for the student. . . .
Significance of the Möbius Strip
is far better than any written description.)
you make a Möbius
strip by cutting a band of about two inches in width and at least 15
in length. Give the band a half twist, and re-attach the
ends. Then draw a line down the middle of the band.
scissors, cut the band along the pencil mark. Voilà!
One long cut produces two divisions but results in only one new
The half-twist results in a one-sided surface.
giving it additional twists, and reconnecting the ends produce figures
called "paradromic rings" that are studied in topology.
There is an
to the history of the now famous strip. In 1847, Johann Benedict
Listing published Vorstudien zur Topologie. This was the first
use of the word "topology." Nearing bankruptcy in 1858,
due to a wife who could not control her spending, Listing discovered
properties of the Möbius strip at almost the same time as, and
of, Möbius. His publication included the results of various
twists, half-twists, cuts, divisions and lengths. Four years
he extended Euler's formula for the Euler characteristic of oriented
polyhedra to the case of certain four-dimensional simplicial
Today, a far
of mathematicians now knows the name of "Möbius" and can recall
topics, a student
should also investigate the extensive literature on the Klein Bottle, Roman
Surface, Boy Surface, Cross-Cap, and Torus.