Mo%CC%88biusbandtransparentThe Mobius Strip

Visualizing the Universe, one pixel at a time.

The Mobius Strip: Visualizing a 2 dimensional object in 3 dimensional space

Introduction

Mo´bi`us strip`
n. 1. A mathematical object, or a physical representation of it, which is a two-dimensional sheet with only one surface. It is constructed or visualized as a rectangle, one end of which is held fixed while the opposite end is twisted through a 180 degree angle and joined to the fixed end. It is a two-dimensional object that can only exist in a three-dimensional space.

Material Needed

  • Several strips of paper ( a good size is 2 inches wide and 11 or 14 inches long)
  • Sticky tape
  • scissors
  • a felt tip pen

Step 1

mobiushalfmarkedMake a cylinder.

Bend a paper strip into a loop with no twists. Draw a line down the middle of your loop.
First Guess what you will get when you cut the cylinder along center line.

Cut the loop and observe what happens.

No surprise here.

Make a Mobius strip

Make a 1/2 twist (180 degrees) in the paper and tape it together at the ends

Draw a line down the center the Mobius strip.

Guess what the result will be when you cut the Mobius strip along the center line.

Are you surprised by the result?

What you get is described in this limerick:

A mathematician confided
That a Mobius band is one-sided,
And you'll get quite a laugh,
If you cut one in half,
For it stays in one piece when divided.

moebius_stripCut a New Mobius strip in Thirds (1/3)

First, guess what will happen…

Start your cut one-third of the way in from one edge. Keep going until you get back to where you started cutting!

What did you get? Did it surprise you?


Try cutting the strip 1/4 and/or 1/5 of the distance in from the edge. Before you cut predict what the result will be.

Other experiments

Make a strip with 2 or 3 half-twists... First predict the result then do the cuts.

They are called Paradromic Rings
Cutting a Möbius strip, giving it extra twists, and reconnecting the ends produces unexpected figures called paradromic rings

If you bisect a strip with an even number, n, of half twists you get two loops each with n half-twists. So a loop with 2 half-twists splits into two loops each with 2 half-twists.

If n is odd, you get one loop with 2n + 2 half-twists. So a Mobius strip with 1 half-twist becomes a loop with 2+2 = 4 half-twists.

For more on Mobius Strips and Klein bottles… check out these references
wolfram.com

wiki Mobius strip