# Problem Solving

## CONTENTS OF CURRICULUM UNIT 80.07.08

## Topology

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## 1. TOPOLOGICAL TRANSFORMATIONS

- 1. (a) A list of the ways in which the drawings differ from one another. Example: Some drawings are made only from segments of straight lines, others are not.
- (b) A list of the ways in which the drawings are alike. Example: Each drawing divides the page into the same number of cells or regions.
- 1. (a) and (b) These questions are intended for class discussion, but the discussion is sometimes more fruitful if it is preceded by short group discussion or individual consideration of the questions. There follows an example of such a discussion on (b).

Question: | In what way are the figures alike? |

Answer: | They all have four regions. |

Question: | Regions? |

Answer: | They divide the page into four parts. |

Answer: | Five parts. |

Question: | Why five? |

Answer: | There’s the outside too. |

Question: | Do all the drawings have five regions or parts? |

Answer: | Yes. |

Answer: | They are all joined up. |

Question: | What do you mean by joined up? |

Question: | Draw some figures that are not joined up. |

Answer 1: | See figure 10. |

Answer 2: | See figure 11. |

Answer 3: | See figure 12. |

Answer 4: | The ones in figure 9 have a boundary, these haven’t. |

Answer 5: | They have a boundary and two lines cross inside. |

Question: | Will any of these do? See figure 13. |

Answer: | No. |

Question: | But two lines cross inside. |

Answer: | The lines must join the boundary at different points. |

Question: | Draw some other figures like the ones in figure 9. |

Students will probably express themselves in informal and imprecise language, but the discussion should bring out the following ideas:

Topology is about points, lines and the figures they make, but length, area, curvature and angle can be altered as much as you wish.

- 1. Distance, angle and direction are not preserved.
- 2. Any one of the figures could be stretched and pulled into any one of the others if distance, angle and direction are ignored.
- 3. The number of lines meeting at a point and the order of points on a line do not change.

(i) We call bending and stretching (BUT NOT TEARING OR JOINING) TOPOLOGICAL TRANSFORMATIONS.

(ii) Properties that are still true about a drawing or network after it has been transformed are called invariant because they do not change or vary. The distance between two points is not invariant.

(iii) Two curves such that each is a topological transformation of the other are said to be equivalent.

### EXERCISE A

(figure available in print form)

- 1. Which of the drawings in figure 14 are SIMPLE CLOSED CURVES? Provide each student a copy of figure 14 and add more diagrams. (USE OF CLAY OR PLASTICENE CAN BE FUN AND HELPFUL TO FIND CORRECT ANSWERS.)
- 2. Can the first curve of each pair in figure 15 be transformed topologically into the second curve? NOTE: ADD MORE DIAGRAMS.
- 3. Which of the curves in figure 16 are topologically equivalent to the straight line segment A B?
- 4. State which of the drawings in figure 17 are topologically equivalent to each other?
- 5. Which of the pairs of drawings in figure 18 are equivalent? If you think a pair is equivalent, copy the second member of the pair and mark on it a possible position for A, the image of A under the transformation. If more than one position is possible, mark all of them. You are allowed to turn the drawings over.
- 6. What can you turn a beetle into? The original insect and one suggestion is shown in figure 19.
- 7. State which of the shapes (i)(v) are topologically equivalent to each of the shapes (a), (b), (c) and (d). See figure 20.
- 8. Draw some topological transformations of each of the drawings in figure 21.