Batch 2 - Class 29 - Topology

• Moscow - 160 - There is a 12x9 rectangular rug, which has got spoiled by acid, resulting in a 8x1 rectangular hole in centre of the rug. Can you cut and stitch the rug back into a perfect square with no holes?

  

• MartinShCol 8.1 - Draw a closed curve of any shape. Now with another color, draw another closed curve of any shape as long as it does not pass through a previous intersection. What is true of the number of points of intersection between the two color curves?

  Even number. Why?
• MartinShCol 8.7 - See figure below and draw a line which intersects each of the 16 segments exactly once

  Answer: Can not be done because each of A, B, C have 5 sides each so a line can't enter and leave (even points of intersection) - note that two of these could be start and end pointsCan you do it on a Torus (Introduce the torus to kids)Yes, by having one point of entry "under" the drawing! (See bottom right solution)

• Fold a "sock torus" inside out - what happens?
• It becomes another torus, but the length becomes the circle and vice-versa!

• Ask kids what would happen if they cut a circular strip through the center

  Answer: Will split into two strips
• Show a Mobius strip

  How many surfaces does the Mobius strip have?Answer: 1 (Can demonstrate by drawing a line through the middle)How many edges does the Mobius strip have?Answer: 1 (Trace an edge using a pen)
• Cutting a Mobius strip

  Ask them to see what happens if they cut it through the centerAnswer: Remains in one piece, twice as long, with twice the number of twistsAsk them to draw a line about a third along the width and see what happens - Cut along the lineTwo loops - one inside the other, with one and two twists respectively
• Ask kids what would happen if they cut two circular strips joined at right angles

  Answer: Becomes a squareWhat if one of those bands was twisted into a mobius strip?Still the same square :-)What if you trisected the Mobius strip?Square with another mobius strip inside it!

• Ask kids to draw a shape on a plane which can be colored using three colors. Explain the notion of coloring of maps
• Can all shapes on a plane be colored using 3 colors?

  Ask them to draw a shape which can'tCan all shapes on a plane be colored using 4 colors?
• Ask kids think of surfaces where 4 colors might not be enough
• Show a mobius strip and ask kids to think about how many colors may be required?

  Answer: 6 colors

• Show kids a torus and ask kids to think about how many colors may be required?

  Answer: 7 colors  (The first figure below folds into a torus; the second is another example of a torus map that requires 7 colors)