The Moebius Strip (or Band)

The Moebius Strip (or Band) is an unusual and very interesting geometric figure - this three-dimensional object has only one surface! If you draw any other geometric figure on a piece of paper, such a circle, and then cut it out, you will see that it has a back side and a front side, so it seems to have two surfaces. So you may wonder how we can say that a Moebius Band can have only one surface. Well, when you make your own Moebius band, in this project, you will see, particularly if you cut your band out of a piece of paper the has one color on the front and a different color on the back, that it has only one surface. If you draw a line starting anywhere on the band, and continue drawing without lifting your pencil from the paper, you will end up right back where you started.

You can create a moebius band by taking a strip of paper, making one twist, and attaching the two ends:

Step 1: Cut a long strip of paper:

Step 2: Take end AD in one hand and end BC in the other hand, and give the paper one twist:

Step 3: Tape of glue side AD to side BC:

Now you have a moebius band, as shown in the drawing below:

The Moebius band can be used as a "review sheet" or "study guide". If you write the circle theorems on the band, they might look something like this:

If 2 chords are congruent, then their arcs are congruent . . . If 2 arcs are congruent, then their chords are congruent . . . If 2 central angles are congruent, then their arcs are congruent ..If 2 arcs are congruent then their central angles are congruent . . .An angle formed by a chord and a tangent = 1/2 its intercepted arc . An angle formed by 2 chords in a circle = 1/2 the sum of its intercepted arcs . . . etc.

Then you can carry your moebius strip with you, and study while you are waiting in a line, or riding home from school. Frequent short periods of studying can be very helpful in learning mathematics.

There is another interesting project that you can do related to the Moebius Band. Cut 2 strips of paper the same length and the same width. Take the first strip of paper, give it one twist, and tape the 2 ends together. Then take the second strip of paper, but when you give it a twist, twist it in the opposite direction than you did the first. Now take these to strips and tape them together at right angles to each other. Now cut along the length of each strip as you did before. You will find that you have 2 adjoining bands and if you hold them vertically in the air, letting them hang loosely, they will form 2 joined hearts. This might make a nice project for Valentine's Day!

"The mathematician's best work is art, a high perfect art, as daring as the most secret dreams of imagination, clear and limpid. Mathematical genius and artistic genius touch one another." Göstam Mittag-Leffler