Background
for the student. . . .
Significance of the Möbius Strip
(The
handson 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 reattach 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 halftwist results in a onesided 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, halftwists, cuts, divisions and lengths. Four years
later
he extended Euler's formula for the Euler characteristic of oriented
threedimensional
polyhedra to the case of certain fourdimensional 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, CrossCap, and Torus. 
Brazil
Mathematical
Colloquim
1967
