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 Software Architecture

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Two Ways to Create a Famous Surface:  Maple and Java 3D

The Möbius Strip

NCB Deposit  # 31

MAPLE logo
 

You must own a copy of Maple Software to run this code. 
Moebius code and image
Instructions:

This interactive animation requires three downloads. First, your computer must have Java and Java 3D. Then you must download the JAR file to your desktop. Please note that the Java3D software is only available for Windows and Solaris.

For download assistance click here.
Java link and image
___

Having now downloaded Java and Java 3D, you may run the animation for the Möbius Strip as shown to the right. Download the following:

File image and animation download
Möbius Animation Download 

After the download, simply double click on the JAR file to see the animation.

Tech savy programmers may enjoy the following source code:

Animation code
Möbius Animation Source Code 

Moebius image

 
 
 
 
  Background for the student. . . .

Significance of the Möbius Strip
(The hands-on experiment is far better than any written description.)

We suggest you make a Möbius strip by cutting a band of about two inches in width and at least 15 inches in length.  Give the band  a half twist, and re-attach the two ends.  Then draw a line down the middle of the band.   With scissors, cut the band along the pencil mark.  Voilà!  One long cut produces two divisions but results in only one new band.  The half-twist results in a one-sided surface.

Cutting a Möbius strip, giving it additional twists, and reconnecting the ends produce figures called "paradromic rings" that are studied in topology. 

There is an additional "twist" to the history of the now famous strip.  In 1847, Johann Benedict Listing published Vorstudien zur Topologie. This was the first published use of the word "topology."   Nearing bankruptcy in 1858, largely due to a wife who could not control her spending, Listing discovered the properties of the Möbius strip at almost the same time as, and independently of, Möbius.  His publication included the results of various twists, half-twists, cuts, divisions and lengths.  Four years later he extended Euler's formula for the Euler characteristic of oriented three-dimensional polyhedra to the case of certain four-dimensional simplicial complexes. 

Today, a far larger audience of mathematicians now knows the name of "Möbius" and can recall the Euler Formula.

For related topics, a student should also investigate the extensive literature on the Klein Bottle, Roman Surface, Boy Surface, Cross-Cap, and Torus.


Brazilian stamp

Brazil
Mathematical
Colloquim
1967


Links and Printed References



For the PoV3 code see  <  http://www.uib.no/people/nfytn/moebtxt.htm  >.

For a movie of the Möbius Strip and a selection of mathematical games see
< http://www.cut-the-knot.org/do_you_know/moebius.shtml  >.

W. W. R. Ball and H. S. M. Coxeter, Mathematical Recreations and Essays,13th ed. Dover, 1987.

For Mathematica® code that will create many variations of these graphs see
Gray, A.,  MODERN DIFFERENTIAL GEOMETRY of Curves and Surfaces with Mathematica®,  2nd. ed., CRC Press, 1998. 

For a biography of August Möbius see  < http://www-history.mcs.st-and.ac.uk/history/Mathematicians/Mobius.html >.

For a biography of Johann Benedict Listing see  < http://www-history.mcs.st-and.ac.uk/history/Mathematicians/Listing.html >.
Moebius picture
August Ferdinand Möbius
1790 - 1868

Get Java
Warning:
This animation requires three downloads. First, your computer must have Java and Java 3D. Then you must download the JAR file to your desktop. Please note that the Java3D software is only available for Windows and Solaris. Installation will take several minutes. 

Java Software

  • If you do not have the free Java software, go to http://www.java.com/ and click on "Get It Now."  Follow the on-screen instructions to download and install all components.
  • Now that you have the Java software, go to http://java.sun.com/products/java-media/3D/download.html to also download Java 3D. 
    • Click on the Download button for Java 3D Software.
    • Accept the terms of the License Agreement.
    • Select the Java 3D Runtime for the JRE appropriate for your computer. 
    • Once the download is complete, run the executable file to install. 
  • Now you're ready to run this animation of the Möbius Strip.
November 6, 2003

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Dr. Paul Chabot
Dr. Russell Abbott
Students of CS 390, Summer, 2003
Matthew Nelson, FITSC Lab, CSULA