Spherical Geometry Exploration

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Objective: Discover principles of geometry on the sphere.

Materials

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

Procedure

Geodesics

  • Pick two points on a sphere. What is the shortest way to get from one to the other?
  • You live on a sphere. If you walk "straight" (or fly), what will your path look like?
  • A piece of string pulled tight between two points is "straight". Try this with a ball.
  1. Describe the "straight" lines (i.e., shortest paths) on a sphere.

These "straight" lines are called geodesics. Draw some geodesics. Do they work well with all three of the above ideas?

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.

Polygons

  1. Create an equilateral triangle. In Euclidean geometry the angles must all be 60 degrees. What are the possible angle measures for an equilateral triangle on the sphere?
  2. Using a piece of rope, see if you can construct a rhombus on the sphere. Suggestion: Mark equidistant spots on a rope and - with the help of your team mates - see if you can close the 4-gon and make it into a polygon on the surface of the sphere. Draw a sketch of what this rhombus looks like.
  3. Convince yourself that the rhombus can be turned into a regular 4-gon. Draw a picture of what this looks like. In Euclidean geometry a regular 4-gon is a square. There are different ways to define a square (Definitions of Square). Is the regular 4-gon a square? Explain carefully.

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?
  2. In plane geometry, two triangles which have equal side lengths are congruent. This is called Side-Side-Side. Is SSS still true in spherical geometry? What about Side-Angle-Side, is SAS still true? ASA? In plane geometry, there's no AAA.. why not? Is there AAA in spherical geometry?

Handin: A sheet with answers to all questions.

Instructor:Spherical Geometry Exploration Solutions