Short Stories
    4. The Bald Soprano
    5. The Gift
    6. Homework Exercises about Literature
  5. Music
    1. Bach and Schoenberg
    2. The Moebius Strip Tease
    3. If You're Musically Inclined...
    4. Homework Exercises about Music
  6. Other Topics 
    1. Knit Hats and Scarves
    2. Fun Toys on the Internet
    3. Non-Orientable Housing
    4. The Marvelous Moebius Molecule
    5. Moebius Mistakes
    6. Non-Orientable Surfaces in Art
    7. Homework Exercises on These Topics
  7. Bibliography 

Knitting Non-Orientable Surfaces

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? 

Here are some other ideas for how to knit Moebius band scarves and a Klein bottle hat. If you know how to knit, you may want to try one of these patterns. If a Klein bottle is just two Moebius bands attached at their common boundry, why can't you create one by taking two Moebius bands and attaching them? Equally, why can't you make a projective plane by attaching a Moebius band to a disc? 

Fun Toys on the Internet

"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

and http://www.mhri.edu.au/~pdb/geometry/kleincycloid/ 
This is is a fairly math-oriented site describing Klein bottles.  Do you understand the equations the pages gives for the bottle?  What do they mean? 

A similar site for Moebius strips. Can you understand its denfinition? 

The Geometry Center's Topological Zoo provides a lot of complicated information about various manifolds and lots of pretty pictures and movies.  Visit the site and write a one-to-two paragraph long discription of what you find there. 

Non-Orientable Housing

Robert HeinleinA young Robert A. Heinlein wrote a short story called ". . . And He Built A Crooked House," which describes what happens when an eccentric architect tries to build a house in the shape of an unfolded hypercube. (A hypercube is a four-dimensional version of a square or a cube.) The architect and his clients have an adventure exploring the house and learning about what happens when three-dimensional people live are stuck in a four-dimensional box. Can you imagine what it would be like living in a house that was non-orientable? What about pouring water into a Klein bottle? Would that be possible? Write a short story describing someone like you playing with--or playing in--one of the surfaces we've studied. 

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. 

The Marvelous Moebius Molecule

At just about the same time that our friend Mobius was “discovering” his famous surface in Leipzig, a chemist named Kekule, at the nearby University of Bonn, was busy “discovering” benzene. The carbon chains that compose this complex molecule supposedly came to Kekule in a daydream on a London Bus. “One fine summer evening,” Kekule mused, “I was returning by the last omnibus...and lo! the atoms were gamboling before my eyes...”. 

Cover of Archimedes' RevengeKekule 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: 

  • Explore the geometry of other complex molecules.  What, for instance, is a Lewis structure (find a chemistry textbook and look it up)?  How was the double helix structure of DNA deduced by Watson and Crick?  Who was Rosalind Franklin?
  • Think more aobut the ways math whows up in nature and the everyday world.  Can you find examples of the golden ration, imaginary numbers, the fourth dimension, and guaternions in the ingineering, industrial, and natural worlds?
  • Look into the current debate on the ulitmate fate of the universe.  Could the universe indeed by shaped like a Klein bottle?  Could we be headed for a Big Crunch?

Moebius Mistakes

A lot of people like calling what they--or others--do Moebius bands.  Check out some of these mistakes (from Moebius and His Band)! 

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. 

Non-Orientable Surfaces in Art

Artist M.C. Escher is known for his paradoxical and confusing pictures.  One of them is "Mobius Strip II," which depicts red ants following each other around.  Take a look at Escher's picture: 
Escher's Mobius Strip II

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: 
  • Try knitting a non-orientable surface. Make either a Moebius scarf or a Klein hat.
  • Can you describe what the equations for different Klein bottle immersions mean? What are they describing? Why does one surface (the Klein bottle) have different possibilities for the equation?
  • Write a paragraph or two describing what you find at the Geometry Center's Topological Zoo.
  • Try your hand at some poetry. Does the beauty of the shapes we've studied inspire you? What do you think about when you see a Klein bottle? Come up with a witty limerick about a projective plane. (Is that possible!?)
  • What if, after his attempt at building a tesseract house, Quintus Teal (Heinlein's architect character from ". . . And He Built a Crooked House") tried to build a house in the shape of a Klein bottle, or on the surface of a Klein bottle? What would happen? Try writing a sequel to Heinlein's story.
  • Write a paragraph on Archimedes' Revenge.  You can also use the book as a starting point for the questions posed to the right.
  • Look at the Moebius Mistakes and describe why and how the writers were confused.
  • 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?
  • Try to find some other artwork that uses non-orientable surfaces.

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: 

Heinlein's story can also be found online
The World of Escher has a page about "Mobius Strip II." 

For more information on sources and other ideas for further reading, see the bibliography.

 Created 981125 by jacr. Updated 981201 by jacr. URL is ./random.htm