# Problem Solving

## CONTENTS OF CURRICULUM UNIT 80.07.08

## Topology

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## COURSE OUTLINE

### 1. Topological Transformations

A topological transformation is the changing of one shape into another without cutting, breaking, filling up holes or joining, for example, when we make a doughnut from plasticene or clay and transform it into a cup. See figure 1.Two shapes such that each is a topological transformation of the other are said to be equivalent. See figure 1 for example.

The result of applying a topological transformation of a CIRCLE is called a SIMPLE CLOSED CURVE. See figure 2 for example. Drawings (a), (b), and (c) are SIMPLE CLOSED CURVES.

### 2. Nodes and Networks

A point with at least one path leading from it is called a NODE. The order of the node is the number of paths leading from the node. In figure 3, for example, B is 4node, A is 1node, and C is 1 node.A NETWORK is a collection of points and curves joining some of the points. See figure 3 for example.

### 3. Arcs and Regions

A line joining two nodes is an ARC. An area bounded by arcs is a REGION. The area outside a figure is also a region. See figure 3 for example. In this network there are 3 arcs, 3 nodes and 2 regions.### 4. Traversable Networks or Graphs

A network or graph is said to be TRAVERSABLE if it can be drawn with one sweep of the pencil, without lifting the pencil from the paper, and without tracing the same arc twice. It is permitted to pass through nodes several times. See figure 4 for example. Network 5 is NONTRAVERSABLE.### 5. Inside or Outside

A line segment that starts outside a simple closed curve and finishes inside, crosses (TOUCHING IS NOT COUNTED) in an ODD number of points. See figure 6 for example.### 6. Coloring Regions

Regions of a network are colored so that regions with a common arc have different colors. (Regions with a common node may have the same color.) No one has yet found a plane design which needs more than four colors. If you think you have found a map which needs five colors, see if someone else can color it in four. In figure 7, for example, 3 colors are needed.### 7. The Moebius Band and Other Surfaces

See figure 8. There is something remarkable about the drawing.This BAND, called a MOEBIUS BAND, is named after one of the pioneers in the subject of topology, a German mathematician who was born in 1790 and wrote a paper about its properties.

You will discover that the band with a half twist has unusual properties.

(figure available in print form)