Half a Klein bottle. If you have a real Klein bottle, though, you probably don't want to try this! So just take 2 Moebius strips and glue them together along their edge, and you'll get a Klein bottle!

A Möbius strip has only one surface and one edge. To make one take a strip of paper, rotate one end through 180 degrees, so you have a twisted piece of paper. Bring both ends together to form a loop, et voila! A Möbius strip!

Knock yourself out, this object you just created has one edge (follow it around) and one surface (follow it around).

An entertaining thing to do with your paper moebius strip is to cut it in half along its length. Use a pair of scissors to pierce the paper at any point and then start cutting along the strip until you reach the point where you started.

I would love to tell you what you'll create, but I believe it is more rewarding to do this for yourself.

/msg me if you are too lazy to try it or if you think I should include the 'answer' in this writeup.

An intellectual curiosity that was discovered in 1858 by the German mathematician and astronomer, Augustus Ferdinand Möbius. It was believed that no practical application could be found for the Möbius band beyond the creation of a decorative Möbius band Turk's head (a fancy knot that forms a turban shape - looks like a doughnut made from string. Also seen in jewelry or in dead animal carcasses as MAGGOTS EAT THE EYEBALLS AND SUCK OUT THE TASTY JUICES OF LOVE!).

It does however have real-life applications in industry and in art. For example, the Goodrich Tire Company, created a conveyor belt that is twisted into a Möbius band ensuring even wear of the belt. The concept has also been used to create continuous-loop recording tapes to double recording time and in electronic resistors.

A Möbius band can be formed by taking a rectangular strip of paper, rotating one of the two ends 180 degrees and joining the two ends of the strip of paper. Don't try this without parental supervision. This results in two extraordinary properties:

It has only one surface. If a line is drawn from a point on the surface parallel to the edges of the strip, the line will eventually pass through a point beneath the starting point, and then back to the starting point

It has only one edge. When cut in half along a line down its middle, the result is not two bands but a single larger band.

Sources:
Knots: A 2000 Calendar. The Ink Group
http://www.geom.umn.edu/zoo/features/mobius/
http://www.cpr.it/logo.html
http://www.middlebury.edu/~zambrose/explain.html

In addition to FlameBoy's writeup, you may also want to try to cut 1/3 from an edge, and continue cutting 1/3 from that edge until your scissors reach the point where you started. This also produces a very interesting result. Try various things, like cutting the result again. You get all sorts of unexpected suprises, or maybe they are expected if you are a topologist.

M.C. Escher drew (etched?) an interesting lattice moebius strip, with ants crawling on it. While one ant may look as if it is on an opposite side compared to another ant, they are actually all on the same side. This drawing is animated in the xscreensaver GL screensaver collection, where you can watch the ants walk around to a place on the "other side" of their starting position.

A parametric representation of a Mobius strip, with its centre-line a circle of radius r in the xy plane, and with a width of (2 * w), is as follows:
x = ((r + (t * cos(u / 2))) * cos(u))
y = ((r + (t * cos(u / 2))) * sin(u))
z = (t * sin(u / 2))
0 ≤ tw
(0 ≤ u ≤ (4 * π))
r = 5 and w = 1 draws a pretty good Mobius strip.

The parameterisation also works with the intervals:
-wtw
(0 ≤ u ≤ (2 * π))

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