Geometry versus Topology
Objectives:
- Discover topological properties of surfaces.
- Classify properties as topological or geometrical.
Materials Needed:
- Stretch Cloth Or Stretchable Plastic
- Colored Pens Or Markers
- Round Balloons
- Scissors
Launch:
What aspects of a shape remain unchanged if the surface it is drawn on is
deformed without tearing or cutting?
Activity 1:
Cut out a square piece of material from stretch cloth or stretch plastic.
Draw various polygons on the piece using colored pens or markers. Bend or
fold the piece of material and list common properties of the shapes that
change on the material. List common properties of the shapes that do not
change. Stretch the piece of material in various directions. Edit your
list to include this deformation of the surface. Twist the piece of
material. Edit your list to include this deformation of the surface.
Activity 2:
Draw various shapes on a deflated balloon. Inflate the balloon in various
stages observing what properties of the shapes change and which properties
do not change. Edit your list from Activity 1 to include the observations
you have made. Deform the inflated balloon to any shape except by cutting
or tearing. Edit your list to include any additional observations you
made.
Comments:
A surface's geometry consists of those properties of the shapes drawn on
it which change when the surface is deformed without cutting or tearing.
A surface's topology consists of those properties of the shapes drawn on
it which do not change when the surface is deformed without cutting or
tearing.
Question:
1. Indicate whether the following is a geometrical property or a
topological property of a surface.
- Length of segments and arcs?
- Size of angles?
- Shortest distance between two points?
- Straightness of lines?
- An enclosed or open region?
- Number of closed regions?
- Number of vertices?
- Number of arcs between pairs of vertices?
- Area of an enclosed region?
- Perimeter of an enclosed region?
- Total number of arcs (straight or curved)?
- Number of arcs enclosing a region?
- Curvature of shape on surface?
Extension:
If you were also allowed to cut and/or glue surfaces to alter them, which
of the above properties would remain unchanged?