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 


The close connection which exists between music and mathematics takes on a new dimension, literally, when it comes to Moebius strips and music.  If you are at all confused when going through our examples, just remember: a Moebius strip is a loop with a twist.  If you take a piece of music, and it can be played through from the beginning to the end sounding harmonically and melodically correct (basically, sounding nice), you have gone around the Moebius strip once.  Then, if you can go through it a second time, yet you start at the end of the piece, so that the last note of the piece becomes the first note of the piece, and it still sounds nice, you have Moebius music.  Still confused?  Take a look at the examples we found.  Print them, cut them out, glue them into Moebius trips, and see for yourself! 

Bach and Schoenberg and Moebius Strips

Johan Sebastian Bach [Corbis]Johann Sebastian Bach wrote a lot of canons, and experimented with all sorts of different forms.  The one which is most interesting for our purposes is from his “Musical Offering.”   His canon for two violins, also called “crab canon,” is written so that you can play it normally, from one end to the other, and then you can take the score of music, flip it upside down, and play it again.  Use the buttons below to listen to the music and look at the score at the same time. Keep in mind these questions as you listen.  (The button will play only the first half of the song; it will not play the retrograde form.) 


Arnold SchoenbergArnold Schoenberg, several centuries later, experimented with crab canons too.  He wrote an entire treatise called Style and Idea which discusses, among other things,  different ways of rearranging musical scores.  He wrote an example of a crab canon which has four parts instead of Bach’s two.  He called it mirror canon.  Use the buttons below to listen to the music and look at the score at the same time. Keep in mind these questions as you listen. 


Moebius Strip Tease

Nicolas Slonimsky was a theorist and composer who specialized in shocking his audience.  You can judge for yourself when you listen to his piece "Moebius Strip Tease."  He is famous for his Lexicon of Musical Invective, which is a "a multilingual Schimpflexicon of violent criticisms leveled at composers since Beethoven's time" as he explains himself. Nicholas Slonimsky"Moebius Strip Tease" was first presented on May 5, 1965, at Nicolas Slonimsky's Arriere-Garde Coffee House at the University of California, Los Angeles, when he was on the faculty.  It is scored, originally, for two singers and a piano non-obligatto.  The words read such: 

Ach! Professor Möbius, glörious Möbius  
Ach, we love your topological,  
And, ach, so logical strip! 
One-sided inside and two-sided outside! 
Ach!  euphörius, glörius Möbius Strip-Tease! 

The music is rotated around the head of each performer endlessly, (as the instructions specify), as the inside goes outside and the outside goes inside.  "The piece is a unilateral perpetual rondo in a linearly dodecaphonic vertically consonant counterpoint" says Slonimsky.  The instructions on the score are: "Copy the music for each performer on a strip of 110-b card stock, 68" by 6".  Give the strip a half twist to turn it into a Moebius strip." 

Slonimsky, unlike Schoenberg and Bach, was perfectly aware that his piece was shaped like a Moebius strip.  Again, you can listen to the music while you look at the music. 

If You're Musically Inclined...

If you're particularly interested in the use of Moebius bands in music, you may note that mucians are often using the word Moebius without really having any idea what it means.  Look at a selection of the sites below and decide if they're really Moebius-related or if someone just like the sound of the word. 
  • Korean musician Jo Yun has a CD called Mobius Strip.  There are sound files available, so see if you can hear a Moebius band.
  • Allen Sapp wrote a piece called Sonata VI.  According to the website, "it is in a two-module form, the two parts of which involve altered perspectives on the same material. Each movement is a kind of Moebius strip of the other."  Listen to the sound files provided and see if you agree with the discription.
  • Mike Metlay released an 18-minute-long CD designed to be played in a loop.  Is it a real Moebius band or just a normal loop?
  • Look at these lyrics from musician Peter Hamill.  What does this song have to do with a Moebius band?
Indicision and uncertainty
catch you now as they never have before
How come you didn't recognize
the revolving door?
Oh, you're gonna take sides
on the checkered floor 
It used to be so easy
You saw everything in black and white
but when you lost track of all the moves you'd made
you lost faith in wrong and right 
It doesn't seem conceivable
look what's handing in your hands
Oh, is it a trick of comprehension
or a master plan? 
Oh, a change in your perspective
from the gutter now used to
How come you didn't recognize
the fiery room
How you're gonna take sides 
Now you're on the Moebius loop.

For homework, answer the following questions for both the Bach and the Schoenberg: 
  • Why do you suppose they're called a crab canons?
  • What differences can you hear between the two versions in the Bach work?  What similarities can you hear between the two versions?  What differences or similarities are there in the Schoenberg work's versions?
  • Compare Bach’s and Schoenberg’s canons.  What do you like more or less about the two? 

  • This section written by MLB. 

    Bach's illustration is owned by Corbis.  The picture of Schoenberg comes from Belmont Music Publishers.  Peter Hammill's lyrics are provided by Henk den Bok.  Special thanks to Roman Ivanovitch of the Yale Music Department for his ideas and help on Moebius music and Dan Sharfman for his help creating the MIDI files. 

    For further reading, look at the following books: 

    Schoenberg, Arnold.  Style and Idea.  Berkeley, Calif.:  University of California Press, 1989. 

    Slonimsky, Nicolas.  "Moebius Strip Tease."  Source.  Vol. V/1.  Nov. 1971. pp. 64-66. 

    On the web, you can look at the following sites: 

    Belmont Music Publishers' The Legacy of Arnold Schoenberg

    Other Minds is a good resource for information on Slonimsky. 

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

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