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 50 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?