Spherical Geometry Exploration
- Smooth plastic ball
- Water soluble marker and wet sponge for erasing
On a flat surface:
- What is the shortest way to get from one point to another?
- If you walk without turning what will your path look like?
- If you pull a piece of string tight between two points, what will it look like?
On a sphere:
- 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.
- Describe the resulting geodesic curves.
- 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.
- 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.
- 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.
- 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).
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?