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## A Review of Basic Geometry—Activity 1

# Möbius Strips, Klein Bottles, *etc.*

Cut a strip of paper (about 1"×11"), twist it once,
and tape the ends together.
It should look like the illustration below to the right.
(By twist we mean 180°, sometimes called a half-twist.)

How many sides does this strip have?
This strip of paper, the **Möbius strip**,
is named after a professor of mathematics at the German University of Leipzig,
August Ferdinand Möbius, who invented it in 1858.
Check for yourself that it is one-sided by drawing a line down the
middle of "one-side"—surprise!

What happens to the Möbius strip when it is cut
in "half" lengthwise?
Can you predict what happens when you cut it in quarters?
Another variation is to cut the original Möbius strip in thirds!

*
A mathematician confided*

That a Möbius strip is one-sided,

And you'll get quite a laugh

If you cut one in half,

for it stays in one piece when divided.
## Paul Bunyan and the Hard Rock Mine Conveyor Belt

Michigan is awash in Paul Bunyan tales of his logging exploits, but
perhaps less well known are his ventures into mining and
his extensive mathematical knowledge.
While on his western tour, Paul developed a deep mine and
was extracting high quality ore.
Our story begins when the mine was a horizontal shaft
about a quarter of a mile back into the mountains
and had a fairly high volume of ore coming out the
wind- and Babe (the blue ox)-powered main conveyor belt four feet wide.
The mine branched at that point, but the best ore was found
straight ahead and the mine shaft had developed such that
a longer but narrower conveyor belt would increase efficiency.
Specifically it was decided that a conveyor belt twice as
long and half as wide as the original would be the best solution.
Paul got out his sharp ax and proceeded to slice the
conveyor belt down the middle. When he finished they
installed the new converyor belt without having to cut it anymore
or even splice it, much to everyone's surprise. Paul explained that
he had put the half twist in it so it would wear evenly on both sides.

A few years went by. Mining operations continued. Although the best
ore vein curved slightly and narrowed down, they could still
improve operations with a narrower, but longer conveyor.
As Paul got out his new chain saw, he sent his
trusty assistant to town to get a couple splicing kits.
His assistant bulked having seen the results of the prior modification.
He went anyway and was surprised when he came back to find the kits
were not only needed, but the two resulting rings were linked together.

## Please Don't eat the Tori

What most people call a doughnut, mathematicians call a
**torus**—the plural being **tori**.
If a similar but three dimensional process of adding a twist is carried out
on a torus, a strange 3-D figure will result.
Below are a joke and another limerick in that regard.
Felix Klein (1849-1925), professor of mathematics at several
German universities, invented it in 1882.
You can order your own (or a calibrated stein for me) at
www.kleinbottle.com.

*
A mathematician named Klein*

Thought the Möbius strip divine.

Said he, "If you glue

The edges of two

You'll get a weird bottle like mine."