# Spherical Geometry Exploration

From EscherMath

**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:

- 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?

### Geodesics

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.

### Between

- 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?

### 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.

- 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.

### Squares

- 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).

### Postulates

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?

**Handin:**
A sheet with answers to all questions.