Möbius Strip
Challenge!
Questioning is the foundation of all learning.
The first step in rejecting not knowing is to ask, why?
Sweetland
Exploration!
To explore a Möbius strip, create a loop by using a strip of paper 3 centimeters X 40 centimeters. Take it and make a single 180° twist, and tape the ends together.
This topological marvel will reval some mind bending secrets through the following experiments.
Let's explore its geometry!
Exploration 1
- Take a marker, pen, or pencil and start in the middle of the strip.
- Draw a continuous line in the center, without ever lifting your pen all around the loop.
Focus questions:
- Did you end up back where you started?
- Did you actually trace a line around the entire length of the strip.
- Is there two sides or one side?
- Did you cross the edge?
- Then how can there be two sides if you didn't move from one to the other?
- What is your definition of a side?
Exploration 2
- Use the tip of a highlighter flat against the side edge of the loop.
- Rotate the strip slowly, keeping the highlighter still, to trace the very edge.
Focus questions
- Is one side of the loop highlighted?
- Or have both edges of the loop been highlighted?
Again,
- Is there two sides or one side?
- Did you cross the edge?
- Then how can there be two sides if you didn't move from one to the other?
- What is your definition of a side?
- Is it possible to have only one continuous edge?
Exporation 3
- Take a pair of scissors.
- Poke a hole in the center line you drew in Step 1.
- Carefully cut all the way around the strip.
Focus questions
- Did it break into two separate loops?
- Did you get two smaller loops?
- Or did the strip open into a single, massive loop that is twice as long and twisted?
Exploration 4
Twist, Cut, Tape, a fresh Möbius strip together.
This time, start cutting exactly one-third of the way from the outer edge, traveling all the way around the loop.
Focus questions
- Did you get two physically interlinked loops?
- One smaller Möbius strip inside a larger, twice-as-long twisted loop?
Mobius after one cut
Notice the twist? It is easier to see with actual exploration, but until you have the opportunity to explore your own mobius, explore the picture!
Mobius after two cuts
Notice the twists? It is easier to see with actual exploration, but until you have the opportunity to explore your own mobius, explore the picture!
What would happen if it were cut again?
And again?
Could there be a pattern?
Another cut
It's about as hard to see what happens after the third cut with the actual strip as it is from the picture.
One thing for certain, the path the scissors took went around one loop and cut it in half.
You can make your own and check to see how many twists are in it.
Some Mobius inspired applications!
Another application of Möbius strips is a mechanical belts that wears evenly on both sides.
However, ompanies like Goodyear Rubber Products, do not sell these anymore. The belt fell out of favor for a few simple reasons:
- Extra Space: The half-twist required a massive amount of physical room on the factory floor to let the heavy belt flip over without catching and became a safety riskN
- New Materials: Modern companies sell standard flat belts made of synthetic, multi-layered materials. These new materials are so incredibly durable on their own that factories no longer need to use the physical Möbius twist trick to extend their lifespan.
How about a Mobius inspired roof? At the NASCAR Hall of Fame.
See more about the history of Mobius.
