Introduction to Non-Orientability:  A Fable

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 
Once upon a time, in a land that looked and seemed very much like yours and mine, there lived a plentiful race of people named the Lapaloopans who had ruled their world for many years. They called it The Strip, except there were no gambling casinos or bright neon lights. To the Lapaloopans, it was simply their home. The Strip had local movie theatres and local delicatessens just like our world, where the Lapaloopans went to eat and unwind. 

The Lapaloopan sun rose over the Lapaloopan horizon just as our Sun does, and, except for the mountains and hills, the Lapaloopan land was broad and flat, as far as the eye could see. All around, the Lapaloopans lived a happy, contended life. 

Things went very well on The Strip for many centuries. The Lapaloopans had built a straight and even white fence at the edge of their Lapaloopan border for the protection of the children. No Lapaloopan had ever been past the tall white spires of this fence, and that served them very well. No citizen of The Strip, young or old, had ever been lost, and everyone knew at all times where everyone else was, within the local territory. 

One day, however, there was a terrific change on The Strip. One bright young Lapaloopan announced that she had scaled the even white fence and seen beyond. “There is more for every Lapaloopan outside the straight and even white fence!” cried the young lass. “There is more to see!” 

The Lapaloopan Council convened over this difficult matter, and decided that Progress would be difficult to quell. At their nervous bidding, the young lass set out over the even white fence for The Great Beyond. All Lapaloopans watched her departure, watched the Lapaloopan sun set over the horizon and cast long, flat shadows against the land. 

Many weeks and months and years passed before the young lass returned. The white and even fence was built up many orders higher, and no Lapaloopan, young or old, was allowed near its spires. The young Lapaloopan lass was assumed surely dead. 

One day, however, as the colors faded from the sky, she returned again, although from the other direction from the one in which she had left. Many changes had overcome the young lass, though she seemed unfazed. She could no longer read writing left-to-right, as was the Lapaloopan custom. “It all looks like gibberish!” she cried. Even more peculiarly, she came back left-handed instead of right-handed. The Lapaloopan elders were dumbstruck. What had happened to their young lass? 

An emergency meeting of the Lapaloopan council was set, and the leaders of The Strip discussed exactly what to do. It was decided, after much debate, that a larger team of Lapaloopan explorers would be sent out to uncover the mysterious phenomena the poor lass had already been subjected to. 

This party was sent out, then another, then another. The same strange things kept happening. Each member of the party came back left-handed if he had left right-handed, and right-handed if he had left left-handed. No explorer could read writing left to right. Each was a mirror image of his former self! 

These explorers had children, and their children had children, and after many years all of the explorers had returned. They became celebrities, and soon the white and even fence was crushed. Every Lapaloopan wanted to see the Great Beyond. Opportunities for “Round the Strip” cruises and vacations were offered en masse. 

Eventually, there was no Lapaloopan who had not traveled around his homeland at least once, and ended up back where he belonged, queerly reversed. It became a regular matter of life. Lapaloopans began making straight-handled tools and used only circles, squares, and lines in their writing. 
A Moebius Band--Much like what the Lapaloopans lived on 
After many years, another meeting of the council was called, this time for the purpose of changing the world’s name. After all, so many fantastic events had occurred since its first naming. The elders on the council and many Lapaloopans voted on the title for their world, and it was Moebius. (The Lapaloopans evidently thought this had a nice ring to it.) Soon after, The Strip became The Moebius Strip

The local movie theatres and delicatessens still stand there, but the white and even fence is gone forever. Life for Lapaloopans has ever since been grand. The sun still rises and sets every day on the Lapaloopan horizon, and the land looks broad and flat. But things will never be the same. 

And the Lapaloopans seem to like it that way. 

Welcome to "Math That Makes You Go Wow."  In this brief course, we hope you'll learn what happened to that lass and the other reversed Lapaloopans.  You'll also come into contact with some beautiful and strange shapes.  One of them, the familiar Moebius strip, is pictured above.  You'll also meet the Klein bottle and the projective plane, neither of which can be created in our space.  Does that sound weird?  You betcha.  Welcome to the world of non-orientability.

This section written by EEC. 

The rainbow-colored Moebius band is from the Geometry Center

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

Created 981125 by jacr. Updated 981203 by jacr. URL is ./intro.htm