1. Topology Möbius Strip Torus Klein Bottle Doug Rohrer
Mathematical study of geometric properties unaffected by twisting and stretching.
Torus
Klein Bottle
Möbius Strip
Doug Rohrer
Images from Wikipedia, MathWorld, and elsewhere.

2. Pop Quiz. What is the difference between a donut and a coffee cup?
Topologist: Nothing! They are “topologically equivalent.”
As it transforms,
any two “neighboring points” on its surface (or inside it) remain neighbors.
And non-neighbors remain non-neighbors (e.g., the hole doesn’t close).

3. The inside surface of this bottle is also its outside
Klein Bottles

4. Try This Cut 1.5” x 11” strip of paper.
Twist the strip 180 (half twist).
Connect the ends with one-sided invisible tape.
This surface has a special name…

5. August Ferdinand Möbius
Möbius Strip
August Ferdinand Möbius
(1790–1868)

6. Defining Property of Möbius Strip
Whereas an untwisted loop has an inside and an outside,
a Möbius Strip has only one side.

7. Ants on a Möbius Strip
Suppose that ants are afraid to walk over the “edge” of a sheet of paper.
With an untwisted loop, an ant on the inside cannot meet an ant on the outside.
On a Möbius Strip, two ants can always find each other.

8. Möbius Strip inspired this work of art.
M. C. Escher (1898 – 1972)
38-sec YouTube Video of Escher’s Ants

9. Möbius Conveyor Belt
Unlike conventional conveyor belt, wear and tear is distributed across “both sides.”
Patented by B. F. Goodrich Company, 1957

10. Task 1 Suppose a Möbius strip is cut along its center. What happens?
Try it!
One twisted loop
Click for Answer

11. Task 2
What happens if you twist the strip twice (i.e., two half twists) before cutting it?
Try it.
Two linked twisted loops
Click for Answer

12. Task 3 What happens if you cut the strip along an
imaginary line that is one-third across?
Short loop linked with long loop
Click for Answer

13. Task 4 Now try two half-twists and a one-third cut.
Narrow loop linked with wide loop
Answer