Spherical Geometry Exploration

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Time-50.svg

Objective: Discover principles of geometry on the sphere.

Materials

  • Smooth plastic ball
  • Water soluble marker and wet sponge for erasing
  • String

Procedure

Straight Lines

On a flat surface:

  1. What is the shortest way to get from one point to another?
  2. If you walk without turning what will your path look like?
  3. If you pull a piece of string tight between two points, what will it look like?

Geodesics

On a sphere:

  1. Use a ball, marker and string to answer questions 1-3 for the surface of a sphere.

These "straight" lines are called geodesics. Draw more geodesics on the sphere, extending the curves as far as possible.

  1. Describe the resulting geodesic curves.

Between

  1. In the plane, if three points are on a line then one is always between the other two. Is this true for on a sphere?
  2. Can you give a definition of "between" for points on a sphere?

Circles

A circle is the curve of all points which are the same distance from a given point, called the center. We can use the same definition in spherical geometry.

  • Draw some circles on the sphere, by marking a center point and using a piece of string to find all points at a fixed distance from it.
  1. As the radius gets larger, what happens to the circle? Then what happens? Then what happens?
  2. In the plane, a circle has an inside which is finite and an outside which is infinite. Do circles have insides and outsides on the sphere? Explain.

Squares

  1. Recall that a rhombus (or diamond) is a quadrilateral with four equal sides. Using a piece of string, see if you can construct a rhombus on the sphere as follows: Mark a segment on the string with two marks and try to create a 4-gon with all four sides given by the marked length. Draw a sketch of what this rhombus looks like.
  2. Now draw a rhombus with all four corner angles the same. That is, make a regular 4-gon.
    Draw a picture of what this looks like.
  3. In Euclidean geometry a regular 4-gon is a square. On the sphere, is the regular 4-gon a square? (You may want to consider various ways to define a square).

Postulates

Five postulates (assumptions) for Euclidean geometry are:

  1. There is exactly one straight line joining any two points.
  2. Any straight line can be extended forever.
  3. There is a circle with any given center and radius.
  4. The plane looks the same at every point.
  5. Given a line and a point not on the line, there is exactly one line through the given point which is parallel to the given line.
  1. If you replace “straight line” with “geodesic”, most of these are wrong for spherical geometry. How would you adapt them to spherical geometry?

Handin: A sheet with answers to all questions.

Instructor:Spherical Geometry Exploration Solutions