Spherical Geometry Exploration
- Smooth plastic ball
- Water soluble marker and wet sponge for erasing
- 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.
- 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?
- 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?
- Can you give a definition of "between" for points on a sphere?
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.
- As the radius gets larger, what happens to the circle? Then what happens? Then what happens?
- 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.
- 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?
- 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.
- 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.
Five postulates (assumptions) for Euclidean geometry are:
- There is exactly one straight line joining any two points.
- Any straight line can be extended forever.
- There is a circle with any given center and radius.
- The plane looks the same at every point.
- 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.
- If you replace “straight line” with “geodesic”, most of these are wrong for spherical geometry. How would you adapt them to spherical geometry?
- 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?