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Unchanged: What is Topology?

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Author: Janice VanCleave

Problem

What is topology?

Materials

  • Walnut-size piece of modeling clay

Procedure

  1. Shape the clay into a ball.
  2. Tap the clay ball against a hard surface, such as a table, to shape it into a cube.
  3. Mold the cube into the shape of a gingerbread man without breaking the clay.
  4. Make a diagram of the changes in the clay similar to the one shown here.

Unchanged

Results

Different figures are made by molding the clay.

Why?

Topology is a branch of geometry that studies properties of geometric figures that remain unchanged when the shape of the figures is gradually changed. Figures are said to be topologically equivalent if they can be obtained from one another without cutting the figure or punching a hole. In this experiment, the ball is molded into a cube that is molded into a gingerbread man. All three figures are topologically equivalent. The property that remains unchanged about all three figures is that each has zero holes. In topology, the number of holes in a figure is called its genus. The figures in this experiment all have a genus of zero.

Let's Explore

A figure with a genus of one indicates that it has one hole through it. Repeat the experiment, starting with a clay figure in the shape of a donut. Form topologically equivalent figures, such as a needle (hole is the eye), cup (hole is the handle), or funnel (hole through the center). Science Fair Hint: Prepare a poster showing figures with different genus.

Show Time!

  1.  

    Unchanged

    Unchanged

    1. Another example of what topology studies is the Möbius strip. This loop, named after its discoverer, the German mathematician August Ferdinand Möbius (1790-1868), appears to have two sides, but actually has only one. This property does not change if the strip's shape is changed (within the restrictions that it is not cut or its genus changed). To make a Möbius strip, take a 36-inch (1-m) strip of adding machine tape, give one end half a turn, and tape the ends together. Prove that the strip has only one side by laying the strip over the corner of a table. Starting where the edges are taped together, draw a zigzag line back and forth down the strip until you return to the starting point.
    2. The Möbius strip has only one side and one edge. Mark the strip again, this time using a colored pen to mark along one edge of the paper.
    3. The Möbius strip has one half-twist, which is an odd number. If the number of half-twists is odd, will a loop with more than one half-twist have more than one surface and one edge, as does the Möbius strip? Repeat steps 1a and 1b twice, first using a loop with 3 half-twists, then using a loop with 5 half-twists.
    4. How would an even number of halftwists in a loop affect its number of surfaces and edges? Repeat steps 1a and 1b two or more times, using a loop with 2, 4, 6, or any even number of half-twists.
    5. Let T represent the number of half-twists in a loop. Make and display a poster with drawings for loops where T = 0, T = 1, T = 2. Indicate the number of surfaces and edges for each drawing.

Check it Out!

One of the branches of topology deals with the drawing of figures with one continuous stroke of a pencil—that is, without lifting the pencil or tracing a line twice. How can one determine if a figure can be traced in this manner? For information, see pages 95-102 in Janice VanCleave's Geometry for Every Kid (New York: Wiley, 1994).

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