Batch 2 - Class 306 - Topology: Cutting Mobius strips

• Mrs. Smith and Mrs. Jones had equal number of apples but Mrs. Jones had larger fruits and was selling hers at the rate of two for a penny, while Mrs. Smith sold three of hers for a penny. Mrs. Smith was for some reason called away and asked Mrs. Jones to dispose of her stock. Upon accepting the responsibility of disposing her friend's stock, Mrs. Jones mixed them together and sold them of at the rate of five apples for two pence. When Mrs. Smith returned the next day the apples had all been disposed of, but when they came to divide the proceeds they found that they were just seven pence short, and it is this shortage in the apple or financial market which has disturbed the mathematical equilibrium for such a long period. Supposing that they divided the money equally, each taking one-half, the problem is to tell just how much money Mrs. Jones lost by the unfortunate partnership?
• Answer: 21 pence

• Take a strip of paper and fold the ends to create a cylinder - cut it down the middle - what shape do you get?
• How many surfaces does this cylinder have?
• How many edges does it have
• On the same strip, if you give it a single twist before pasting the ends, you get a Mobius strip 
• How many surfaces does the Mobius strip have?
• How many edges does it have?
• What do you expect to happen if you cut the mobius strip down the middle? Try it out. (later, note that this is a four-twist mobius strip)
• What happens if you cut the mobius strip at one third width?
• Applications: Giant mobius strips are used as conveyor belt, because all surfaces get equal wear and tear. Also in typewriter ribbons because it allows use of both sides (you use bands that are twice the width of typing head). There are also applications in physics and chemistry.
• Demonstrate this by drawing a line and tracing it along an edge
• What if you do two twists, how many surfaces and edges do you have? What happens if you cut it down the middle?
• What if you do three twists? (recycling logo) - we land up with one strip with a knot in it - notice that this creates a knot from something that didn't have a knot. This is called a Trefoil knot. This is also how the knot gets into DNA strands. 
• Now lets take two cylinders, stick them at right angles and now cut both in half. What do you expect to happen? Try it out!
• What do you think happens if you take two half twist mobius strips, stick them at right angles and then cut both strips through the middle? Did some students not get the interconnected hearts? why?

• What if you took one cylinder and stuck it to a mobius strip?
• In the above, what if you cut the mobius strip at a third width?